Expanding, factorising and equivalent expressions

Expand and factorise expressions, recognise equivalent forms and simplify algebraic fractions.

Also known as: expanding brackets, factorising, polynomials, algebraic fractions.

SAT · Advanced Math GCSE Maths · Algebra IGCSE Maths · Algebra and graphs

Key ideas

Expand $(ax + b)(cx + d) = acx^2 + (ad + bc)x + bd$ .
Difference of two squares: $a^2 - b^2 = (a - b)(a + b)$ .
Two expressions are equivalent if they are equal for every value of the variable.
Factorise fully before cancelling algebraic fractions.

Expanding

Multiply every term in the first bracket by every term in the second: $(2x + 3)(x - 5) = 2x^2 - 10x + 3x - 15 = 2x^2 - 7x - 15.$ Matching coefficients is a fast way to answer "which expression is equivalent" questions.

Factorising

Take out common factors first: $6x^2 + 9x = 3x(2x + 3)$ . For quadratics, find two numbers that multiply to $ac$ and add to $b$ . Watch for the difference of two squares: $x^2 - 49 = (x - 7)(x + 7)$ .

Algebraic fractions

Factorise, then cancel common factors (never individual terms): $\frac{x^2 - 9}{x - 3} = \frac{(x - 3)(x + 3)}{x - 3} = x + 3, \quad x \neq 3.$

Worked examples

Example 1 · Matching coefficients

If $(3x + k)(x - 2) = 3x^2 + x - 14$ for all $x$ , find $k$ .

Step 1.Expand: $3x^2 + (k - 6)x - 2k$ .
Step 2.Match the $x$ -coefficient: $k - 6 = 1$ , so $k = 7$ . Check the constant: $-2(7) = -14$ ✓.

Answer: $k = 7$

Example 2 · Simplifying

Simplify $\dfrac{x^2 + 5x + 6}{x^2 - 4}$ .

Step 1.Numerator: $(x + 2)(x + 3)$ . Denominator: $(x - 2)(x + 2)$ .
Step 2.Cancel $(x + 2)$ .

Answer: $\dfrac{x + 3}{x - 2}$

Common mistakes

Forgetting the middle term: $(x + 3)^2 \neq x^2 + 9$ .
Cancelling terms instead of factors: $\frac{x + 6}{x}$ is not $6$ .
Sign errors when multiplying two negative terms.

Try it yourself

Original practice questions. Your answer is checked instantly.

Question 1Expanding, factorising and equivalent expressionsEasier

Which expression is equivalent to $(2x + 3)(x - 5)$ ?

Question 2Expanding, factorising and equivalent expressionsEasier

When $(x + 4)^2$ is written as $x^2 + bx + c$ , what is the value of $b$ ?