Algebraic Expressions – GCSE Maths

Here is everything you need to know about algebraic expressions for GCSE maths (Edexcel, AQA and OCR). You’ll learn what algebraic expressions are, how to simplify algebraic expressions, how to expand brackets and how to rearrange equations.

Look out for the algebraic expression worksheets, word problems and exam questions at the end.

What is an algebraic expression?

An algebraic expression is a set of terms that are combined using addition (+), subtraction (-), multiplication (✕) and division (÷).

An expression that contains two terms is called a binomial.

\[E.g. 2x+3y\quad or \quad 2-5y^{2}\quad etc.\]

An expression that contains three terms is called a trinomial.

\[E.g. 2x+3y-5\quad or \quad 2-5y^{2}+6xy\quad etc.\]

Keywords

Let’s define some of these keywords when using algebraic notation:

A variable is a symbol (often a letter) that is used to represent an unknown quantity.

\[E.g. x\quad or \quad y\quad or\quad a\quad etc.\]

Variables can also have exponents (be raised to a certain power).

\[E.g. x^{2}\quad or \quad y^{3}\]

A coefficient is the value that is before a variable. It tells us how many lots of the variable there is.

\[ \begin{aligned} E.g. 5x&=x+x+x+x+x\\ &=5\times x \end{aligned}\]

Here 5 is the coefficient and x is the variable.

A term is a number by itself, a variable by itself, or a combination of numbers and letters. If the term includes a variable it is called an algebraic term.

\[E.g. 2\quad or\quad 5xy\quad or\quad 12x^{2}\quad or \quad 12xy\quad etc.\]

An expression that contains one term is called a monomial.

A polynomial expression consists of two or more algebraic terms.

How to use algebraic expressions

We can use algebraic expressions in a variety of different ways.

Algebraic expressions methods

Within algebraic expressions, you will find lessons on:

Multiple methods of using algebraic expressions are summarised below. For detailed examples, practice questions and worksheets on each one follow the links to the step by step guides.

1. Simplifying algebraic expressions

a) Example. Collecting like terms

Example of simplifying an algebraic expression:

\[5x+3y+4-2x+8y-7\]

\[5x-2x=3x\qquad 3y+8y=11y\qquad 4-7=-3\]

\[3x+11y-3\]

b) Example. Write and simplify algebraic expressions:

Write an expression for the perimeter of the shape.

\[\begin{aligned} Perimeter&=2x+3+x-2+2x+3+x-2\\ &=6x+2 \end{aligned} \]

c) Example. Simplify algebraic fractions:

\[\frac{14 x y}{8 x}\]

Divide the numerator and the denominator by 2x.

\[\frac{7y}{4}\]

Step by step guide: Simplifying algebraic expressions

2. Expanding brackets

Example of expanding brackets:

\[2(x+3)\]
x+ 3
22x+ 6

\[=2x+6\]

Step by step guide: Expanding brackets

3. Expand and simplify

Example of expand and simplify:

\[2(x+5)+3(x-2)\]

x+ 5
22x+ 10

x- 2
+ 3+ 3x- 6

\[2x+10+3x-6\]

\[2x+3x=5x\qquad 10-6=4\]

\[=5x+4\]

Step by step guide: Expand and simplify

Common misconceptions

  • Expressions, Equations and Formulas

An algebraic equation is different from an expression, an equation and a formula.

An expression is a set of terms that are combined using arithmetic operations:

(+, -, ✕, ÷)

\[2x+5\]

An equation is an expression that equals something.

\[2x+5=15\]

We can solve equations to work out the value of the unknown variable (x).

Step by step guide: Solving equations (coming soon)
Step by step guide: Quadratic equations (coming soon)

Algebraic formulae are a set of instructions which give a desired result.

E.g. Area of a circle = πr2
Circumference of a Circle = πd

  • Brackets terminology

Brackets are sometimes referred to as parentheses.

Practice algebraic expressions questions

1. Simplify: 6x2y - 2x2 + 4x2y - 5x2

Show answer

= 10x2y - 7x2

2. Write an expression for the area and the perimeter of the parallelogram.

Algebraic expressions questions

Show answer

\[\begin{aligned}Perimeter&=3x+1+2x+5+3x+1+2x+5\\&=10x+12\\\\ Area &=(3x+1)(2x+1)\\&=6x^{2}+5x+1\end{aligned}\]

3. Simplify:

\[\frac{9x^{2}y}{15x^{3}}\]

Show answer

\[=\frac{3y}{5x}\]

4. Simplify:

\[\frac{x^{2}-3x-10}{x^{2}-25}\]

Show answer

\[=\frac{x+2}{x+5}\]

5. Simplify: 3x(4 - 5x + 2y)

Show answer

=12x - 15x2 + 6xy

6. Simplify and simplify: 4(2x - 1) - 3(x + 6)

Show answer

= 5x - 22

Algebraic expressions GCSE questions

1. Simplify: 4f - 2e + 3f + 5e

Show answer

7f + 3e

(2 marks)

2. Expand and simplify: 4x(2x - 7)

Show answer

8x2 - 28x

(2 marks)

3. Simplify:

\[\frac{15x^{3}y^{2}}{5xy^{3}}\]

Show answer

\[=\frac{3x^{2}}{y}\]

(2 marks)

Algebraic expressions worksheets

Get your free algebraic expressions worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Learning checklist

  • simplify expressions
  • use language and properties precisely to analyse algebraic expressions
  • simplify and manipulate algebraic expressions to maintain equivalence by:
    - collecting like terms
    - multiplying a single term over a bracket
    - taking out common factors
  • translate simple situations or procedures into algebraic expressions

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