Algebraic Expressions

What is an Algebraic Expression?

An algebraic expression contains numbers, variables (letters representing unknown values), and operations. Unlike equations, expressions do not have an equals sign.

Examples: 3x + 5, 2a² - b, 4(x + 3)

Expressions represent a quantity whose value depends on the variable. When you substitute a value for the variable, you can calculate the result.

Key Vocabulary

Term	Meaning	Example in 5x² - 3x + 7
Term	A single number, variable, or product	5x², -3x, 7
Coefficient	Number multiplying the variable	5 in 5x², -3 in -3x
Variable	The unknown letter	x
Constant	A fixed number with no variable	7
Like terms	Same variable and same power	3x and 5x

Identifying Like Terms

Like terms have the same variable(s) raised to the same power. Only like terms can be combined.

• 3x and 5x are like terms ✓ (both have x to the power 1)
• 4x² and 2x² are like terms ✓ (both have x²)
• 3x and 3x² are NOT like terms ✗ (different powers)
• 3x and 5y are NOT like terms ✗ (different variables)

Simplifying Expressions

Collect like terms by adding or subtracting their coefficients:

Example: 4x + 3y + 2x - y

• Group x-terms: 4x + 2x = 6x
• Group y-terms: 3y - y = 2y
• Answer: 6x + 2y

Example: 5a - 3b + 2a + 4b - a

• a-terms: 5a + 2a - a = 6a
• b-terms: -3b + 4b = b
• Answer: 6a + b

You cannot simplify unlike terms — 3x + 5y cannot be simplified further.

Expanding Brackets

Multiply every term inside the bracket by the term outside:

Example: 3(2x + 5)

• 3 × 2x = 6x
• 3 × 5 = 15
• Answer: 6x + 15

Example: -2(x - 4)

• -2 × x = -2x
• -2 × (-4) = +8
• Answer: -2x + 8

Take special care with negative signs outside the bracket — they change the sign of every term inside.

Expanding and Simplifying

Expand all brackets first, then collect like terms.

Example: 2(x + 4) + 3(x - 1)

• Expand: 2x + 8 + 3x - 3
• Collect: (2x + 3x) + (8 - 3) = 5x + 5

Substitution

Replace each variable with its given value and calculate. Follow the order of operations (BIDMAS/BODMAS).

Example: Find 2x² - x + 4 when x = 3

• 2(3²) - 3 + 4
• 2(9) - 3 + 4
• 18 - 3 + 4 = 19

Example: Find a² + 2b when a = 2, b = -3

• (2²) + 2(-3)
• 4 + (-6) = -2

Factorising Expressions

Factorising is the reverse of expanding — find the highest common factor and write it outside a bracket.

Example: 6x + 9

• HCF of 6 and 9 is 3
• 6x + 9 = 3(2x + 3)

Example: 10x² + 15x

• HCF of 10 and 15 is 5; both terms have x
• 10x² + 15x = 5x(2x + 3)

Writing Expressions from Words

Translating word problems into algebra is a key skill:

Word phrase	Algebraic expression
5 more than x	x + 5
4 less than y	y - 4
3 times a number n	3n
A number divided by 7	n/7
Square of a number	n²

Common Mistakes

• Combining unlike terms: 3x + 4y ≠ 7xy. Never add terms with different variables.
• Forgetting to multiply all terms in the bracket: 3(x + 2) ≠ 3x + 2.
• Sign errors when expanding: -2(x - 3) = -2x + 6, not -2x - 6.
• Squaring only the number, not the variable: 2x² means 2(x²), not (2x)².

Tips and Tricks

• Underline like terms in the same way before collecting them (e.g., underline all x-terms, circle all y-terms).
• When substituting negative numbers, always put them in brackets: (−3)² = 9, not −3² = −9.
• To check a factorisation, expand your answer — you should get back the original expression.
• The number of terms in a simplified expression tells you something: one term = monomial, two = binomial, three = trinomial.