Algebraic Manipulation Notes

Edexcel GCSE Maths: Revision notes

Key facts

Collect like terms: the same letters to the same powers.
Expand by multiplying every term in one bracket by every term in the other.
Factorise by taking the highest common factor outside brackets, then check by expanding.
The difference of two squares: $x^2 - a^2 = (x + a)(x - a)$ .
Simplify algebraic fractions by factorising, then cancelling factors, never terms.

Collecting like terms

Like terms have the same letters to the same powers, so you can add or subtract their coefficients.

Like terms have the same variable raised to the same power. To simplify, find the like terms, group them, add or subtract the coefficients and write the result.

$2x$ and $2x^2$ are not like terms because the powers differ.

2x + 3x is the same line as 5x, but 2x² is a different curve: only terms with the same power combine.

3x + 5x + 2y

	Like terms	Unlike terms
In $3x + 5x + 2y$	$3x$ and $5x$ make $8x$	$3x$ and $2y$
In $4a^2 + 2a + 3a^2$	$4a^2$ and $3a^2$ make $7a^2$	$a^2$ and $a$
In $6xy + 2y + xy$	$6xy$ and $xy$ make $7xy$	$xy$ and $y$

Like terms

In $3x + 5x + 2y$ :: $3x$ and $5x$ make $8x$
In $4a^2 + 2a + 3a^2$ :: $4a^2$ and $3a^2$ make $7a^2$
In $6xy + 2y + xy$ :: $6xy$ and $xy$ make $7xy$

Unlike terms

In $3x + 5x + 2y$ :: $3x$ and $2y$
In $4a^2 + 2a + 3a^2$ :: $a^2$ and $a$
In $6xy + 2y + xy$ :: $xy$ and $y$

Worked example

Simplify $4x + 3y - 2x + 5y + 1$ .

1
Group like terms: $(4x - 2x) + (3y + 5y) + 1$ .
2
Combine each group.

Simplify $5a + 2b - 3a + b$ .

Expanding brackets

Multiply every term in one bracket by every term in the other.

For a single bracket, multiply each term inside by the term outside: $3(2x + 5) = 6x + 15$ .

For double brackets, use FOIL (First, Outer, Inner, Last) or a grid. For triple brackets, expand two of them first, then multiply the result by the third.

Watch the negative signs: in $(x - 3)(x + 2)$ the inner term is $-3x$ .

Area model for

(x + 3)(x + 2)

: four rectangles, with x drawn as 3 units.

Worked example: double brackets

Expand and simplify $(x + 2)(x + 3)$ .

1
First: $x \times x = x^2$ .
2
Outer: $x \times 3 = 3x$ .
3
Inner: $2 \times x = 2x$ .
4
Last: $2 \times 3 = 6$ .
5
Collect like terms.

Worked example: triple brackets

Expand $(x + 1)(x + 2)(x + 3)$ .

1
$(x + 1)(x + 2) = x^2 + 3x + 2$ .
2
$(x^2 + 3x + 2)(x + 3) = x^3 + 3x^2 + 3x^2 + 9x + 2x + 6$ .
3
Collect like terms.

Expand and simplify $(x - 3)(x + 2)$ .

Common factors

Take the greatest common factor of all the terms outside a bracket.

Factorisation is the reverse of expanding.

Find the greatest common factor of all the terms, including numbers and letters.
Divide each term by it.
Write it outside brackets.

For $12a^3b + 8ab^2$ the factor is $4ab$ , giving $4ab(3a^2 + 2b)$ .

With a = 1 and b = 1, 12a³b + 8ab² is an area of 12 + 8 = 20, a rectangle 4ab by (3a² + 2b) = 4 by 5.

$6x^2 + 9x$ $3x(2x + 3)$

Worked example

Factorise $6x^2 + 9x$ .

1
The greatest common factor of $6x^2$ and $9x$ is $3x$ .
2
Divide: $6x^2 \div 3x = 2x$ and $9x \div 3x = 3$ .
3
Check by expanding: $3x \times 2x + 3x \times 3 = 6x^2 + 9x$ .

Factorise fully $10x^2 - 15x$ .

Factorising quadratics

Use the difference of two squares, or find numbers that multiply to $ac$ and add to $b$ .

A quadratic has the form $ax^2 + bx + c$ . If $a = 1$ , find $p$ and $q$ with $p + q = b$ and $pq = c$ .

Higher Tier, for $a \neq 1$ : find two numbers that multiply to $ac$ and add to $b$ , split the $bx$ term, then factorise by grouping. Include $a$ in the product.

$x^2 + 25$ cannot be factorised as a difference of two squares.

x² − 5x + 6 = (x − 2)(x − 3): the factors give the x-intercepts.

a = 1

	Difference of two squares	Simple, $a = 1$	General, $a \neq 1$ (Higher)
Method	$x^2 - a^2 = (x + a)(x - a)$	Find $p + q = b$ and $pq = c$	Numbers that multiply to $ac$ and add to $b$
Example	$4x^2 - 9 = (2x + 3)(2x - 3)$	$x^2 + 7x + 12 = (x + 3)(x + 4)$	Split the middle term, then group

Difference of two squares

Method:: $x^2 - a^2 = (x + a)(x - a)$
Example:: $4x^2 - 9 = (2x + 3)(2x - 3)$

Simple, $a = 1$

Method:: Find $p + q = b$ and $pq = c$
Example:: $x^2 + 7x + 12 = (x + 3)(x + 4)$

General, $a \neq 1$ (Higher)

Method:: Numbers that multiply to $ac$ and add to $b$
Example:: Split the middle term, then group

Worked example: general quadratic (Higher Tier)

Factorise $2x^2 + 7x + 3$ .

1
$ac = 6$ and $b = 7$ , so the numbers are 6 and 1.
2
Split: $2x^2 + 6x + x + 3$ .
3
Group: $2x(x + 3) + 1(x + 3)$ .
4
Take out the common bracket.

Factorise $x^2 - 49$ .

Algebraic fractions

Factorise, then cancel common factors, never common terms.

Algebraic fractions contain variables. Factorise the numerator and denominator and cancel common factors. For example $\frac{x^2 - 4}{x + 2} = \frac{(x + 2)(x - 2)}{x + 2} = x - 2$ .

To add or subtract, find a common denominator. To multiply, factorise, cancel and multiply across. To divide, flip the second fraction and multiply.

(x² − 4)/(x + 2) simplifies to x − 2, except at x = −2 where it is undefined (a gap in the line).

Simplifyfactorise, then cancel common factors

Worked example: adding

Write $\frac{3}{x} + \frac{2}{x + 1}$ as a single fraction.

1
Common denominator: $x(x + 1)$ .
2
Rewrite: $\frac{3(x + 1)}{x(x + 1)} + \frac{2x}{x(x + 1)}$ .
3
Combine: $\frac{3x + 3 + 2x}{x(x + 1)}$ .

Worked example: dividing

Simplify $\frac{2x}{x + 3} \div \frac{4x}{x + 1}$ .

1
Flip and multiply: $\frac{2x}{x + 3} \times \frac{x + 1}{4x}$ .
2
Cancel $2x$ .

Simplify $\frac{x^2 - 9}{x + 3}$ .

Try an exam question

(a) Expand and simplify $(x - 3)(x + 2)$ . (b) Factorise $6x^2 + 9x$ .

[4 marks]

That's the notes covered.

Carry on to the next subtopic.