Expand products, factorise expressions including quadratics and the difference of two squares, and simplify algebraic fractions

What this dot point is asking

SEAB wants you to expand products of brackets, factorise a range of expressions (common factors, grouping, the difference of two squares, and quadratic trinomials), and simplify algebraic fractions. Factorising is the reverse of expanding, and it is the key that unlocks solving quadratics and simplifying fractions later.

The answer

Expanding brackets

To expand a single bracket, multiply each term inside by the term outside. To expand two brackets, multiply every term in the first by every term in the second:

Useful identities are $(a + b)^2 = a^2 + 2ab + b^2$ and $(a - b)(a + b) = a^2 - b^2$.

Factorising by common factor

Take out the highest common factor of all terms: $6x^2 + 9x = 3x(2x + 3)$. Always check first whether a common factor exists, because it simplifies everything that follows.

Factorising by grouping

When four terms share factors in pairs, group and factorise each pair, then take out the common bracket:

The difference of two squares

An expression of the form $a^2 - b^2$ factorises as $(a - b)(a + b)$. Spotting this pattern, including with coefficients such as $9x^2 - 16 = (3x - 4)(3x + 4)$, is a frequent exam shortcut.

Factorising quadratics

For $x^2 + bx + c$, find two numbers that multiply to $c$ and add to $b$. For $x^2 + 5x + 6$ the numbers are $2$ and $3$, giving $(x + 2)(x + 3)$. When the coefficient of $x^2$ is not $1$, split the middle term using factors that multiply to give the product of the outer coefficients.

Simplifying algebraic fractions

Factorise the numerator and denominator fully, then cancel any common factors. You can only cancel a factor that multiplies the whole top and the whole bottom, never an individual term.

Examples in context

Example 1. Solving equations. Factorising $x^2 - 2x - 15 = 0$ into $(x - 5)(x + 3) = 0$ immediately gives the solutions $x = 5$ and $x = -3$. Factorisation is the bridge from an expression to the roots of an equation.

Example 2. Simplifying a formula. A rectangle of length $x + 2$ and width $x$ has area $x(x + 2) = x^2 + 2x$. Expanding and factorising let you move between a factorised geometric form and an expanded algebraic one as a problem requires.

Try this

Q1. Expand and simplify $(x - 4)(x + 6)$. [2 marks]

• Cue. $x^2 + 6x - 4x - 24 = x^2 + 2x - 24$.

Q2. Factorise $25 - 16y^2$. [1 mark]

• Cue. Difference of two squares: $(5 - 4y)(5 + 4y)$.

Q3. Factorise $x^2 - 3x - 10$. [2 marks]

• Cue. Two numbers multiplying to $-10$ and adding to $-3$ are $-5$ and $2$, giving $(x - 5)(x + 2)$.