Algebraic Expressions

Spec reference: Algebra - Notation, vocabulary and manipulation
Key idea: Write, simplify and manipulate algebraic expressions.

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Algebraic notation

In algebra, letters represent unknown values or variables.

Expression	Meaning
$3x$	$3\times x$
$\frac{a}{b}$	$a÷b$
$x^2$	$x\times x$
$ab$	$a\times b$
$2(x+3)$	$2\times (x+3)$

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Collecting like terms

Like terms have the same letter(s) and power(s). Only like terms can be added or subtracted.

Example

Simplify $5x+3y-2x+7y$

Group like terms: $(5x-2x)+(3y+7y)=\boxed{{\displaystyle 3x+10y}}$

Example

Simplify $4x^2+3x-x^2+5$

$(4x^2-x^2)+3x+5=\boxed{{\displaystyle 3x^2+3x+5}}$

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Expanding brackets

Multiply everything inside the bracket by the term outside.

$$a(b+c)=ab+ac$$

Example

Expand $4(3x-2)$

$4\times 3x=12x$ and $4\times (-2)=-8$

$$\boxed{{\displaystyle 12x-8}}$$

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Expanding two brackets (FOIL)

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

First, Outer, Inner, Last

Example

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

$x\cdot x+x\cdot 5+3\cdot x+3\cdot 5=x^2+5x+3x+15=\boxed{{\displaystyle x^2+8x+15}}$

Example

Expand $(2x-1)(x+4)$

$2x^2+8x-x-4=\boxed{{\displaystyle 2x^2+7x-4}}$

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Factorising

Factorising is the reverse of expanding - find the HCF and write as a product.

Example

Factorise $12x+8$

HCF of 12 and 8 is 4:

$$\boxed{{\displaystyle 4(3x+2)}}$$

Example

Factorise $6x^2-9x$

HCF is $3x$:

$$\boxed{{\displaystyle 3x(2x-3)}}$$

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Factorising quadratics

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

Example

Factorise $x^2+7x+12$

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

$$\boxed{{\displaystyle (x+3)(x+4)}}$$

Example

Factorise $x^2-x-6$

Need: multiply to $-6$, add to $-1$ → $-3$ and $2$

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

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Difference of two squares

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

Example

Factorise $x^2-25$

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

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Exam tips

Watch out for

• $3x^2$ and $3x$ are NOT like terms
• When expanding, be careful with negative signs: $-(x-3)=-x+3$
• Always check your factorisation by expanding back

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Test Yourself

Question 1 of 5

Simplify 5x + 3y - 2x + 7y