EDEXCEL GCSE MATHS · HIGHER TIER
More expanding and factorising
Algebra · Lesson 7 of 7
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
|
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
|
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
|
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.
|
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
|
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
|
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
|
Expression |
Factorised |
|---|---|
|
2x² + 7x + 3 (2x + 1)(x + 3) |
|
|
3x² − 5x − 2 (3x + 1)(x − 2) |
|
|
4x² − 25 (2x + 5)(2x − 5) |
|
|
2x² − 8 2(x + 2)(x − 2) |
|
|
6x² + x − 1 (3x − 1)(2x + 1) |
|
|
x³ − 4x x(x + 2)(x − 2) |
Key Terms
|
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). |
|
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. |
|
Proof An argument showing something is true in every case, not just the ones tested. |
|
Your Task: Factorise and Cancel
15 minutes
|
(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)
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.
|
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. |