More expanding and factorising - Teacher Slides.pptx

EDEXCEL GCSE MATHS · HIGHER TIER

More expanding and factorising

Algebra

Lesson 7 of 7

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EDEXCEL GCSE MATHS · HIGHER TIER

From Earlier Lessons

Algebra

Lesson 7 of 7

Answer each one, then check.

1

Expand and simplify (x + 3)(x + 5).

2

Factorise x² + 5x + 6.

3

Factorise x² − 25.

4

Simplify 6x² ÷ 2x.

5

Work out 7² − 3².

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EDEXCEL GCSE MATHS · HIGHER TIER

From Earlier Lessons - Answers

Algebra

Lesson 7 of 7

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

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EDEXCEL GCSE MATHS · HIGHER TIER

Learning Objectives

Algebra

Lesson 7 of 7

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.

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Expanding Three Brackets

Algebra

Lesson 7 of 7

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

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Three Brackets with Negatives

Algebra

Lesson 7 of 7

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)

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The Difference of Two Squares, Again

Algebra

Lesson 7 of 7

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).

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Why the Difference of Two Squares Works

Algebra

Lesson 7 of 7

Cut the L-shape and rearrange it: a² − b² = (a + b)(a − b).

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.

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PART ONE

Factorising ax² + bx + c

When the squared term has a number in front.

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Splitting the Middle Term

Algebra

Lesson 7 of 7

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.

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Factorising a Quadratic with a Coefficient

Algebra

Lesson 7 of 7

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)

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Simplifying Algebraic Fractions

Algebra

Lesson 7 of 7

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.

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Simplifying an Algebraic Fraction

Algebra

Lesson 7 of 7

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)

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Algebraic Proof

Algebra

Lesson 7 of 7

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.

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Match the Quadratic to Its Factors

Algebra

Lesson 7 of 7

Expression

Factorised

2x² + 7x + 3

A

(3x − 1)(2x + 1)

1

3x² − 5x − 2

B

x(x + 2)(x − 2)

2

4x² − 25

C

(2x + 1)(x + 3)

3

2x² − 8

D

(3x + 1)(x − 2)

4

6x² + x − 1

E

(2x + 5)(2x − 5)

5

x³ − 4x

F

2(x + 2)(x − 2)

6

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Key Terms

Algebra

Lesson 7 of 7

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.

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Algebra

Lesson 7 of 7

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.

WHAT 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)

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Can I...?

Algebra

Lesson 7 of 7

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.

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Summary & Exam Focus

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.

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Exam Practice: More Expanding and Factorising

Algebra

Lesson 7 of 7

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

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Exam Practice: More Expanding and Factorising (cont.)

Algebra

Lesson 7 of 7

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…

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Question 1 · 2 marks · Non-calculator

Algebra

Lesson 7 of 7

“

Factorise 3x² + 10x + 8

HOW TO ANSWER IT

Command word: Non-calculator. Worth 2 marks, so plan before writing.

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Question 1 · mark scheme

Algebra

Lesson 7 of 7

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

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Question 2 · 3 marks · Non-calculator

Algebra

Lesson 7 of 7

“

Expand and simplify (x + 4)(x − 1)(x + 2)

HOW TO ANSWER IT

Command word: Non-calculator. Worth 3 marks, so plan before writing.

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Question 2 · mark scheme

Algebra

Lesson 7 of 7

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

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Question 3 · 2 marks · Non-calculator

Algebra

Lesson 7 of 7

“

Factorise fully 50 − 2x²

HOW TO ANSWER IT

Command word: Non-calculator. Worth 2 marks, so plan before writing.

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Question 3 · mark scheme

Algebra

Lesson 7 of 7

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

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Question 4 · 3 marks · Non-calculator

Algebra

Lesson 7 of 7

“

Simplify fully (x² + 3x − 10)/(x² − 4)

HOW TO ANSWER IT

Command word: Non-calculator. Worth 3 marks, so plan before writing.

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Question 4 · mark scheme

Algebra

Lesson 7 of 7

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

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Question 5 · 3 marks · Non-calculator · Prove

Algebra

Lesson 7 of 7

“

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.

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Question 5 · mark scheme

Algebra

Lesson 7 of 7

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

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Question 6 · 3 marks · Non-calculator · Show that

Algebra

Lesson 7 of 7

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

Algebra

Lesson 7 of 7

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

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