Grade 9 Algebraic Expressions and Factorising

Key concepts

• Like terms and simplifying expressions
• The distributive law: a(b + c) = ab + ac
• Product of two binomials: (x + a)(x + b) = x² + (a + b)x + ab
• Squaring a binomial: (a + b)² = a² + 2ab + b²
• Factorising by taking out a common factor
• Difference of two squares and trinomials of the form x² + bx + c

Worked examples

Expand and simplify: (x + 3)(x − 5)

Multiply each term in the first bracket by each term in the second: x² − 5x + 3x − 15 = x² − 2x − 15.

Factorise fully: x² − 49

This is a difference of two squares, a² − b² = (a − b)(a + b), with a = x and b = 7. So x² − 49 = (x − 7)(x + 7).

Factorise: x² + 7x + 12

Find two numbers that multiply to 12 and add to 7: 3 and 4. So x² + 7x + 12 = (x + 3)(x + 4).

Expand: (2x − 3)²

Use (a − b)² = a² − 2ab + b²: (2x)² − 2(2x)(3) + 3² = 4x² − 12x + 9.

Factorise fully: 3x² − 12

Take out the common factor 3 first: 3(x² − 4). Then x² − 4 is a difference of two squares, so 3x² − 12 = 3(x − 2)(x + 2).

Common mistakes

• Writing (x + 3)² = x² + 9 and forgetting the middle term 6x.
• Adding unlike terms, such as 3x + 2x² = 5x³.
• Stopping before factorising fully, e.g. leaving 2x² − 8 as 2(x² − 4).
• Sign errors when multiplying negatives.

Exam tips

• Check a factorisation by expanding it again.
• Always look for a common factor first.
• Line up like terms in columns when simplifying long expressions.

Frequently asked questions

What does it mean to factorise?

To write an expression as a product of factors, for example x² + 7x + 12 = (x + 3)(x + 4). It's the reverse of expanding brackets.

How do I recognise a difference of two squares?

There are two terms, both perfect squares, with a minus between them: x² − 25, 4a² − 9, 1 − y².

What is a binomial?

An expression with two terms, such as x + 3 or 2a − 5. A trinomial has three terms.