Expanding and factorising - Teacher Slides.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Expanding and factorising

Algebra

Lesson 2 of 7

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Last Lesson and Before

Algebra

Lesson 2 of 7

Answer each one, then check.

1

Simplify 3a + 5a − 2a.

2

Work out −3 × −4.

3

What is the HCF of 12 and 18?

4

Last lesson: simplify x × x.

5

Last lesson: simplify 2x × 5x².

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Last Lesson and Before - Answers

Algebra

Lesson 2 of 7

1

Simplify 3a + 5a − 2a.

6a

2

Work out −3 × −4.

12

3

What is the HCF of 12 and 18?

6

4

Last lesson: simplify x × x.

x²

5

Last lesson: simplify 2x × 5x².

10x³

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Algebra

Lesson 2 of 7

1

Expand single brackets and simplify.

2

Factorise expressions by taking out the highest common factor.

3

Expand double brackets.

4

Factorise quadratic expressions of the form x² + bx + c.

5

Factorise the difference of two squares, x² − a².

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

The Words You Need

Algebra

Lesson 2 of 7

Term

One part of an expression: 3x, −5 and x² are terms.

Expression

Terms added or subtracted, with no equals sign: 3x² − 5x + 2.

Coefficient

The number in front of a letter: the 3 in 3x².

Like terms

Same letters and powers: 4x and −7x, but not 4x and 4x².

Expand

Multiply out the brackets.

Factorise

Put back into brackets - the opposite of expanding.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Expanding Single Brackets

Algebra

Lesson 2 of 7

Multiply every term inside the bracket by the term outside.

Numbers

4(3x − 7) = 12x − 28.

Letters

x(x + 5) = x² + 5x.

Negative outside

−3(x − 4) = −3x + 12: a negative times a negative is positive.

Expand and simplify

Expand every bracket, then collect like terms.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Expand and Simplify

Algebra

Lesson 2 of 7

Expand and simplify 5(2x + 3) − 3(x − 4)

1

Expand the first bracket

5(2x + 3) = 10x + 15

2

Expand the second: the −3 multiplies both terms

−3(x − 4) = −3x + 12

3

Write it all out

10x + 15 − 3x + 12

4

Collect like terms

10x − 3x = 7x and 15 + 12 = 27

ANSWER

7x + 27

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Factorising Fully

Algebra

Lesson 2 of 7

Factorise fully 12x²y − 18xy²

1

Find the HCF of the numbers

HCF of 12 and 18 = 6

2

Find the HCF of the letters

Both terms have x and y: xy

3

The HCF goes outside

6xy(…)

4

Divide each term by the HCF

12x²y ÷ 6xy = 2x and 18xy² ÷ 6xy = 3y

ANSWER

6xy(2x − 3y)

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

PART ONE

Double Brackets

Every term in the first bracket multiplies every term in the second.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

The Grid Method for (x + 4)(x - 3)

Algebra

Lesson 2 of 7

Multiply each row by each column, then collect like terms: x² − 3x + 4x − 12 = x² + x − 12.

×

x

−3

x

x²

−3x

+4

+4x

−12

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Squaring a Bracket

Algebra

Lesson 2 of 7

Expand and simplify (2x − 5)²

1

Write it as two brackets

(2x − 5)(2x − 5)

2

First times first

2x × 2x = 4x²

3

Outer and inner

2x × (−5) = −10x and (−5) × 2x = −10x

4

Last times last

(−5) × (−5) = +25

5

Collect like terms

4x² − 20x + 25

ANSWER

4x² − 20x + 25 (not 4x² + 25)

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

PART TWO

Factorising Quadratics

Double brackets in reverse.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Factorising x² + bx + c

Algebra

Lesson 2 of 7

Find two numbers that MULTIPLY to give c and ADD to give b.

The brackets

x² + bx + c = (x + p)(x + q), where pq = c and p + q = b.

c positive

Both numbers have the same sign as b: x² − 7x + 12 = (x − 3)(x − 4).

c negative

The numbers have opposite signs: x² + x − 12 = (x + 4)(x − 3).

Check

Expand your answer: it should give the expression you started with.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Factorising a Quadratic

Algebra

Lesson 2 of 7

Factorise x² − 2x − 15

1

List pairs that multiply to −15

1 and −15, −1 and 15, 3 and −5, −3 and 5

2

Pick the pair that adds to −2

3 + (−5) = −2

3

Write the brackets

(x + 3)(x − 5)

4

Check by expanding

x² − 5x + 3x − 15 = x² − 2x − 15

ANSWER

(x + 3)(x − 5)

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

The Difference of Two Squares

Algebra

Lesson 2 of 7

One square number (or square term) minus another factorises straight away.

The rule

x² − a² = (x + a)(x − a).

Examples

x² − 49 = (x + 7)(x − 7). x² − 1 = (x + 1)(x − 1).

Why it works

(x + 7)(x − 7) = x² − 7x + 7x − 49: the middle terms cancel.

Spot it

Two terms, both square, with a MINUS between them. x² + 49 does not factorise.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Match the Expression to Its Factors

Algebra

Lesson 2 of 7

Expression

Factorised

x² + 5x + 6

A

4(x + 3)

1

x² − 9

B

x(x + 3)

2

x² + x − 6

C

(x + 2)(x + 3)

3

x² − 5x + 6

D

(x + 3)(x − 3)

4

4x + 12

E

(x + 3)(x − 2)

5

x² + 3x

F

(x − 2)(x − 3)

6

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Algebra

Lesson 2 of 7

Expand

Multiply out brackets.

Factorise

Write an expression as a product of factors, using brackets.

Factorise fully

Take out the highest common factor, so no common factor is left inside.

Quadratic expression

An expression whose highest power is x², such as x² + 5x + 6.

Difference of two squares

An expression of the form x² − a², which factorises to (x + a)(x − a).

Identity

Two expressions that are equal for every value of x, written with ≡.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Algebra

Lesson 2 of 7

YOUR TASK

Expand, Then Undo

12 minutes

Work in pairs. Partner A expands (a) (x + 5)(x + 2) (b) (x − 4)(x + 6) (c) (x − 3)². Partner B factorises (d) x² + 9x + 14 (e) x² − 4x − 21 (f) x² − 100. Then swap papers and check each other's answers by doing the opposite process.

1

Partner A: expand.

2

Partner B: factorise.

3

Swap and check by reversing.

WHAT A GOOD ANSWER SHOWS

(a) x² + 7x + 10 (b) x² + 2x − 24 (c) x² − 6x + 9 (d) (x + 2)(x + 7) (e) (x − 7)(x + 3) (f) (x + 10)(x − 10)

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Algebra

Lesson 2 of 7

Expand a single bracket.

Expand and simplify two single brackets.

Factorise by taking out the HCF.

Factorise fully.

Expand double brackets.

Square a bracket.

Factorise x² + bx + c.

Factorise the difference of two squares.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Expand: multiply every term inside by every term outside.

Factorise: take out the HCF of the numbers AND the letters.

Double brackets: four multiplications, then collect like terms.

x² + bx + c: find two numbers that multiply to c and add to b.

x² − a² = (x + a)(x − a).

EXAM FOCUS

Factorise x² + 9x + 20 (2 marks)

Always check a factorisation by expanding it again. It takes ten seconds and catches sign errors that would cost the mark.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Expanding and Factorising

Algebra

Lesson 2 of 7

Answer all questions. Show your working. All questions are non-calculator. · 25 minutes

Question 1 · 1 mark · Non-calculator

Expand 4(3x − 7)

Question 2 · 2 marks · Non-calculator

Expand and simplify 3(2y + 5) − 2(y − 4)

Question 3 · 2 marks · Non-calculator

Factorise fully 15y² + 10y

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Expanding and Factorising (cont.)

Algebra

Lesson 2 of 7

Question 4 · 2 marks · Non-calculator

Expand and simplify (x + 6)(x − 2)

Question 5 · 2 marks · Non-calculator

Factorise x² + 9x + 20

Question 6 · 1 mark · Non-calculator

Factorise x² − 64

Question 7 · 3 marks · Non-calculator · Show that

The diagram shows a rectangle. The length is (x + 5) cm and the width is (x − 2) cm. Show that the area of the rectangle, in cm², is x² +…

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · 1 mark · Non-calculator

Algebra

Lesson 2 of 7

“

Expand 4(3x − 7)

HOW TO ANSWER IT

Command word: Non-calculator. Worth 1 mark, so plan before writing.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · mark scheme

Algebra

Lesson 2 of 7

1 mark available. Award a mark for each point made.

12x − 28

B1

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · 2 marks · Non-calculator

Algebra

Lesson 2 of 7

“

Expand and simplify 3(2y + 5) − 2(y − 4)

HOW TO ANSWER IT

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

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · mark scheme

Algebra

Lesson 2 of 7

2 marks available. Award a mark for each point made.

6y + 15 or −2y + 8 correct

M1

4y + 23

A1

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · 2 marks · Non-calculator

Algebra

Lesson 2 of 7

“

Factorise fully 15y² + 10y

HOW TO ANSWER IT

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

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · mark scheme

Algebra

Lesson 2 of 7

2 marks available. Award a mark for each point made.

5y(3y + 2)

B2

A correct partial factorisation, e.g. 5(3y² + 2y) or y(15y + 10)

B1

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · 2 marks · Non-calculator

Algebra

Lesson 2 of 7

“

Expand and simplify (x + 6)(x − 2)

HOW TO ANSWER IT

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

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · mark scheme

Algebra

Lesson 2 of 7

2 marks available. Award a mark for each point made.

At least 3 of the 4 terms correct: x², −2x, +6x, −12

M1

x² + 4x − 12

A1

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · 2 marks · Non-calculator

Algebra

Lesson 2 of 7

“

Factorise x² + 9x + 20

HOW TO ANSWER IT

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

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · mark scheme

Algebra

Lesson 2 of 7

2 marks available. Award a mark for each point made.

(x + 4)(x + 5)

B2

(x ± 4)(x ± 5)

B1

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · 1 mark · Non-calculator

Algebra

Lesson 2 of 7

“

Factorise x² − 64

HOW TO ANSWER IT

Command word: Non-calculator. Worth 1 mark, so plan before writing.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · mark scheme

Algebra

Lesson 2 of 7

1 mark available. Award a mark for each point made.

(x + 8)(x − 8)

B1

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 7 · 3 marks · Non-calculator · Show that

Algebra

Lesson 2 of 7

The diagram shows a rectangle. The length is (x + 5) cm and the width is (x − 2) cm. Show that the area of the rectangle, in cm², is x² + 3x − 10. (3 marks)

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 7 · mark scheme

Algebra

Lesson 2 of 7

3 marks available. Award a mark for each point made.

Area = (x + 5)(x − 2)

M1

At least 3 of the 4 terms correct

M1

Fully correct expansion reaching x² + 3x − 10

A1

openrevise.com