Expanding and factorising
By the end of this chapter you will be able to
- Collect like terms, and simplify expressions with one or two letters
- Multiply terms together, and expand a bracket by multiplying each term inside it
- Expand two brackets by multiplying each pair of terms, and collect like terms
- Take out the highest common factor from an expression, and write the answer in brackets
- Factorise x2 + bx + c, and check the answer by expanding the brackets
You should already know
- Writing and evaluating simple expressions, and substituting numbers for letters (Year 7)
- Multiplying and dividing negative numbers (Year 7)
- Multiplying by a number outside a bracket, and factors and multiples (Year 7, Chapter 3)
Where this leads
- Solving equations with brackets, and forming equations from expressions (Year 8, Chapter 6)
- Factorising quadratics with a coefficient of x2 other than 1, and the difference of two squares (Year 9)
- Algebraic fractions, expanding and factorising in full (GCSE)
National Curriculum for England, Key Stage 3 mathematics, Year 8. Content: Simplify expressions by collecting like terms, Expand single brackets and double brackets such as (x + 3)(x − 2), Factorise by taking out the highest common factor, Factorise quadratics of the form x2 + bx + c into two brackets, and check a factorisation by expanding.
Exercise P Do you remember?
Non-calculatorMethod. Work out the brackets first, then multiply.
- Work out the brackets. The brackets come first.4 + 2 = 6
- Multiply. Multiply the answer by 7.7 × 6 = 42
Check. 7 × 4 + 7 × 2 = 28 + 14 = 42.
Method. Add the numbers in front of the letter a.
- Find the like terms. Both terms have the letter a, so they are like terms.2a and 3a
- Add the numbers. 2 + 3 = 5.5a
- Write the answer. 5a
Check. If a = 1, then 2 + 3 = 5 and 5 × 1 = 5.
Method. Replace n with 4, then work out the answer.
- Replace n with 4. Every n becomes 4.2 × 4 + 1
- Multiply first. Multiplication comes before addition.2 × 4 = 8
- Add. Add 1.8 + 1 = 9
Check. n = 4 gives 9, so 2n + 1 = 9.
Method. A positive number times a negative number is negative.
- Multiply the numbers. 3 × 4 = 12.3 × 4 = 12
- Decide the sign. Positive times negative gives a negative answer.3 × (−4) = −12
- Write the answer. −12
Check. −12 + 12 = 0, so −12 is the opposite of 12.
Method. Four squared means 4 × 4.
- Multiply 4 by itself. The index 2 means two 4s multiplied together.4 × 4 = 16
- Write the answer. 16
Check. 16 is the square of 4.
Simplifying expressions
An expression is a mix of numbers, letters and operations. A term is a part of an expression, such as 3a, −2 or 5xy.
Like terms have the same letters with the same powers. 3a and 2a are like terms. 3a and 2b are not.
To simplify, collect the like terms: add or subtract their numbers in front, keeping the letters the same.
To multiply terms, multiply the numbers together and multiply the letters: 3x × 2y = 6xy.
Key ideas
- Like terms have exactly the same letters.
- Collect like terms by adding or subtracting the numbers in front.
- Keep unlike terms separate. You cannot simplify 3a + 2b.
- When multiplying, multiply the numbers and multiply the letters.
Method. Collect the like terms: add or subtract the numbers in front of the same letters.
- Collect the a terms. Terms with the same letters are like terms, so we can add their numbers.3a − a = 2a
- Collect the b terms. Terms with the same letters are like terms, so we can add their numbers.2b + 5b = 7b
- Write the answer. Put the collected terms together.2a + 7b
Check. Substitute x = 2 (or a = 2) into the question and into the answer. Both give the same number.
Watch out. Only like terms can be collected. a and a2 are not like terms, and 3a + 2b cannot be simplified further.
Method. Collect the like terms: add or subtract the numbers in front of the same letters.
- Collect the x2 terms. Terms with the same letters are like terms, so we can add their numbers.5x2 − 3x2 = 2x2
- Collect the x terms. Terms with the same letters are like terms, so we can add their numbers.2x + x = 3x
- Write the answer. Put the collected terms together.2x2 + 3x
Check. Substitute x = 2 (or a = 2) into the question and into the answer. Both give the same number.
Watch out. Only like terms can be collected. a and a2 are not like terms, and 3a + 2b cannot be simplified further.
Method. Multiply the numbers together, then multiply the letters. Add the powers of the same letter.
- Multiply the numbers. Multiply the numbers in front of the letters.3 × 2 = 6
- Multiply the letters. Write each letter once. If a letter appears twice, add the powers.x × y = xy
- Write the answer. 6xy
Check. Check the numbers: 3 × 2 = 6, and the letters are the same letters as in the answer.
Watch out. Multiply the numbers and the letters separately. 3x × 2y is 6xy, not 5xy.
Watch out
Watch out: 3a + 2b is already as simple as it can be. You cannot add a and b together, because they are different letters.
Exam tip
Exam tip. Substitute a number for each letter into the question and your answer. If they give the same value, your simplifying is probably right.
Exercise 5A Collecting like terms and multiplying
Non-calculatorMethod. Collect the like terms: add or subtract the numbers in front of the same letters.
- Collect the a terms. Terms with the same letters are like terms, so we can add their numbers.3a − a = 2a
- Collect the b terms. Terms with the same letters are like terms, so we can add their numbers.2b + 5b = 7b
- Write the answer. Put the collected terms together.2a + 7b
Check. Substitute x = 2 (or a = 2) into the question and into the answer. Both give the same number.
Watch out. Only like terms can be collected. a and a2 are not like terms, and 3a + 2b cannot be simplified further.
Method. Collect the like terms: add or subtract the numbers in front of the same letters.
- Collect the x terms. Terms with the same letters are like terms, so we can add their numbers.4x + 2x = 6x
- Collect the y terms. Terms with the same letters are like terms, so we can add their numbers.−3y + y = −2y
- Write the answer. Put the collected terms together.6x − 2y
Check. Substitute x = 2 (or a = 2) into the question and into the answer. Both give the same number.
Watch out. Only like terms can be collected. a and a2 are not like terms, and 3a + 2b cannot be simplified further.
Method. Collect the like terms: add or subtract the numbers in front of the same letters.
- Collect the p terms. Terms with the same letters are like terms, so we can add their numbers.5p − 3p = 2p
- Collect the q terms. Terms with the same letters are like terms, so we can add their numbers.−2q − 4q = −6q
- Write the answer. Put the collected terms together.2p − 6q
Check. Substitute x = 2 (or a = 2) into the question and into the answer. Both give the same number.
Watch out. Only like terms can be collected. a and a2 are not like terms, and 3a + 2b cannot be simplified further.
Method. Collect the like terms: add or subtract the numbers in front of the same letters.
- Collect the m terms. Terms with the same letters are like terms, so we can add their numbers.7m − 7m = 0
- Collect the n terms. Terms with the same letters are like terms, so we can add their numbers.3n = 3n
- Write the answer. Put the collected terms together.3n
Check. Substitute x = 2 (or a = 2) into the question and into the answer. Both give the same number.
Watch out. Only like terms can be collected. a and a2 are not like terms, and 3a + 2b cannot be simplified further.
Method. Collect the like terms: add or subtract the numbers in front of the same letters.
- Collect the a terms. Terms with the same letters are like terms, so we can add their numbers.3a + a = 4a
- Collect the number terms. Terms with the same letters are like terms, so we can add their numbers.2 − 5 = −3
- Write the answer. Put the collected terms together.4a − 3
Check. Substitute x = 2 (or a = 2) into the question and into the answer. Both give the same number.
Watch out. Only like terms can be collected. a and a2 are not like terms, and 3a + 2b cannot be simplified further.
Method. Collect the like terms: add or subtract the numbers in front of the same letters.
- Collect the x2 terms. Terms with the same letters are like terms, so we can add their numbers.5x2 − 3x2 = 2x2
- Collect the x terms. Terms with the same letters are like terms, so we can add their numbers.2x + x = 3x
- Write the answer. Put the collected terms together.2x2 + 3x
Check. Substitute x = 2 (or a = 2) into the question and into the answer. Both give the same number.
Watch out. Only like terms can be collected. a and a2 are not like terms, and 3a + 2b cannot be simplified further.
Method. Collect the like terms: add or subtract the numbers in front of the same letters.
- Collect the c terms. Terms with the same letters are like terms, so we can add their numbers.6c − 6c = 0
- Collect the d terms. Terms with the same letters are like terms, so we can add their numbers.−2d + 4d = 2d
- Write the answer. Put the collected terms together.2d
Check. Substitute x = 2 (or a = 2) into the question and into the answer. Both give the same number.
Watch out. Only like terms can be collected. a and a2 are not like terms, and 3a + 2b cannot be simplified further.
Method. Multiply the numbers together, then multiply the letters. Add the powers of the same letter.
- Multiply the numbers. Multiply the numbers in front of the letters.3 × 2 = 6
- Multiply the letters. Write each letter once. If a letter appears twice, add the powers.x × y = xy
- Write the answer. 6xy
Check. Check the numbers: 3 × 2 = 6, and the letters are the same letters as in the answer.
Watch out. Multiply the numbers and the letters separately. 3x × 2y is 6xy, not 5xy.
Method. Multiply the numbers together, then multiply the letters. Add the powers of the same letter.
- Multiply the numbers. Multiply the numbers in front of the letters.4 × 2 = 8
- Multiply the letters. Write each letter once. If a letter appears twice, add the powers.a × b = ab
- Write the answer. 8ab
Check. Check the numbers: 4 × 2 = 8, and the letters are the same letters as in the answer.
Watch out. Multiply the numbers and the letters separately. 3x × 2y is 6xy, not 5xy.
Method. Multiply the numbers together, then multiply the letters. Add the powers of the same letter.
- Multiply the numbers. Multiply the numbers in front of the letters.5 × 3 = 15
- Multiply the letters. Write each letter once. If a letter appears twice, add the powers.p × q = pq
- Write the answer. 15pq
Check. Check the numbers: 5 × 3 = 15, and the letters are the same letters as in the answer.
Watch out. Multiply the numbers and the letters separately. 3x × 2y is 6xy, not 5xy.
Method. Multiply the numbers together, then multiply the letters. Add the powers of the same letter.
- Multiply the numbers. Multiply the numbers in front of the letters.2 × 6 = 12
- Multiply the letters. Write each letter once. If a letter appears twice, add the powers.m × n = mn
- Write the answer. 12mn
Check. Check the numbers: 2 × 6 = 12, and the letters are the same letters as in the answer.
Watch out. Multiply the numbers and the letters separately. 3x × 2y is 6xy, not 5xy.
Method. Multiply the numbers together, then multiply the letters. Add the powers of the same letter.
- Multiply the numbers. Multiply the numbers in front of the letters.3 × -2 = -6
- Multiply the letters. Write each letter once. If a letter appears twice, add the powers.x × y = xy
- Write the answer. −6xy
Check. Check the numbers: 3 × -2 = -6, and the letters are the same letters as in the answer.
Watch out. Multiply the numbers and the letters separately. 3x × 2y is 6xy, not 5xy.
Method. Collect the x terms and the numbers, then substitute.
- Collect the x terms. 4x − x = 3x.3x
- Collect the numbers. 3 + 5 = 8.8
- Write the simplified expression. 3x + 8
- Substitute x = 2. 3 × 2 + 8 = 6 + 8.6 + 8 = 14
- Write the answer. 3x + 8, and 14 when x = 2
Check. Substituting into the original: 4 × 2 + 3 − 2 + 5 = 14.
Method. Collect the like terms: add or subtract the numbers in front of the same letters.
- Collect the a terms. Terms with the same letters are like terms, so we can add their numbers.a + a = 2a
- Collect the b terms. Terms with the same letters are like terms, so we can add their numbers.b + b = 2b
- Write the answer. Put the collected terms together.2a + 2b
Check. Substitute x = 2 (or a = 2) into the question and into the answer. Both give the same number.
Watch out. Only like terms can be collected. a and a2 are not like terms, and 3a + 2b cannot be simplified further.
Method. Multiply the number of each fruit by its price.
- Apples. x apples at 4p each cost 4x pence.4x
- Pears. y pears at 5p each cost 5y pence.5y
- Add the costs. These are not like terms, so keep them separate.4x + 5y
- Write the answer. 4x + 5y
Check. If x = 2 and y = 3, the total is 8 + 15 = 23 pence.
Method. Look at the letters. a and b are not like terms, so they cannot be added.
- Look at the terms. 3a is a term in a and 2b is a term in b.3a and 2b
- Check the letters. Different letters cannot be collected together.a ≠ b
- Write the answer. Ali is not right. 3a + 2b cannot be simplified.No
Check. If a = 1 and b = 1, then 3a + 2b = 5 but 5ab = 5. These are coincidences, not a rule.
Method. Collect the like terms: add or subtract the numbers in front of the same letters.
- Collect the x2 terms. Terms with the same letters are like terms, so we can add their numbers.2x2 − x2 = x2
- Collect the x terms. Terms with the same letters are like terms, so we can add their numbers.5x + 3x = 8x
- Write the answer. Put the collected terms together.x2 + 8x
Check. Substitute x = 2 (or a = 2) into the question and into the answer. Both give the same number.
Watch out. Only like terms can be collected. a and a2 are not like terms, and 3a + 2b cannot be simplified further.
Method. Compare the powers of the letter x.
- Look at the first term. 2x2 has x with a power of 2.x2
- Look at the second term. 2x has x with a power of 1.x1
- Compare. The powers are different, so they are not like terms.x2 ≠ x1
- Write the answer. They cannot be collected together.Different powers
Check. 2x2 + 2x cannot be simplified to one term.
Expanding single brackets
To expand a bracket, multiply every term inside the bracket by the number or letter outside.
For example: 3(x + 4) means 3 × x and 3 × 4, so 3(x + 4) = 3x + 12.
Be careful with negative numbers. −2(x + 4) = −2x − 8, because −2 × 4 = −8.
Key ideas
- Multiply every term inside the bracket by the term outside.
- The sign of each answer depends on the signs of both numbers.
- A letter outside a bracket multiplies each term inside too: x(x + 3) = x² + 3x.
- Collect like terms after expanding, if there are two brackets.
Method. Multiply every term inside the bracket by the term outside it.
- Look at what is outside the bracket. Everything inside the bracket is multiplied by this.3(x + 4)
- Multiply each term inside by the outside term. Keep the signs. A minus times a minus is a plus.3 × x = 3x and 3 × 4 = 12
- Write the answer. Put the two products together.3x + 12
Check. Try x = 2: the bracket gives 6, and the outside gives 3, so the product is 18. The answer gives the same value at x = 2.
Watch out. Be careful with negative numbers: −2 × (−3) = +6, and −2 × (+3) = −6.
Method. Multiply every term inside the bracket by the term outside it.
- Look at what is outside the bracket. Everything inside the bracket is multiplied by this.−2(x + 4)
- Multiply each term inside by the outside term. Keep the signs. A minus times a minus is a plus.−2 × x = −2x and −2 × 4 = −8
- Write the answer. Put the two products together.−2x − 8
Check. Try x = 2: the bracket gives 6, and the outside gives -2, so the product is -12. The answer gives the same value at x = 2.
Watch out. Be careful with negative numbers: −2 × (−3) = +6, and −2 × (+3) = −6.
Method. Multiply every term inside the bracket by the term outside it.
- Look at what is outside the bracket. Everything inside the bracket is multiplied by this.x(x − 3)
- Multiply each term inside by the outside term. Keep the signs. A minus times a minus is a plus.x × x = x2 and x × −3 = −3x
- Write the answer. Put the two products together.x2 − 3x
Check. Try x = 2: the bracket gives -1, and the outside gives 2, so the product is -2. The answer gives the same value at x = 2.
Watch out. Be careful with negative numbers: −2 × (−3) = +6, and −2 × (+3) = −6.
Watch out
Watch out: 3(x + 4) is 3x + 12, not 3x + 4. The 3 multiplies the 4 as well.
Exam tip
Exam tip. Write each product on its own line, so you do not miss a term. Check the signs at the end.
Exercise 5B Expanding single brackets
Non-calculatorMethod. Multiply every term inside the bracket by the term outside it.
- Look at what is outside the bracket. Everything inside the bracket is multiplied by this.3(x + 4)
- Multiply each term inside by the outside term. Keep the signs. A minus times a minus is a plus.3 × x = 3x and 3 × 4 = 12
- Write the answer. Put the two products together.3x + 12
Check. Try x = 2: the bracket gives 6, and the outside gives 3, so the product is 18. The answer gives the same value at x = 2.
Watch out. Be careful with negative numbers: −2 × (−3) = +6, and −2 × (+3) = −6.
Method. Multiply every term inside the bracket by the term outside it.
- Look at what is outside the bracket. Everything inside the bracket is multiplied by this.2(3x − 5)
- Multiply each term inside by the outside term. Keep the signs. A minus times a minus is a plus.2 × 3x = 6x and 2 × −5 = −10
- Write the answer. Put the two products together.6x − 10
Check. Try x = 2: the bracket gives 1, and the outside gives 2, so the product is 2. The answer gives the same value at x = 2.
Watch out. Be careful with negative numbers: −2 × (−3) = +6, and −2 × (+3) = −6.
Method. Multiply every term inside the bracket by the term outside it.
- Look at what is outside the bracket. Everything inside the bracket is multiplied by this.−2(x + 4)
- Multiply each term inside by the outside term. Keep the signs. A minus times a minus is a plus.−2 × x = −2x and −2 × 4 = −8
- Write the answer. Put the two products together.−2x − 8
Check. Try x = 2: the bracket gives 6, and the outside gives -2, so the product is -12. The answer gives the same value at x = 2.
Watch out. Be careful with negative numbers: −2 × (−3) = +6, and −2 × (+3) = −6.
Method. Multiply every term inside the bracket by the term outside it.
- Look at what is outside the bracket. Everything inside the bracket is multiplied by this.5(2x − 1)
- Multiply each term inside by the outside term. Keep the signs. A minus times a minus is a plus.5 × 2x = 10x and 5 × −1 = −5
- Write the answer. Put the two products together.10x − 5
Check. Try x = 2: the bracket gives 3, and the outside gives 5, so the product is 15. The answer gives the same value at x = 2.
Watch out. Be careful with negative numbers: −2 × (−3) = +6, and −2 × (+3) = −6.
Method. Multiply every term inside the bracket by the term outside it.
- Look at what is outside the bracket. Everything inside the bracket is multiplied by this.4(2x + 3)
- Multiply each term inside by the outside term. Keep the signs. A minus times a minus is a plus.4 × 2x = 8x and 4 × 3 = 12
- Write the answer. Put the two products together.8x + 12
Check. Try x = 2: the bracket gives 7, and the outside gives 4, so the product is 28. The answer gives the same value at x = 2.
Watch out. Be careful with negative numbers: −2 × (−3) = +6, and −2 × (+3) = −6.
Method. Multiply every term inside the bracket by the term outside it.
- Look at what is outside the bracket. Everything inside the bracket is multiplied by this.−3(2x − 1)
- Multiply each term inside by the outside term. Keep the signs. A minus times a minus is a plus.−3 × 2x = −6x and −3 × −1 = 3
- Write the answer. Put the two products together.−6x + 3
Check. Try x = 2: the bracket gives 3, and the outside gives -3, so the product is -9. The answer gives the same value at x = 2.
Watch out. Be careful with negative numbers: −2 × (−3) = +6, and −2 × (+3) = −6.
Method. Multiply every term inside the bracket by the term outside it.
- Look at what is outside the bracket. Everything inside the bracket is multiplied by this.x(x − 3)
- Multiply each term inside by the outside term. Keep the signs. A minus times a minus is a plus.x × x = x2 and x × −3 = −3x
- Write the answer. Put the two products together.x2 − 3x
Check. Try x = 2: the bracket gives -1, and the outside gives 2, so the product is -2. The answer gives the same value at x = 2.
Watch out. Be careful with negative numbers: −2 × (−3) = +6, and −2 × (+3) = −6.
Method. Multiply every term inside the bracket by the term outside it.
- Look at what is outside the bracket. Everything inside the bracket is multiplied by this.2x(3x + 1)
- Multiply each term inside by the outside term. Keep the signs. A minus times a minus is a plus.2x × 3x = 6x2 and 2x × 1 = 2x
- Write the answer. Put the two products together.6x2 + 2x
Check. Try x = 2: the bracket gives 7, and the outside gives 4, so the product is 28. The answer gives the same value at x = 2.
Watch out. Be careful with negative numbers: −2 × (−3) = +6, and −2 × (+3) = −6.
Method. Multiply every term inside the bracket by the term outside it.
- Look at what is outside the bracket. Everything inside the bracket is multiplied by this.−3(2x − 1)
- Multiply each term inside by the outside term. Keep the signs. A minus times a minus is a plus.−3 × 2x = −6x and −3 × −1 = 3
- Write the answer. Put the two products together.−6x + 3
Check. Try x = 2: the bracket gives 3, and the outside gives -3, so the product is -9. The answer gives the same value at x = 2.
Watch out. Be careful with negative numbers: −2 × (−3) = +6, and −2 × (+3) = −6.
Method. Expand each bracket first, then collect the like terms.
- Expand the first bracket. Multiply every term inside the bracket by the number outside.2(x + 3) = 2x + 6
- Expand the second bracket. Do the same for the second bracket.3(x − 1) = 3x − 3
- Collect the like terms. The x terms are like terms, and so are the numbers.2x + 6 + 3x − 3
- Write the answer. 5x + 3
Check. Try x = 1: the original expression and the answer both give the same number.
Method. Expand each bracket first, then collect the like terms.
- Expand the first bracket. Multiply every term inside the bracket by the number outside.3(x + 2) = 3x + 6
- Expand the second bracket. Do the same for the second bracket.-2(x − 5) = −2x + 10
- Collect the like terms. The x terms are like terms, and so are the numbers.3x + 6 − 2x + 10
- Write the answer. x + 16
Check. Try x = 1: the original expression and the answer both give the same number.
Method. Multiply every term inside the bracket by the term outside it.
- Look at what is outside the bracket. Everything inside the bracket is multiplied by this.x(x + 2)
- Multiply each term inside by the outside term. Keep the signs. A minus times a minus is a plus.x × x = x2 and x × 2 = 2x
- Write the answer. Put the two products together.x2 + 2x
Check. Try x = 2: the bracket gives 4, and the outside gives 2, so the product is 8. The answer gives the same value at x = 2.
Watch out. Be careful with negative numbers: −2 × (−3) = +6, and −2 × (+3) = −6.
Method. Expand each bracket first, then collect the like terms.
- Expand the first bracket. Multiply every term inside the bracket by the number outside.2(x + 3) = 4x + 6
- Expand the second bracket. Do the same for the second bracket.2(x + 4) = 2x + 8
- Collect the like terms. The x terms are like terms, and so are the numbers.4x + 6 + 2x + 8
- Write the answer. 6x + 14
Check. Try x = 1: the original expression and the answer both give the same number.
Method. Multiply every term inside the bracket by the term outside it.
- Look at what is outside the bracket. Everything inside the bracket is multiplied by this.4(x + 2)
- Multiply each term inside by the outside term. Keep the signs. A minus times a minus is a plus.4 × x = 4x and 4 × 2 = 8
- Write the answer. Put the two products together.4x + 8
Check. Try x = 2: the bracket gives 4, and the outside gives 4, so the product is 16. The answer gives the same value at x = 2.
Watch out. Be careful with negative numbers: −2 × (−3) = +6, and −2 × (+3) = −6.
Method. Multiply every term in the bracket by 3, then compare.
- Multiply the x term. 3 × x = 3x.3x
- Multiply the number term. 3 × 4 = 12, not 4.3 × 4 = 12
- Compare. So 3(x + 4) = 3x + 12, which is not 3x + 4.3x + 12 ≠ 3x + 4
- Write the answer. Ali is not right.No
Check. Try x = 1: 3(1 + 4) = 15 but 3 + 4 = 7.
Method. Multiply every term inside the bracket by the term outside it.
- Look at what is outside the bracket. Everything inside the bracket is multiplied by this.5(2x − 3)
- Multiply each term inside by the outside term. Keep the signs. A minus times a minus is a plus.5 × 2x = 10x and 5 × −3 = −15
- Write the answer. Put the two products together.10x − 15
Check. Try x = 2: the bracket gives 1, and the outside gives 5, so the product is 5. The answer gives the same value at x = 2.
Watch out. Be careful with negative numbers: −2 × (−3) = +6, and −2 × (+3) = −6.
Method. Expand each bracket first, then collect the like terms.
- Expand the first bracket. Multiply every term inside the bracket by the number outside.4(x + 2) = 4x + 8
- Expand the second bracket. Do the same for the second bracket.2(x + 1) = 2x + 2
- Collect the like terms. The x terms are like terms, and so are the numbers.4x + 8 + 2x + 2
- Write the answer. 6x + 10
Check. Try x = 1: the original expression and the answer both give the same number.
Expanding double brackets
To expand two brackets, multiply each term in the first bracket by each term in the second bracket. This is often shown as FOIL: First, Outside, Inside, Last.
Then collect the like terms. For example (x + 3)(x + 2) = x2 + 2x + 3x + 6 = x2 + 5x + 6.
Key ideas
- Multiply every term in the first bracket by every term in the second.
- The x terms come from the outside and inside products.
- Collect the x terms: 2x + 3x = 5x.
- Check with a number: if x = 1, both sides should give the same value.
Method. Multiply each term in the first bracket by each term in the second bracket, then collect like terms.
- Multiply the first terms. First term in each bracket.x × x = x2
- Multiply the outside terms. The first term of the first bracket with the second term of the second bracket.x × 5 = 5x
- Multiply the inside terms. The second term of the first bracket with the first term of the second bracket.3 × x = 3x
- Multiply the last terms. The second term of each bracket.3 × 5 = 15
- Collect the like terms. The two x terms are like terms, so add them.x2 + 8x + 15
- Write the answer. x2 + 8x + 15
Check. Try x = 1: the brackets give 24, and the answer gives 24.
Watch out. The x terms come from the outside and inside products. Do not forget the signs.
Method. Multiply each term in the first bracket by each term in the second bracket, then collect like terms.
- Multiply the first terms. First term in each bracket.x × x = x2
- Multiply the outside terms. The first term of the first bracket with the second term of the second bracket.x × −2 = −2x
- Multiply the inside terms. The second term of the first bracket with the first term of the second bracket.3 × x = 3x
- Multiply the last terms. The second term of each bracket.3 × −2 = −6
- Collect the like terms. The two x terms are like terms, so add them.x2 + x − 6
- Write the answer. x2 + x − 6
Check. Try x = 1: the brackets give -4, and the answer gives -4.
Watch out. The x terms come from the outside and inside products. Do not forget the signs.
Method. Multiply each term in the first bracket by each term in the second bracket, then collect like terms.
- Multiply the first terms. First term in each bracket.2x × x = 2x2
- Multiply the outside terms. The first term of the first bracket with the second term of the second bracket.2x × 4 = 8x
- Multiply the inside terms. The second term of the first bracket with the first term of the second bracket.1 × x = x
- Multiply the last terms. The second term of each bracket.1 × 4 = 4
- Collect the like terms. The two x terms are like terms, so add them.2x2 + 9x + 4
- Write the answer. 2x2 + 9x + 4
Check. Try x = 1: the brackets give 15, and the answer gives 15.
Watch out. The x terms come from the outside and inside products. Do not forget the signs.
Watch out
Watch out: (x + 3)(x + 3) is not x2 + 9. You must include the outside and inside products: x2 + 6x + 9.
Exam tip
Exam tip. Write all four products before you collect, and count them: there are always four products before you collect.
Exercise 5C Expanding double brackets
Non-calculatorMethod. Multiply each term in the first bracket by each term in the second bracket, then collect like terms.
- Multiply the first terms. First term in each bracket.x × x = x2
- Multiply the outside terms. The first term of the first bracket with the second term of the second bracket.x × 5 = 5x
- Multiply the inside terms. The second term of the first bracket with the first term of the second bracket.3 × x = 3x
- Multiply the last terms. The second term of each bracket.3 × 5 = 15
- Collect the like terms. The two x terms are like terms, so add them.x2 + 8x + 15
- Write the answer. x2 + 8x + 15
Check. Try x = 1: the brackets give 24, and the answer gives 24.
Watch out. The x terms come from the outside and inside products. Do not forget the signs.
Method. Multiply each term in the first bracket by each term in the second bracket, then collect like terms.
- Multiply the first terms. First term in each bracket.x × x = x2
- Multiply the outside terms. The first term of the first bracket with the second term of the second bracket.x × −2 = −2x
- Multiply the inside terms. The second term of the first bracket with the first term of the second bracket.3 × x = 3x
- Multiply the last terms. The second term of each bracket.3 × −2 = −6
- Collect the like terms. The two x terms are like terms, so add them.x2 + x − 6
- Write the answer. x2 + x − 6
Check. Try x = 1: the brackets give -4, and the answer gives -4.
Watch out. The x terms come from the outside and inside products. Do not forget the signs.
Method. Multiply each term in the first bracket by each term in the second bracket, then collect like terms.
- Multiply the first terms. First term in each bracket.x × x = x2
- Multiply the outside terms. The first term of the first bracket with the second term of the second bracket.x × −1 = −x
- Multiply the inside terms. The second term of the first bracket with the first term of the second bracket.−4 × x = −4x
- Multiply the last terms. The second term of each bracket.−4 × −1 = 4
- Collect the like terms. The two x terms are like terms, so add them.x2 − 5x + 4
- Write the answer. x2 − 5x + 4
Check. Try x = 1: the brackets give 0, and the answer gives 0.
Watch out. The x terms come from the outside and inside products. Do not forget the signs.
Method. Multiply each term in the first bracket by each term in the second bracket, then collect like terms.
- Multiply the first terms. First term in each bracket.2x × x = 2x2
- Multiply the outside terms. The first term of the first bracket with the second term of the second bracket.2x × 4 = 8x
- Multiply the inside terms. The second term of the first bracket with the first term of the second bracket.1 × x = x
- Multiply the last terms. The second term of each bracket.1 × 4 = 4
- Collect the like terms. The two x terms are like terms, so add them.2x2 + 9x + 4
- Write the answer. 2x2 + 9x + 4
Check. Try x = 1: the brackets give 15, and the answer gives 15.
Watch out. The x terms come from the outside and inside products. Do not forget the signs.
Method. Multiply each term in the first bracket by each term in the second bracket, then collect like terms.
- Multiply the first terms. First term in each bracket.3x × x = 3x2
- Multiply the outside terms. The first term of the first bracket with the second term of the second bracket.3x × 5 = 15x
- Multiply the inside terms. The second term of the first bracket with the first term of the second bracket.−2 × x = −2x
- Multiply the last terms. The second term of each bracket.−2 × 5 = −10
- Collect the like terms. The two x terms are like terms, so add them.3x2 + 13x − 10
- Write the answer. 3x2 + 13x − 10
Check. Try x = 1: the brackets give 6, and the answer gives 6.
Watch out. The x terms come from the outside and inside products. Do not forget the signs.
Method. Multiply each term in the first bracket by each term in the second bracket, then collect like terms.
- Multiply the first terms. First term in each bracket.x × x = x2
- Multiply the outside terms. The first term of the first bracket with the second term of the second bracket.x × −6 = −6x
- Multiply the inside terms. The second term of the first bracket with the first term of the second bracket.6 × x = 6x
- Multiply the last terms. The second term of each bracket.6 × −6 = −36
- Collect the like terms. The two x terms are like terms, so add them.x2 − 36
- Write the answer. x2 − 36
Check. Try x = 1: the brackets give -35, and the answer gives -35.
Watch out. The x terms come from the outside and inside products. Do not forget the signs.
Method. Multiply each term in the first bracket by each term in the second bracket, then collect like terms.
- Multiply the first terms. First term in each bracket.2x × 2x = 4x2
- Multiply the outside terms. The first term of the first bracket with the second term of the second bracket.2x × −1 = −2x
- Multiply the inside terms. The second term of the first bracket with the first term of the second bracket.3 × 2x = 6x
- Multiply the last terms. The second term of each bracket.3 × −1 = −3
- Collect the like terms. The two x terms are like terms, so add them.4x2 + 4x − 3
- Write the answer. 4x2 + 4x − 3
Check. Try x = 1: the brackets give 5, and the answer gives 5.
Watch out. The x terms come from the outside and inside products. Do not forget the signs.
Method. Multiply each term in the first bracket by each term in the second bracket, then collect like terms.
- Multiply the first terms. First term in each bracket.x × x = x2
- Multiply the outside terms. The first term of the first bracket with the second term of the second bracket.x × 2 = 2x
- Multiply the inside terms. The second term of the first bracket with the first term of the second bracket.2 × x = 2x
- Multiply the last terms. The second term of each bracket.2 × 2 = 4
- Collect the like terms. The two x terms are like terms, so add them.x2 + 4x + 4
- Write the answer. x2 + 4x + 4
Check. Try x = 1: the brackets give 9, and the answer gives 9.
Watch out. The x terms come from the outside and inside products. Do not forget the signs.
Method. Multiply each term in the first bracket by each term in the second bracket, then collect like terms.
- Multiply the first terms. First term in each bracket.2x × 3x = 6x2
- Multiply the outside terms. The first term of the first bracket with the second term of the second bracket.2x × 2 = 4x
- Multiply the inside terms. The second term of the first bracket with the first term of the second bracket.−5 × 3x = −15x
- Multiply the last terms. The second term of each bracket.−5 × 2 = −10
- Collect the like terms. The two x terms are like terms, so add them.6x2 − 11x − 10
- Write the answer. 6x2 − 11x − 10
Check. Try x = 1: the brackets give -15, and the answer gives -15.
Watch out. The x terms come from the outside and inside products. Do not forget the signs.
Method. Multiply each term in the first bracket by each term in the second bracket, then collect like terms.
- Multiply the first terms. First term in each bracket.4x × x = 4x2
- Multiply the outside terms. The first term of the first bracket with the second term of the second bracket.4x × −3 = −12x
- Multiply the inside terms. The second term of the first bracket with the first term of the second bracket.1 × x = x
- Multiply the last terms. The second term of each bracket.1 × −3 = −3
- Collect the like terms. The two x terms are like terms, so add them.4x2 − 11x − 3
- Write the answer. 4x2 − 11x − 3
Check. Try x = 1: the brackets give -10, and the answer gives -10.
Watch out. The x terms come from the outside and inside products. Do not forget the signs.
Method. Multiply each term in the first bracket by each term in the second bracket, then collect like terms.
- Multiply the first terms. First term in each bracket.x × x = x2
- Multiply the outside terms. The first term of the first bracket with the second term of the second bracket.x × 3 = 3x
- Multiply the inside terms. The second term of the first bracket with the first term of the second bracket.3 × x = 3x
- Multiply the last terms. The second term of each bracket.3 × 3 = 9
- Collect the like terms. The two x terms are like terms, so add them.x2 + 6x + 9
- Write the answer. x2 + 6x + 9
Check. Try x = 1: the brackets give 16, and the answer gives 16.
Watch out. The x terms come from the outside and inside products. Do not forget the signs.
Method. Multiply each term in the first bracket by each term in the second bracket, then collect like terms.
- Multiply the first terms. First term in each bracket.x × x = x2
- Multiply the outside terms. The first term of the first bracket with the second term of the second bracket.x × 3 = 3x
- Multiply the inside terms. The second term of the first bracket with the first term of the second bracket.5 × x = 5x
- Multiply the last terms. The second term of each bracket.5 × 3 = 15
- Collect the like terms. The two x terms are like terms, so add them.x2 + 8x + 15
- Write the answer. x2 + 8x + 15
Check. Try x = 1: the brackets give 24, and the answer gives 24.
Watch out. The x terms come from the outside and inside products. Do not forget the signs.
Method. Multiply each term in the first bracket by each term in the second bracket, then collect like terms.
- Multiply the first terms. First term in each bracket.x × x = x2
- Multiply the outside terms. The first term of the first bracket with the second term of the second bracket.x × −4 = −4x
- Multiply the inside terms. The second term of the first bracket with the first term of the second bracket.4 × x = 4x
- Multiply the last terms. The second term of each bracket.4 × −4 = −16
- Collect the like terms. The two x terms are like terms, so add them.x2 − 16
- Write the answer. x2 − 16
Check. Try x = 1: the brackets give -15, and the answer gives -15.
Watch out. The x terms come from the outside and inside products. Do not forget the signs.
Method. Expand the brackets, then compare with x2 + 6.
- Expand the brackets. Multiply each pair of terms.x2 + 3x + 2x + 6
- Collect the x terms. 3x + 2x = 5x.x2 + 5x + 6
- Compare. The expansion has 5x and +6, but x2 + 6 has neither.x2 + 5x + 6 ≠ x2 + 6
- Write the answer. No
Check. Try x = 1: (1 + 2)(1 + 3) = 12, but 1 + 6 = 7.
Method. Expand both, and compare.
- Expand (x + 4)(x + 3). Multiply each pair of terms.x2 + 3x + 4x + 12
- Expand (x + 3)(x + 4). Multiply each pair of terms in the other order.x2 + 4x + 3x + 12
- Compare. Both give x2 + 7x + 12.x2 + 7x + 12
- Write the answer. The order of the brackets does not change the answer.Same
Check. Try x = 2: both give 6 × 7 = 42.
Method. Multiply each term in the first bracket by each term in the second bracket, then collect like terms.
- Multiply the first terms. First term in each bracket.2x × 3x = 6x2
- Multiply the outside terms. The first term of the first bracket with the second term of the second bracket.2x × 2 = 4x
- Multiply the inside terms. The second term of the first bracket with the first term of the second bracket.−5 × 3x = −15x
- Multiply the last terms. The second term of each bracket.−5 × 2 = −10
- Collect the like terms. The two x terms are like terms, so add them.6x2 − 11x − 10
- Write the answer. 6x2 − 11x − 10
Check. Try x = 1: the brackets give -15, and the answer gives -15.
Watch out. The x terms come from the outside and inside products. Do not forget the signs.
Factorising by taking out a common factor
To factorise an expression, write it as a product of two factors. Take out the highest common factor (HCF) of all the terms and write the rest inside a bracket.
Look for the highest common factor of the numbers, and any letters that are in every term. For example 6x + 9 has the HCF 3, so 6x + 9 = 3(2x + 3).
You can check your answer by expanding the brackets. You should get back the original expression.
Key ideas
- The highest common factor (HCF) is the biggest number, and the highest shared letter power.
- Divide each term by the HCF, and write the answers inside the brackets.
- The factorised form is HCF(... + ...).
- Check by expanding the brackets.
Method. Find the highest common factor of the numbers and any letters that every term has. Take it outside the brackets.
- Find the common factor of the numbers. Find the biggest number that divides into every number.HCF of the numbers = 3
- Find any letters that every term has. The lowest power of each letter is in every term.common letter part = none
- Divide each term by the common factor. Write the results inside the brackets.6x + 9 ÷ (3) = 2x + 3
- Write the answer. Check by expanding the brackets.3(2x + 3)
Check. Expanding the brackets gives the original expression back.
Watch out. Take out the highest common factor, not just any factor. Check that every term has been divided.
Method. Find the highest common factor of the numbers and any letters that every term has. Take it outside the brackets.
- Find the common factor of the numbers. Find the biggest number that divides into every number.HCF of the numbers = 4
- Find any letters that every term has. The lowest power of each letter is in every term.common letter part = x
- Divide each term by the common factor. Write the results inside the brackets.8x2 − 12x ÷ (4x) = 2x − 3
- Write the answer. Check by expanding the brackets.4x(2x − 3)
Check. Expanding the brackets gives the original expression back.
Watch out. Take out the highest common factor, not just any factor. Check that every term has been divided.
Method. Find the highest common factor of the numbers and any letters that every term has. Take it outside the brackets.
- Find the common factor of the numbers. Find the biggest number that divides into every number.HCF of the numbers = 1
- Find any letters that every term has. The lowest power of each letter is in every term.common letter part = x
- Divide each term by the common factor. Write the results inside the brackets.x2 + 4x ÷ (1x) = x + 4
- Write the answer. Check by expanding the brackets.x(x + 4)
Check. Expanding the brackets gives the original expression back.
Watch out. Take out the highest common factor, not just any factor. Check that every term has been divided.
Watch out
Watch out: take out the highest common factor. 6x + 9 = 3(2x + 3) is correct. 6x + 9 = 1(6x + 9) is true, but it does not help.
Exam tip
Exam tip. If the answer is 3(2x + 3), expand it to check. If you get 6x + 9 back, your answer is right.
Exercise 5D Factorising by common factors
Non-calculatorMethod. Find the highest common factor of the numbers and any letters that every term has. Take it outside the brackets.
- Find the common factor of the numbers. Find the biggest number that divides into every number.HCF of the numbers = 3
- Find any letters that every term has. The lowest power of each letter is in every term.common letter part = none
- Divide each term by the common factor. Write the results inside the brackets.6x + 9 ÷ (3) = 2x + 3
- Write the answer. Check by expanding the brackets.3(2x + 3)
Check. Expanding the brackets gives the original expression back.
Watch out. Take out the highest common factor, not just any factor. Check that every term has been divided.
Method. Find the highest common factor of the numbers and any letters that every term has. Take it outside the brackets.
- Find the common factor of the numbers. Find the biggest number that divides into every number.HCF of the numbers = 4
- Find any letters that every term has. The lowest power of each letter is in every term.common letter part = none
- Divide each term by the common factor. Write the results inside the brackets.8x − 12 ÷ (4) = 2x − 3
- Write the answer. Check by expanding the brackets.4(2x − 3)
Check. Expanding the brackets gives the original expression back.
Watch out. Take out the highest common factor, not just any factor. Check that every term has been divided.
Method. Find the highest common factor of the numbers and any letters that every term has. Take it outside the brackets.
- Find the common factor of the numbers. Find the biggest number that divides into every number.HCF of the numbers = 5
- Find any letters that every term has. The lowest power of each letter is in every term.common letter part = none
- Divide each term by the common factor. Write the results inside the brackets.10x + 15 ÷ (5) = 2x + 3
- Write the answer. Check by expanding the brackets.5(2x + 3)
Check. Expanding the brackets gives the original expression back.
Watch out. Take out the highest common factor, not just any factor. Check that every term has been divided.
Method. Find the highest common factor of the numbers and any letters that every term has. Take it outside the brackets.
- Find the common factor of the numbers. Find the biggest number that divides into every number.HCF of the numbers = 6
- Find any letters that every term has. The lowest power of each letter is in every term.common letter part = none
- Divide each term by the common factor. Write the results inside the brackets.12x − 18 ÷ (6) = 2x − 3
- Write the answer. Check by expanding the brackets.6(2x − 3)
Check. Expanding the brackets gives the original expression back.
Watch out. Take out the highest common factor, not just any factor. Check that every term has been divided.
Method. Find the highest common factor of the numbers and any letters that every term has. Take it outside the brackets.
- Find the common factor of the numbers. Find the biggest number that divides into every number.HCF of the numbers = 1
- Find any letters that every term has. The lowest power of each letter is in every term.common letter part = x
- Divide each term by the common factor. Write the results inside the brackets.x2 + 4x ÷ (1x) = x + 4
- Write the answer. Check by expanding the brackets.x(x + 4)
Check. Expanding the brackets gives the original expression back.
Watch out. Take out the highest common factor, not just any factor. Check that every term has been divided.
Method. Find the highest common factor of the numbers and any letters that every term has. Take it outside the brackets.
- Find the common factor of the numbers. Find the biggest number that divides into every number.HCF of the numbers = 3
- Find any letters that every term has. The lowest power of each letter is in every term.common letter part = x
- Divide each term by the common factor. Write the results inside the brackets.3x2 + 6x ÷ (3x) = x + 2
- Write the answer. Check by expanding the brackets.3x(x + 2)
Check. Expanding the brackets gives the original expression back.
Watch out. Take out the highest common factor, not just any factor. Check that every term has been divided.
Method. Find the highest common factor of the numbers and any letters that every term has. Take it outside the brackets.
- Find the common factor of the numbers. Find the biggest number that divides into every number.HCF of the numbers = 4
- Find any letters that every term has. The lowest power of each letter is in every term.common letter part = x
- Divide each term by the common factor. Write the results inside the brackets.8x2 − 12x ÷ (4x) = 2x − 3
- Write the answer. Check by expanding the brackets.4x(2x − 3)
Check. Expanding the brackets gives the original expression back.
Watch out. Take out the highest common factor, not just any factor. Check that every term has been divided.
Method. Find the highest common factor of the numbers and any letters that every term has. Take it outside the brackets.
- Find the common factor of the numbers. Find the biggest number that divides into every number.HCF of the numbers = 2
- Find any letters that every term has. The lowest power of each letter is in every term.common letter part = x
- Divide each term by the common factor. Write the results inside the brackets.2x2 − 6x ÷ (2x) = x − 3
- Write the answer. Check by expanding the brackets.2x(x − 3)
Check. Expanding the brackets gives the original expression back.
Watch out. Take out the highest common factor, not just any factor. Check that every term has been divided.
Method. Find the highest common factor of the numbers and any letters that every term has. Take it outside the brackets.
- Find the common factor of the numbers. Find the biggest number that divides into every number.HCF of the numbers = 5
- Find any letters that every term has. The lowest power of each letter is in every term.common letter part = none
- Divide each term by the common factor. Write the results inside the brackets.5x + 10 ÷ (5) = x + 2
- Write the answer. Check by expanding the brackets.5(x + 2)
Check. Expanding the brackets gives the original expression back.
Watch out. Take out the highest common factor, not just any factor. Check that every term has been divided.
Method. Find the highest common factor of the numbers and any letters that every term has. Take it outside the brackets.
- Find the common factor of the numbers. Find the biggest number that divides into every number.HCF of the numbers = 3
- Find any letters that every term has. The lowest power of each letter is in every term.common letter part = none
- Divide each term by the common factor. Write the results inside the brackets.9x − 6 ÷ (3) = 3x − 2
- Write the answer. Check by expanding the brackets.3(3x − 2)
Check. Expanding the brackets gives the original expression back.
Watch out. Take out the highest common factor, not just any factor. Check that every term has been divided.
Method. Find the highest common factor of the numbers and any letters that every term has. Take it outside the brackets.
- Find the common factor of the numbers. Find the biggest number that divides into every number.HCF of the numbers = 4
- Find any letters that every term has. The lowest power of each letter is in every term.common letter part = none
- Divide each term by the common factor. Write the results inside the brackets.4x2 + 8 ÷ (4) = x2 + 2
- Write the answer. Check by expanding the brackets.4(x2 + 2)
Check. Expanding the brackets gives the original expression back.
Watch out. Take out the highest common factor, not just any factor. Check that every term has been divided.
Method. Find the highest common factor of the numbers and any letters that every term has. Take it outside the brackets.
- Find the common factor of the numbers. Find the biggest number that divides into every number.HCF of the numbers = 3
- Find any letters that every term has. The lowest power of each letter is in every term.common letter part = x
- Divide each term by the common factor. Write the results inside the brackets.6x2 − 9x ÷ (3x) = 2x − 3
- Write the answer. Check by expanding the brackets.3x(2x − 3)
Check. Expanding the brackets gives the original expression back.
Watch out. Take out the highest common factor, not just any factor. Check that every term has been divided.
Method. Find the highest common factor of the numbers and any letters that every term has. Take it outside the brackets.
- Find the common factor of the numbers. Find the biggest number that divides into every number.HCF of the numbers = 4
- Find any letters that every term has. The lowest power of each letter is in every term.common letter part = x
- Divide each term by the common factor. Write the results inside the brackets.12x2 + 8x ÷ (4x) = 3x + 2
- Write the answer. Check by expanding the brackets.4x(3x + 2)
Check. Expanding the brackets gives the original expression back.
Watch out. Take out the highest common factor, not just any factor. Check that every term has been divided.
Method. Find the highest common factor of the numbers and any letters that every term has. Take it outside the brackets.
- Find the common factor of the numbers. Find the biggest number that divides into every number.HCF of the numbers = 2
- Find any letters that every term has. The lowest power of each letter is in every term.common letter part = none
- Divide each term by the common factor. Write the results inside the brackets.4x + 6 ÷ (2) = 2x + 3
- Write the answer. Check by expanding the brackets.2(2x + 3)
Check. Expanding the brackets gives the original expression back.
Watch out. Take out the highest common factor, not just any factor. Check that every term has been divided.
Method. Divide the area by the length to find the width.
- Write the area as a product. Area = length × width, so the area must be the length times the width.6x + 9 = 3(2x + 3)
- Use the length. The length is 3 cm, so the width is the bracket.3 × (2x + 3)
- Write the answer. The width is (2x + 3) cm.2x + 3 cm
Check. 3 × (2x + 3) = 6x + 9.
Method. Find the highest common factor of the numbers and any letters that every term has. Take it outside the brackets.
- Find the common factor of the numbers. Find the biggest number that divides into every number.HCF of the numbers = 4
- Find any letters that every term has. The lowest power of each letter is in every term.common letter part = x
- Divide each term by the common factor. Write the results inside the brackets.12x2 + 8x ÷ (4x) = 3x + 2
- Write the answer. Check by expanding the brackets.4x(3x + 2)
Check. Expanding the brackets gives the original expression back.
Watch out. Take out the highest common factor, not just any factor. Check that every term has been divided.
Method. Expand the bracket and compare with the original expression.
- Expand 3(2x + 3). 3 × 2x = 6x and 3 × 3 = 9.6x + 9
- Compare. This matches 6x + 9.6x + 9 = 6x + 9
- Write the answer. Ali is right.Yes
Check. Try x = 1: 6 + 9 = 15, and 3(2 + 3) = 15.
Method. Look at the common factor.
- Find the highest common factor. The biggest number that divides 6 and 9 is 3.HCF = 3
- Compare the two forms. 3(2x + 3) has the highest common factor outside. 1(6x + 9) does not.3 is the HCF
- Write the answer. 3(2x + 3) is the full factorisation
Check. The bracket 2x + 3 has no common factor left.
Method. Factorise first, then substitute.
- Find the HCF. HCF of 5 and 15 is 5, and the lowest power of a is a.5a
- Divide each term by 5a. 5a2 ÷ 5a = a, and −15a ÷ 5a = −3.5a(a − 3)
- Substitute a = 4. 5 × 4 × (4 − 3) = 20 × 1.20
- Write the answer. 5a(a − 3), and 20 when a = 4
Check. 5a2 − 15a at a = 4 is 80 − 60 = 20.
Factorising quadratics
A quadratic such as x2 + 5x + 6 can often be written as two brackets: (x + p)(x + q).
Find two numbers p and q that multiply to make c (the number at the end) and add to make b (the number in front of x).
For x2 + 5x + 6: the numbers must multiply to 6 and add to 5. They are 2 and 3, so x2 + 5x + 6 = (x + 2)(x + 3).
Key ideas
- For x2 + bx + c, find p and q with pq = c and p + q = b.
- Write the answer as (x + p)(x + q).
- Watch the signs: a negative c means one of p and q is negative.
- Check your answer by expanding the brackets.
Method. Find two numbers that multiply to make c and add to make b. Then write them in two brackets with x.
- Write what we need. Two numbers that multiply to 6 and add to 5.multiply to 6, add to 5
- Find the two numbers. 2 and 3 work: 2 × 3 = 6 and 2 + 3 = 5.2 × 3 = 6, 2 + 3 = 5
- Write the brackets. Each number goes in its own bracket with x.(x + 2)(x + 3)
- Write the answer. (x + 2)(x + 3)
Check. Expanding (x + 2)(x + 3) gives x2 + 5x + 6, which is the original expression.
Watch out. Watch the signs. The two numbers can be negative. Check both the sum and the product.
Method. Find two numbers that multiply to make c and add to make b. Then write them in two brackets with x.
- Write what we need. Two numbers that multiply to -6 and add to 1.multiply to -6, add to 1
- Find the two numbers. -2 and 3 work: -2 × 3 = -6 and -2 + 3 = 1.-2 × 3 = -6, -2 + 3 = 1
- Write the brackets. Each number goes in its own bracket with x.(x − 2)(x + 3)
- Write the answer. (x − 2)(x + 3)
Check. Expanding (x − 2)(x + 3) gives x2 + 1x − 6, which is the original expression.
Watch out. Watch the signs. The two numbers can be negative. Check both the sum and the product.
Method. Find two numbers that multiply to make c and add to make b. Then write them in two brackets with x.
- Write what we need. Two numbers that multiply to 4 and add to -4.multiply to 4, add to -4
- Find the two numbers. -2 and -2 work: -2 × -2 = 4 and -2 + -2 = -4.-2 × -2 = 4, -2 + -2 = -4
- Write the brackets. Each number goes in its own bracket with x.(x − 2)(x − 2)
- Write the answer. (x − 2)(x − 2)
Check. Expanding (x − 2)(x − 2) gives x2 − 4x + 4, which is the original expression.
Watch out. Watch the signs. The two numbers can be negative. Check both the sum and the product.
Watch out
Watch out: the two numbers must multiply to c and add to b. For x2 + 5x + 6, the pair 1 and 6 multiplies to 6 but adds to 7, so it is wrong.
Exam tip
Exam tip. Expand your brackets to check. The middle term must come out as 5x, not something else.
Exercise 5E Factorising quadratics
Non-calculatorMethod. Find two numbers that multiply to make c and add to make b. Then write them in two brackets with x.
- Write what we need. Two numbers that multiply to 6 and add to 5.multiply to 6, add to 5
- Find the two numbers. 2 and 3 work: 2 × 3 = 6 and 2 + 3 = 5.2 × 3 = 6, 2 + 3 = 5
- Write the brackets. Each number goes in its own bracket with x.(x + 2)(x + 3)
- Write the answer. (x + 2)(x + 3)
Check. Expanding (x + 2)(x + 3) gives x2 + 5x + 6, which is the original expression.
Watch out. Watch the signs. The two numbers can be negative. Check both the sum and the product.
Method. Find two numbers that multiply to make c and add to make b. Then write them in two brackets with x.
- Write what we need. Two numbers that multiply to 10 and add to 7.multiply to 10, add to 7
- Find the two numbers. 2 and 5 work: 2 × 5 = 10 and 2 + 5 = 7.2 × 5 = 10, 2 + 5 = 7
- Write the brackets. Each number goes in its own bracket with x.(x + 2)(x + 5)
- Write the answer. (x + 2)(x + 5)
Check. Expanding (x + 2)(x + 5) gives x2 + 7x + 10, which is the original expression.
Watch out. Watch the signs. The two numbers can be negative. Check both the sum and the product.
Method. Find two numbers that multiply to make c and add to make b. Then write them in two brackets with x.
- Write what we need. Two numbers that multiply to 6 and add to -5.multiply to 6, add to -5
- Find the two numbers. -3 and -2 work: -3 × -2 = 6 and -3 + -2 = -5.-3 × -2 = 6, -3 + -2 = -5
- Write the brackets. Each number goes in its own bracket with x.(x − 3)(x − 2)
- Write the answer. (x − 3)(x − 2)
Check. Expanding (x − 3)(x − 2) gives x2 − 5x + 6, which is the original expression.
Watch out. Watch the signs. The two numbers can be negative. Check both the sum and the product.
Method. Find two numbers that multiply to make c and add to make b. Then write them in two brackets with x.
- Write what we need. Two numbers that multiply to -6 and add to 1.multiply to -6, add to 1
- Find the two numbers. -2 and 3 work: -2 × 3 = -6 and -2 + 3 = 1.-2 × 3 = -6, -2 + 3 = 1
- Write the brackets. Each number goes in its own bracket with x.(x − 2)(x + 3)
- Write the answer. (x − 2)(x + 3)
Check. Expanding (x − 2)(x + 3) gives x2 + 1x − 6, which is the original expression.
Watch out. Watch the signs. The two numbers can be negative. Check both the sum and the product.
Method. Find two numbers that multiply to make c and add to make b. Then write them in two brackets with x.
- Write what we need. Two numbers that multiply to -15 and add to -2.multiply to -15, add to -2
- Find the two numbers. -5 and 3 work: -5 × 3 = -15 and -5 + 3 = -2.-5 × 3 = -15, -5 + 3 = -2
- Write the brackets. Each number goes in its own bracket with x.(x − 5)(x + 3)
- Write the answer. (x − 5)(x + 3)
Check. Expanding (x − 5)(x + 3) gives x2 − 2x − 15, which is the original expression.
Watch out. Watch the signs. The two numbers can be negative. Check both the sum and the product.
Method. Find two numbers that multiply to make c and add to make b. Then write them in two brackets with x.
- Write what we need. Two numbers that multiply to 12 and add to 8.multiply to 12, add to 8
- Find the two numbers. 2 and 6 work: 2 × 6 = 12 and 2 + 6 = 8.2 × 6 = 12, 2 + 6 = 8
- Write the brackets. Each number goes in its own bracket with x.(x + 2)(x + 6)
- Write the answer. (x + 2)(x + 6)
Check. Expanding (x + 2)(x + 6) gives x2 + 8x + 12, which is the original expression.
Watch out. Watch the signs. The two numbers can be negative. Check both the sum and the product.
Method. Find two numbers that multiply to make c and add to make b. Then write them in two brackets with x.
- Write what we need. Two numbers that multiply to -9 and add to 0.multiply to -9, add to 0
- Find the two numbers. -3 and 3 work: -3 × 3 = -9 and -3 + 3 = 0.-3 × 3 = -9, -3 + 3 = 0
- Write the brackets. Each number goes in its own bracket with x.(x − 3)(x + 3)
- Write the answer. (x − 3)(x + 3)
Check. Expanding (x − 3)(x + 3) gives x2 + 0x − 9, which is the original expression.
Watch out. Watch the signs. The two numbers can be negative. Check both the sum and the product.
Method. Find two numbers that multiply to make c and add to make b. Then write them in two brackets with x.
- Write what we need. Two numbers that multiply to 4 and add to -4.multiply to 4, add to -4
- Find the two numbers. -2 and -2 work: -2 × -2 = 4 and -2 + -2 = -4.-2 × -2 = 4, -2 + -2 = -4
- Write the brackets. Each number goes in its own bracket with x.(x − 2)(x − 2)
- Write the answer. (x − 2)(x − 2)
Check. Expanding (x − 2)(x − 2) gives x2 − 4x + 4, which is the original expression.
Watch out. Watch the signs. The two numbers can be negative. Check both the sum and the product.
Method. Find two numbers that multiply to make c and add to make b. Then write them in two brackets with x.
- Write what we need. Two numbers that multiply to 8 and add to 6.multiply to 8, add to 6
- Find the two numbers. 2 and 4 work: 2 × 4 = 8 and 2 + 4 = 6.2 × 4 = 8, 2 + 4 = 6
- Write the brackets. Each number goes in its own bracket with x.(x + 2)(x + 4)
- Write the answer. (x + 2)(x + 4)
Check. Expanding (x + 2)(x + 4) gives x2 + 6x + 8, which is the original expression.
Watch out. Watch the signs. The two numbers can be negative. Check both the sum and the product.
Method. Find two numbers that multiply to make c and add to make b. Then write them in two brackets with x.
- Write what we need. Two numbers that multiply to 12 and add to -7.multiply to 12, add to -7
- Find the two numbers. -4 and -3 work: -4 × -3 = 12 and -4 + -3 = -7.-4 × -3 = 12, -4 + -3 = -7
- Write the brackets. Each number goes in its own bracket with x.(x − 4)(x − 3)
- Write the answer. (x − 4)(x − 3)
Check. Expanding (x − 4)(x − 3) gives x2 − 7x + 12, which is the original expression.
Watch out. Watch the signs. The two numbers can be negative. Check both the sum and the product.
Method. Find two numbers that multiply to make c and add to make b. Then write them in two brackets with x.
- Write what we need. Two numbers that multiply to 3 and add to 4.multiply to 3, add to 4
- Find the two numbers. 1 and 3 work: 1 × 3 = 3 and 1 + 3 = 4.1 × 3 = 3, 1 + 3 = 4
- Write the brackets. Each number goes in its own bracket with x.(x + 1)(x + 3)
- Write the answer. (x + 1)(x + 3)
Check. Expanding (x + 1)(x + 3) gives x2 + 4x + 3, which is the original expression.
Watch out. Watch the signs. The two numbers can be negative. Check both the sum and the product.
Method. Find two numbers that multiply to make c and add to make b. Then write them in two brackets with x.
- Write what we need. Two numbers that multiply to -10 and add to 3.multiply to -10, add to 3
- Find the two numbers. -2 and 5 work: -2 × 5 = -10 and -2 + 5 = 3.-2 × 5 = -10, -2 + 5 = 3
- Write the brackets. Each number goes in its own bracket with x.(x − 2)(x + 5)
- Write the answer. (x − 2)(x + 5)
Check. Expanding (x − 2)(x + 5) gives x2 + 3x − 10, which is the original expression.
Watch out. Watch the signs. The two numbers can be negative. Check both the sum and the product.
Method. Find two numbers that multiply to make c and add to make b. Then write them in two brackets with x.
- Write what we need. Two numbers that multiply to 8 and add to 6.multiply to 8, add to 6
- Find the two numbers. 2 and 4 work: 2 × 4 = 8 and 2 + 4 = 6.2 × 4 = 8, 2 + 4 = 6
- Write the brackets. Each number goes in its own bracket with x.(x + 2)(x + 4)
- Write the answer. (x + 2)(x + 4)
Check. Expanding (x + 2)(x + 4) gives x2 + 6x + 8, which is the original expression.
Watch out. Watch the signs. The two numbers can be negative. Check both the sum and the product.
Method. Find two numbers that multiply to make c and add to make b. Then write them in two brackets with x.
- Write what we need. Two numbers that multiply to 6 and add to 5.multiply to 6, add to 5
- Find the two numbers. 2 and 3 work: 2 × 3 = 6 and 2 + 3 = 5.2 × 3 = 6, 2 + 3 = 5
- Write the brackets. Each number goes in its own bracket with x.(x + 2)(x + 3)
- Write the answer. (x + 2)(x + 3)
Check. Expanding (x + 2)(x + 3) gives x2 + 5x + 6, which is the original expression.
Watch out. Watch the signs. The two numbers can be negative. Check both the sum and the product.
Method. Expand the two brackets to check the factorisation.
- Multiply the first terms. x × x = x2.x2
- Multiply the outside and inside terms. x × -3 and 4 × x.−3x and 4x
- Multiply the last terms. 4 × -3 = -12.−12
- Collect the like terms. The two x terms add to 1x.x2 + x − 12
- Write the answer. This is the expanded form of the brackets.x2 + x − 12
Check. The result matches the original expression.
Method. Expand (x + 1)(x + 6) and compare.
- Expand the brackets. x × x = x2, x × 6 = 6x, 1 × x = x, 1 × 6 = 6.x2 + 6x + x + 6
- Collect the x terms. 6x + x = 7x.x2 + 7x + 6
- Compare. 7x is not 5x, so the answer is different.7x ≠ 5x
- Write the answer. Ali is not right. (x + 2)(x + 3) is the correct factorisation.No
Check. The numbers 1 and 6 add to 7, not 5.
Method. Find two numbers that multiply to make c and add to make b. Then write them in two brackets with x.
- Write what we need. Two numbers that multiply to -9 and add to 0.multiply to -9, add to 0
- Find the two numbers. -3 and 3 work: -3 × 3 = -9 and -3 + 3 = 0.-3 × 3 = -9, -3 + 3 = 0
- Write the brackets. Each number goes in its own bracket with x.(x − 3)(x + 3)
- Write the answer. (x − 3)(x + 3)
Check. Expanding (x − 3)(x + 3) gives x2 + 0x − 9, which is the original expression.
Watch out. Watch the signs. The two numbers can be negative. Check both the sum and the product.
Method. Find two numbers that multiply to make c and add to make b. Then write them in two brackets with x.
- Write what we need. Two numbers that multiply to -15 and add to -2.multiply to -15, add to -2
- Find the two numbers. -5 and 3 work: -5 × 3 = -15 and -5 + 3 = -2.-5 × 3 = -15, -5 + 3 = -2
- Write the brackets. Each number goes in its own bracket with x.(x − 5)(x + 3)
- Write the answer. (x − 5)(x + 3)
Check. Expanding (x − 5)(x + 3) gives x2 − 2x − 15, which is the original expression.
Watch out. Watch the signs. The two numbers can be negative. Check both the sum and the product.
Method. Find two numbers that multiply to make c and add to make b. Then write them in two brackets with x.
- Write what we need. Two numbers that multiply to 12 and add to -7.multiply to 12, add to -7
- Find the two numbers. -4 and -3 work: -4 × -3 = 12 and -4 + -3 = -7.-4 × -3 = 12, -4 + -3 = -7
- Write the brackets. Each number goes in its own bracket with x.(x − 4)(x − 3)
- Write the answer. (x − 4)(x − 3)
Check. Expanding (x − 4)(x − 3) gives x2 − 7x + 12, which is the original expression.
Watch out. Watch the signs. The two numbers can be negative. Check both the sum and the product.
Method. Find two numbers that multiply to make c and add to make b. Then write them in two brackets with x.
- Write what we need. Two numbers that multiply to 9 and add to 6.multiply to 9, add to 6
- Find the two numbers. 3 and 3 work: 3 × 3 = 9 and 3 + 3 = 6.3 × 3 = 9, 3 + 3 = 6
- Write the brackets. Each number goes in its own bracket with x.(x + 3)(x + 3)
- Write the answer. (x + 3)(x + 3)
Check. Expanding (x + 3)(x + 3) gives x2 + 6x + 9, which is the original expression.
Watch out. Watch the signs. The two numbers can be negative. Check both the sum and the product.
Chapter summary
| Skill | Method |
|---|---|
| Collect like terms | Add or subtract the numbers in front of terms with the same letters |
| Expand a single bracket | Multiply every term inside by the term outside: 3(x + 4) = 3x + 12 |
| Expand double brackets | Multiply each term in one bracket by each term in the other, then collect |
| Factorise by a common factor | Take out the highest common factor: 6x + 9 = 3(2x + 3) |
| Factorise x2 + bx + c | Find p and q with pq = c and p + q = b, then write (x + p)(x + q) |
| Check | Expand your factorised answer to see you get the original expression back |
Can you…?
- I can collect like terms and simplify an expression
- I can expand a single bracket, including with a negative number outside
- I can expand two brackets, and collect the like terms
- I can take out the highest common factor of an expression
- I can factorise x2 + bx + c into two brackets
- I can check a factorisation by expanding it
Key terms
- term
- a part of an expression, such as 3a, −2 or 5xy
- like terms
- terms with exactly the same letters and powers
- expression
- a collection of terms with numbers, letters and operations, with no equals sign
- expand
- multiply out the brackets
- factorise
- write an expression as a product, using brackets
- highest common factor
- the biggest number and highest letter power that divides every term
- quadratic
- an expression with x2 as its highest power
Exercise 5Z Check-up
Non-calculatorMethod. Collect the like terms: add or subtract the numbers in front of the same letters.
- Collect the p terms. Terms with the same letters are like terms, so we can add their numbers.4p − 2p = 2p
- Collect the q terms. Terms with the same letters are like terms, so we can add their numbers.3q + q = 4q
- Write the answer. Put the collected terms together.2p + 4q
Check. Substitute x = 2 (or a = 2) into the question and into the answer. Both give the same number.
Watch out. Only like terms can be collected. a and a2 are not like terms, and 3a + 2b cannot be simplified further.
Method. Multiply the numbers together, then multiply the letters. Add the powers of the same letter.
- Multiply the numbers. Multiply the numbers in front of the letters.4 × 3 = 12
- Multiply the letters. Write each letter once. If a letter appears twice, add the powers.a × b = ab
- Write the answer. 12ab
Check. Check the numbers: 4 × 3 = 12, and the letters are the same letters as in the answer.
Watch out. Multiply the numbers and the letters separately. 3x × 2y is 6xy, not 5xy.
Method. Multiply every term inside the bracket by the term outside it.
- Look at what is outside the bracket. Everything inside the bracket is multiplied by this.5(x − 2)
- Multiply each term inside by the outside term. Keep the signs. A minus times a minus is a plus.5 × x = 5x and 5 × −2 = −10
- Write the answer. Put the two products together.5x − 10
Check. Try x = 2: the bracket gives 0, and the outside gives 5, so the product is 0. The answer gives the same value at x = 2.
Watch out. Be careful with negative numbers: −2 × (−3) = +6, and −2 × (+3) = −6.
Method. Multiply every term inside the bracket by the term outside it.
- Look at what is outside the bracket. Everything inside the bracket is multiplied by this.−4(3x + 1)
- Multiply each term inside by the outside term. Keep the signs. A minus times a minus is a plus.−4 × 3x = −12x and −4 × 1 = −4
- Write the answer. Put the two products together.−12x − 4
Check. Try x = 2: the bracket gives 7, and the outside gives -4, so the product is -28. The answer gives the same value at x = 2.
Watch out. Be careful with negative numbers: −2 × (−3) = +6, and −2 × (+3) = −6.
Method. Multiply every term inside the bracket by the term outside it.
- Look at what is outside the bracket. Everything inside the bracket is multiplied by this.x(2x + 5)
- Multiply each term inside by the outside term. Keep the signs. A minus times a minus is a plus.x × 2x = 2x2 and x × 5 = 5x
- Write the answer. Put the two products together.2x2 + 5x
Check. Try x = 2: the bracket gives 9, and the outside gives 2, so the product is 18. The answer gives the same value at x = 2.
Watch out. Be careful with negative numbers: −2 × (−3) = +6, and −2 × (+3) = −6.
Method. Multiply each term in the first bracket by each term in the second bracket, then collect like terms.
- Multiply the first terms. First term in each bracket.x × x = x2
- Multiply the outside terms. The first term of the first bracket with the second term of the second bracket.x × 7 = 7x
- Multiply the inside terms. The second term of the first bracket with the first term of the second bracket.2 × x = 2x
- Multiply the last terms. The second term of each bracket.2 × 7 = 14
- Collect the like terms. The two x terms are like terms, so add them.x2 + 9x + 14
- Write the answer. x2 + 9x + 14
Check. Try x = 1: the brackets give 24, and the answer gives 24.
Watch out. The x terms come from the outside and inside products. Do not forget the signs.
Method. Multiply each term in the first bracket by each term in the second bracket, then collect like terms.
- Multiply the first terms. First term in each bracket.2x × x = 2x2
- Multiply the outside terms. The first term of the first bracket with the second term of the second bracket.2x × −4 = −8x
- Multiply the inside terms. The second term of the first bracket with the first term of the second bracket.3 × x = 3x
- Multiply the last terms. The second term of each bracket.3 × −4 = −12
- Collect the like terms. The two x terms are like terms, so add them.2x2 − 5x − 12
- Write the answer. 2x2 − 5x − 12
Check. Try x = 1: the brackets give -15, and the answer gives -15.
Watch out. The x terms come from the outside and inside products. Do not forget the signs.
Method. Find the highest common factor of the numbers and any letters that every term has. Take it outside the brackets.
- Find the common factor of the numbers. Find the biggest number that divides into every number.HCF of the numbers = 5
- Find any letters that every term has. The lowest power of each letter is in every term.common letter part = none
- Divide each term by the common factor. Write the results inside the brackets.10x + 15 ÷ (5) = 2x + 3
- Write the answer. Check by expanding the brackets.5(2x + 3)
Check. Expanding the brackets gives the original expression back.
Watch out. Take out the highest common factor, not just any factor. Check that every term has been divided.
Method. Find the highest common factor of the numbers and any letters that every term has. Take it outside the brackets.
- Find the common factor of the numbers. Find the biggest number that divides into every number.HCF of the numbers = 3
- Find any letters that every term has. The lowest power of each letter is in every term.common letter part = x
- Divide each term by the common factor. Write the results inside the brackets.9x2 − 6x ÷ (3x) = 3x − 2
- Write the answer. Check by expanding the brackets.3x(3x − 2)
Check. Expanding the brackets gives the original expression back.
Watch out. Take out the highest common factor, not just any factor. Check that every term has been divided.
Method. Find two numbers that multiply to make c and add to make b. Then write them in two brackets with x.
- Write what we need. Two numbers that multiply to 12 and add to 7.multiply to 12, add to 7
- Find the two numbers. 3 and 4 work: 3 × 4 = 12 and 3 + 4 = 7.3 × 4 = 12, 3 + 4 = 7
- Write the brackets. Each number goes in its own bracket with x.(x + 3)(x + 4)
- Write the answer. (x + 3)(x + 4)
Check. Expanding (x + 3)(x + 4) gives x2 + 7x + 12, which is the original expression.
Watch out. Watch the signs. The two numbers can be negative. Check both the sum and the product.
Total 28 marks. Allow about 28 minutes.
Exercise 5X Exam-style questions
Non-calculatorMethod. Collect the like terms: add or subtract the numbers in front of the same letters.
- Collect the x terms. Terms with the same letters are like terms, so we can add their numbers.3x − x = 2x
- Collect the y terms. Terms with the same letters are like terms, so we can add their numbers.5y + 2y = 7y
- Write the answer. Put the collected terms together.2x + 7y
Check. Substitute x = 2 (or a = 2) into the question and into the answer. Both give the same number.
Watch out. Only like terms can be collected. a and a2 are not like terms, and 3a + 2b cannot be simplified further.
Method. Multiply every term inside the bracket by the term outside it.
- Look at what is outside the bracket. Everything inside the bracket is multiplied by this.4(2x − 3)
- Multiply each term inside by the outside term. Keep the signs. A minus times a minus is a plus.4 × 2x = 8x and 4 × −3 = −12
- Write the answer. Put the two products together.8x − 12
Check. Try x = 2: the bracket gives 1, and the outside gives 4, so the product is 4. The answer gives the same value at x = 2.
Watch out. Be careful with negative numbers: −2 × (−3) = +6, and −2 × (+3) = −6.
Method. Multiply each term in the first bracket by each term in the second bracket, then collect like terms.
- Multiply the first terms. First term in each bracket.x × x = x2
- Multiply the outside terms. The first term of the first bracket with the second term of the second bracket.x × 5 = 5x
- Multiply the inside terms. The second term of the first bracket with the first term of the second bracket.4 × x = 4x
- Multiply the last terms. The second term of each bracket.4 × 5 = 20
- Collect the like terms. The two x terms are like terms, so add them.x2 + 9x + 20
- Write the answer. x2 + 9x + 20
Check. Try x = 1: the brackets give 30, and the answer gives 30.
Watch out. The x terms come from the outside and inside products. Do not forget the signs.
Method. Expand each bracket first, then collect the like terms.
- Expand the first bracket. Multiply every term inside the bracket by the number outside.3(x + 2) = 3x + 6
- Expand the second bracket. Do the same for the second bracket.4(x − 1) = 4x − 4
- Collect the like terms. The x terms are like terms, and so are the numbers.3x + 6 + 4x − 4
- Write the answer. 7x + 2
Check. Try x = 1: the original expression and the answer both give the same number.
Method. Find the highest common factor of the numbers and any letters that every term has. Take it outside the brackets.
- Find the common factor of the numbers. Find the biggest number that divides into every number.HCF of the numbers = 5
- Find any letters that every term has. The lowest power of each letter is in every term.common letter part = x
- Divide each term by the common factor. Write the results inside the brackets.15x2 − 10x ÷ (5x) = 3x − 2
- Write the answer. Check by expanding the brackets.5x(3x − 2)
Check. Expanding the brackets gives the original expression back.
Watch out. Take out the highest common factor, not just any factor. Check that every term has been divided.
Method. Find two numbers that multiply to make c and add to make b. Then write them in two brackets with x.
- Write what we need. Two numbers that multiply to 15 and add to 8.multiply to 15, add to 8
- Find the two numbers. 3 and 5 work: 3 × 5 = 15 and 3 + 5 = 8.3 × 5 = 15, 3 + 5 = 8
- Write the brackets. Each number goes in its own bracket with x.(x + 3)(x + 5)
- Write the answer. (x + 3)(x + 5)
Check. Expanding (x + 3)(x + 5) gives x2 + 8x + 15, which is the original expression.
Watch out. Watch the signs. The two numbers can be negative. Check both the sum and the product.
Method. Find two numbers that multiply to make c and add to make b. Then write them in two brackets with x.
- Write what we need. Two numbers that multiply to -12 and add to -1.multiply to -12, add to -1
- Find the two numbers. -4 and 3 work: -4 × 3 = -12 and -4 + 3 = -1.-4 × 3 = -12, -4 + 3 = -1
- Write the brackets. Each number goes in its own bracket with x.(x − 4)(x + 3)
- Write the answer. (x − 4)(x + 3)
Check. Expanding (x − 4)(x + 3) gives x2 − 1x − 12, which is the original expression.
Watch out. Watch the signs. The two numbers can be negative. Check both the sum and the product.
Method. Multiply each term in the first bracket by each term in the second bracket, then collect like terms.
- Multiply the first terms. First term in each bracket.x × x = x2
- Multiply the outside terms. The first term of the first bracket with the second term of the second bracket.x × 2 = 2x
- Multiply the inside terms. The second term of the first bracket with the first term of the second bracket.6 × x = 6x
- Multiply the last terms. The second term of each bracket.6 × 2 = 12
- Collect the like terms. The two x terms are like terms, so add them.x2 + 8x + 12
- Write the answer. x2 + 8x + 12
Check. Try x = 1: the brackets give 21, and the answer gives 21.
Watch out. The x terms come from the outside and inside products. Do not forget the signs.
Method. Multiply every term inside the bracket by the term outside it.
- Look at what is outside the bracket. Everything inside the bracket is multiplied by this.2(x + 5)
- Multiply each term inside by the outside term. Keep the signs. A minus times a minus is a plus.2 × x = 2x and 2 × 5 = 10
- Write the answer. Put the two products together.2x + 10
Check. Try x = 2: the bracket gives 7, and the outside gives 2, so the product is 14. The answer gives the same value at x = 2.
Watch out. Be careful with negative numbers: −2 × (−3) = +6, and −2 × (+3) = −6.
Method. Multiply each term in the first bracket by each term in the second bracket, then collect like terms.
- Multiply the first terms. First term in each bracket.x × x = x2
- Multiply the outside terms. The first term of the first bracket with the second term of the second bracket.x × −2 = −2x
- Multiply the inside terms. The second term of the first bracket with the first term of the second bracket.4 × x = 4x
- Multiply the last terms. The second term of each bracket.4 × −2 = −8
- Collect the like terms. The two x terms are like terms, so add them.x2 + 2x − 8
- Write the answer. x2 + 2x − 8
Check. Try x = 1: the brackets give -5, and the answer gives -5.
Watch out. The x terms come from the outside and inside products. Do not forget the signs.
Method. Expand the brackets first, then subtract 2x2.
- Expand the brackets. The four products are 2x2, 6x, x and 3.2x2 + 7x + 3
- Subtract 2x2. The x2 terms cancel.2x2 + 7x + 3 − 2x2 = 7x + 3
- Write the answer. 7x + 3
Check. At x = 1: (3)(4) − 2 = 10, and 7 + 3 = 10.