Expanding & Factorising

Spec reference: Algebra - Notation, vocabulary and manipulation
Key idea: Expand brackets and factorise expressions including quadratics.

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Expanding a single bracket

Multiply everything inside the bracket by the term outside.

Example

Expand $3(2x-5)$

$$3\times 2x-3\times 5=\boxed{{\displaystyle 6x-15}}$$

Signs

$-2(3x-4)=-6x+8$ (multiplying a negative by a negative gives a positive).

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Expanding two brackets (FOIL)

First, Outer, Inner, Last

$$(a+b)(c+d)=ac+ad+bc+bd$$

Example

Expand $(x+3)(x-5)$

$$x^2-5x+3x-15=\boxed{{\displaystyle x^2-2x-15}}$$

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Difference of two squares

$$(a+b)(a-b)=a^2-b^2$$

Example

Expand $(x+4)(x-4)=x^2-16$

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Factorising - common factor

Find the HCF of all terms, take it outside the bracket.

Example

Factorise $12x^2-8x$

HCF is $4x$:

$$12x^2-8x=\boxed{{\displaystyle 4x(3x-2)}}$$

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Factorising quadratics

For $x^2+bx+c$: find two numbers that multiply to $c$ and add to $b$.

Example

Factorise $x^2+7x+12$

Need two numbers that multiply to 12 and add to 7: 3 and 4

$$x^2+7x+12=\boxed{{\displaystyle (x+3)(x+4)}}$$

Example

Factorise $x^2-x-6$

Need two numbers that multiply to $-6$ and add to $-1$: -3 and 2

$$x^2-x-6=\boxed{{\displaystyle (x-3)(x+2)}}$$

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Difference of two squares (factorising)

$$a^2-b^2=(a+b)(a-b)$$

Example

Factorise $x^2-25=(x+5)(x-5)$

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Test Yourself

Question 1 of 5

Expand $4(3x-2)$