EDEXCEL GCSE MATHS · HIGHER TIER
More expanding and factorising
Algebra · Lesson 7 of 7
Teacher copy - includes the notes for whoever is teaching from it.
From Earlier Lessons
Answer each one, then check.
1. Expand and simplify (x + 3)(x + 5).
x² + 8x + 15
2. Factorise x² + 5x + 6.
(x + 2)(x + 3)
3. Factorise x² − 25.
(x + 5)(x − 5)
4. Simplify 6x² ÷ 2x.
3x
5. Work out 7² − 3².
49 − 9 = 40
Learning Objectives
1. Expand the product of three binomials.
2. Factorise quadratics of the form ax² + bx + c.
3. Factorise the difference of two squares with coefficients, such as 9x² − 16y².
4. Simplify algebraic fractions by factorising.
5. Use expanding and factorising in algebraic proof.
Expanding Three Brackets
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Expand and simplify (x + 1)(x + 2)(x + 3) |
1. Expand the first two brackets
(x + 1)(x + 2) = x² + 3x + 2
2. Multiply every term by x
x(x² + 3x + 2) = x³ + 3x² + 2x
3. Multiply every term by +3
3(x² + 3x + 2) = 3x² + 9x + 6
4. Collect like terms
x³ + 6x² + 11x + 6
Answer: x³ + 6x² + 11x + 6
Three Brackets with Negatives
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Expand and simplify (x − 2)(x + 3)(2x − 1) |
1. Expand the first two
(x − 2)(x + 3) = x² + x − 6
2. Multiply by 2x
2x³ + 2x² − 12x
3. Multiply by −1
−x² − x + 6
4. Collect like terms
2x³ + x² − 13x + 6
Answer: 2x³ + x² − 13x + 6 (check x = 1: (−1)(4)(1) = −4 and 2 + 1 − 13 + 6 = −4)
The Difference of Two Squares, Again
a² − b² = (a + b)(a − b) works for any two squares, not just x² and a number.
▸ With coefficients. 9x² − 16y² = (3x)² − (4y)² = (3x + 4y)(3x − 4y).
▸ Take out a factor first. 2x² − 18 = 2(x² − 9) = 2(x + 3)(x − 3).
▸ With numbers. 101² − 99² = (101 + 99)(101 − 99) = 200 × 2 = 400 - no squaring needed.
▸ Watch for it. 50 − 2x² = 2(25 − x²) = 2(5 + x)(5 − x).
Why the Difference of Two Squares Works
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Take a square of side a and cut a square of side b from its corner. What is left has area a² − b². Slice it in two and rearrange the pieces, and they make a rectangle a + b long and a − b wide. Same area, two ways of writing it. |
Cut the L-shape and rearrange it: a² − b² = (a + b)(a − b). |
PART ONE
Factorising ax² + bx + c
When the squared term has a number in front.
Splitting the Middle Term
A reliable method for any quadratic ax² + bx + c.
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1 Multiply Work out a × c. |
2 Find the pair Two numbers that multiply to ac and add to b. |
3 Split Write bx as the sum of those two x terms. |
4 Pairs Factorise the first two terms and the last two terms. |
5 Common bracket Take out the bracket they share. |
Factorising a Quadratic with a Coefficient
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Factorise 6x² + 11x − 10 |
1. a × c
6 × (−10) = −60
2. Two numbers that multiply to −60 and add to 11
15 and −4
3. Split the middle term
6x² + 15x − 4x − 10
4. Factorise in pairs
3x(2x + 5) − 2(2x + 5)
5. Take out the common bracket
(2x + 5)(3x − 2)
Answer: (2x + 5)(3x − 2) (check: 6x² − 4x + 15x − 10)
Simplifying Algebraic Fractions
Factorise the top and the bottom, then cancel any bracket they share.
▸ Cancel brackets, not terms. In (x + 3)/(x + 5), nothing cancels: the 3 and the 5 are not factors.
▸ Factorise everything. Take out common factors, use the difference of two squares, factorise quadratics.
▸ Then cancel. A bracket on the top and the bottom divides to 1.
Simplifying an Algebraic Fraction
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Simplify fully (x² − 9)/(2x² + 5x − 3) |
1. Top: difference of two squares
x² − 9 = (x + 3)(x − 3)
2. Bottom: ac = −6; 6 and −1 multiply to −6 and add to 5
2x² + 6x − x − 3 = (x + 3)(2x − 1)
3. Write it factorised
(x + 3)(x − 3)/((x + 3)(2x − 1))
4. Cancel (x + 3)
(x − 3)/(2x − 1)
Answer: (x − 3)/(2x − 1)
Algebraic Proof
Algebra can prove something is true for every number at once.
▸ Even and odd. An even number is 2n; an odd number is 2n + 1, where n is an integer.
▸ Consecutive numbers. n, n + 1, n + 2, ...
▸ The method. Write the expression, expand and simplify, then factorise to show the property.
▸ Example. (n + 3)² − (n + 1)² = n² + 6n + 9 − n² − 2n − 1 = 4n + 8 = 4(n + 2), which is always a multiple of 4.
Match the Quadratic to Its Factors
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Expression |
Factorised |
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2x² + 7x + 3 (2x + 1)(x + 3) |
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3x² − 5x − 2 (3x + 1)(x − 2) |
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4x² − 25 (2x + 5)(2x − 5) |
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2x² − 8 2(x + 2)(x − 2) |
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6x² + x − 1 (3x − 1)(2x + 1) |
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x³ − 4x x(x + 2)(x − 2) |
Key Terms
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Binomial An expression with two terms, such as (x + 3). |
Product of three binomials Three brackets multiplied, such as (x + 1)(x + 2)(x + 3). |
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Splitting the middle term Writing bx as two terms so that ax² + bx + c can be factorised in pairs. |
Algebraic fraction A fraction with algebra on the top, the bottom or both. |
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Proof An argument showing something is true in every case, not just the ones tested. |
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Your Task: Factorise and Cancel
15 minutes
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(a) Factorise 2x² + 9x + 4. (b) Factorise 3x² − 10x + 8. (c) Factorise fully 3x² − 75. (d) Simplify (x² + 3x − 10)/(x² − 4). (e) Expand (x + 2)³. (f) Prove that the sum of three consecutive integers is always a multiple of 3. 1. Look for a common factor first. 2. Then the difference of two squares or splitting the middle term. 3. Check by expanding. |
A good answer shows: (a) (2x + 1)(x + 4) (b) (3x − 4)(x − 2) (c) 3(x + 5)(x − 5) (d) (x + 5)(x − 2)/((x + 2)(x − 2)) = (x + 5)/(x + 2) (e) x³ + 6x² + 12x + 8 (f) n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1)
Note: Part (c): many students stop at 3(x² − 25). Ask "can anything in the bracket be factorised again?"
Can I...?
☐ Expand three brackets.
☐ Factorise ax² + bx + c by splitting the middle term.
☐ Factorise the difference of two squares with coefficients.
☐ Take out a common factor before factorising.
☐ Simplify algebraic fractions.
☐ Use 2n and 2n + 1 for even and odd numbers.
☐ Write an algebraic proof.
☐ Check a factorisation by expanding.
Summary
✓ Three brackets: expand two, then multiply every term by the third.
✓ ax² + bx + c: find two numbers that multiply to ac and add to b, split, factorise in pairs.
✓ a² − b² = (a + b)(a − b), with any squares; take out common factors first.
✓ Algebraic fractions: factorise top and bottom, then cancel brackets.
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EXAM FOCUS Factorise 3x² + 10x + 8 (2 marks) In a proof, finish with a sentence: "4(n + 2) is a multiple of 4, so the expression is always a multiple of 4". The last mark is for saying what you have shown. |
Exam Practice: More Expanding and Factorising
Answer all questions. Show your working. All questions are non-calculator. · 30 minutes
▸ Question 1 · 2 marks · Non-calculator. Factorise 3x² + 10x + 8
▸ Question 2 · 3 marks · Non-calculator. Expand and simplify (x + 4)(x − 1)(x + 2)
▸ Question 3 · 2 marks · Non-calculator. Factorise fully 50 − 2x²
▸ Question 4 · 3 marks · Non-calculator. Simplify fully (x² + 3x − 10)/(x² − 4)
▸ Question 5 · 3 marks · Non-calculator · Prove. Prove that (2n + 1)² − (2n − 1)² is a multiple of 8 for all positive integer values of n.
▸ Question 6 · 3 marks · Non-calculator · Show that. The diagram shows a square of side (2x + 3) cm with a square of side (x + 1) cm cut from one corner. Show that the shaded area, in cm², can…
Question 1 · 2 marks · Non-calculator
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“Factorise 3x² + 10x + 8” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 2 marks, so plan before writing.
Question 1 · mark scheme
2 marks available. Award a mark for each point made.
▸ (3x ± 4)(x ± 2), or a correct split of the middle term. M1
▸ (3x + 4)(x + 2). A1
▸ Model answer. ac = 24; 6 and 4 multiply to 24 and add to 10. 3x² + 6x + 4x + 8 = 3x(x + 2) + 4(x + 2) = (3x + 4)(x + 2)
Question 2 · 3 marks · Non-calculator
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“Expand and simplify (x + 4)(x − 1)(x + 2)” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 3 marks, so plan before writing.
Question 2 · mark scheme
3 marks available. Award a mark for each point made.
▸ Two brackets expanded correctly. M1
▸ At least 6 correct terms from multiplying by the third bracket. M1
▸ x³ + 5x² + 2x − 8. A1
▸ Model answer. (x + 4)(x − 1) = x² + 3x − 4. (x² + 3x − 4)(x + 2) = x³ + 2x² + 3x² + 6x − 4x − 8 = x³ + 5x² + 2x − 8
Question 3 · 2 marks · Non-calculator
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“Factorise fully 50 − 2x²” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 2 marks, so plan before writing.
Question 3 · mark scheme
2 marks available. Award a mark for each point made.
▸ 2(25 − x²), or (10 + 2x)(5 − x). M1
▸ 2(5 + x)(5 − x). A1
▸ Model answer. 2(25 − x²) = 2(5 + x)(5 − x)
Question 4 · 3 marks · Non-calculator
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“Simplify fully (x² + 3x − 10)/(x² − 4)” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 3 marks, so plan before writing.
Question 4 · mark scheme
3 marks available. Award a mark for each point made.
▸ (x + 5)(x − 2). M1
▸ (x + 2)(x − 2). M1
▸ (x + 5)/(x + 2). A1
▸ Model answer. x² + 3x − 10 = (x + 5)(x − 2) and x² − 4 = (x + 2)(x − 2). Cancelling (x − 2): (x + 5)/(x + 2).
Question 5 · 3 marks · Non-calculator · Prove
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“Prove that (2n + 1)² − (2n − 1)² is a multiple of 8 for all positive integer values of n.” |
HOW TO ANSWER IT Command word: Non-calculator · Prove. Worth 3 marks, so plan before writing.
Question 5 · mark scheme
3 marks available. Award a mark for each point made.
▸ One bracket expanded correctly. M1
▸ 8n. A1
▸ A concluding statement: 8n is a multiple of 8. C1
▸ Model answer. (2n + 1)² = 4n² + 4n + 1 and (2n − 1)² = 4n² − 4n + 1. Subtracting: 8n. 8n = 8 × n, so it is always a multiple of 8.
Question 6 · 3 marks · Non-calculator · Show that
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The diagram shows a square of side (2x + 3) cm with a square of side (x + 1) cm cut from one corner. Show that the shaded area, in cm², can be written as (3x + 4)(x + 2). (3 marks) |
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Question 6 · mark scheme
3 marks available. Award a mark for each point made.
▸ (2x + 3)² − (x + 1)². M1
▸ Uses the difference of two squares, or expands both to reach 3x² + 10x + 8. M1
▸ Correctly reaches (3x + 4)(x + 2). A1
▸ Model answer. Shaded area = (2x + 3)² − (x + 1)². This is a difference of two squares: ((2x + 3) + (x + 1))((2x + 3) − (x + 1)) = (3x + 4)(x + 2).