IMAT Mathematics · Equations and Inequalities

Expanding, Factorising and Algebraic Fractions

Expanding multiplies brackets out, factorising puts them back, and an algebraic fraction is simplified by cancelling common factors, never common terms.

All Equations and Inequalities notes

Why this matters

Simplifying an algebraic fraction by factorising appeared in 2016, 2021 and 2022, and the 2025 paper asked you to recognise a perfect square trinomial. The factor theorem has not been asked yet, but it is the quickest way to factorise a cubic.

Concept 1 of 4: Expanding brackets and the special products

Expanding means every term in one bracket multiplies every term in the other. A few products come up so often that recognising them saves the work in both directions: spotting

a^2 - b^2

or a perfect square is the first step of most factorising.

Definition

Perfect squares: $(a + b)^2 = a^2 + 2ab + b^2$ and $(a - b)^2 = a^2 - 2ab + b^2$ .
Difference of two squares: $(a + b)(a - b) = a^2 - b^2$ . It also speeds up arithmetic: $61^2 - 39^2 = 22 \times 100$ .
Cubes (in outline): $(a + b)^3 = a^3 + 3a^2 b + 3ab^2 + b^3$ , $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ and $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ .
A trinomial is a perfect square when the middle term is twice the product of the square roots of the outer terms: $9x^2 - 12xy + 4y^2 = (3x - 2y)^2$ .

Special products

(a \pm b)^2 = a^2 \pm 2ab + b^2 \qquad (a+b)(a-b) = a^2 - b^2

Worked example

(a) Expand

(3x - 2y)^2

. (b) Evaluate

61^2 - 39^2

without a calculator. (c) Expand

(x + 1)^3

Practice this conceptself-check · 4 quick reps

The square of a sum is not the sum of the squares

(a + b)^2 = a^2 + 2ab + b^2

, not

a^2 + b^2

. Dropping the middle term

2ab

is the commonest expansion error, and IMAT offers the result as an option.

Concept 2 of 4: Factorising: common factors, grouping and trinomials

Factorising writes an expression as a product, which is what lets you cancel fractions and read off the solutions of an equation. Work in a fixed order: common factor first, then a special product, then a trinomial, then grouping.

Definition

Work through these steps in order, and check by expanding back:

Take out any common factor: $6x^3 - 24x = 6x(x^2 - 4)$ .
Look for a special product (difference of squares, perfect square).
Trinomial $x^2 + sx + p$ : find two numbers with sum $s$ and product $p$ .
Trinomial $ax^2 + bx + c$ : find two numbers with sum $b$ and product $ac$ , split the middle term, and group.
Grouping for four terms: $xy - 3x + 2y - 6 = x(y - 3) + 2(y - 3) = (x + 2)(y - 3)$ .
A sum of two squares, $x^2 + k^2$ , does not factorise over the real numbers.

Monic trinomial

x^2 + sx + p = (x + m)(x + n) \quad\text{where}\quad m + n = s,\; mn = p

Worked example

Factorise completely: (a)

6x^3 - 24x

, (b)

x^2 - x - 12

, (c)

2x^2 + 7x + 3

Practice this conceptself-check · 4 quick reps

A sum of two squares does not factorise

x^2 - 9 = (x - 3)(x + 3)

, but

x^2 + 9

has no real factors:

(x + 3)(x - 3)

expands to

x^2 - 9

, and

(x + 3)^2

has a middle term.

Concept 3 of 4: Simplifying and combining algebraic fractions

Algebraic fractions follow the rules of number fractions. You may cancel a factor that multiplies the whole top and the whole bottom, so factorise both first. To add or subtract, bring them to a common denominator, which is quickest when the denominators are already factorised.

Definition

Simplify: factorise numerator and denominator, then cancel common factors.
Add or subtract: use the lowest common denominator, built from the factorised denominators.
Divide: multiply by the reciprocal.
Any value that makes an original denominator zero stays excluded, even if its factor cancels.

Adding fractions

\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}

Worked example

(a) Simplify

\dfrac{x^2 - 9}{x^2 + x - 6}

. (b) Write

\dfrac{3}{x^2 - 4} - \dfrac{1}{x - 2}

as a single fraction.

Practice this conceptself-check · 3 quick reps

Cancel factors, never terms

\frac{x + 5}{x + 7}

is not

\frac{5}{7}

: the

x

is a term added on, not a factor of the whole top and bottom. Factorise first, then cancel only whole brackets that multiply everything.

Concept 4 of 4: Polynomial division and the factor theorem

Dividing polynomials works like long division of numbers: dividend = divisor × quotient + remainder. When the divisor is

x - a

, putting

x = a

kills the first part and leaves the remainder. So if

p(a) = 0

, then

x - a

divides exactly: it is a factor.

Definition

Division (in outline): $p(x) = d(x)\,q(x) + r(x)$ , where the remainder has a lower degree than the divisor.
Remainder theorem: the remainder when $p(x)$ is divided by $x - a$ is $p(a)$ .
Factor theorem: $x - a$ is a factor of $p(x)$ exactly when $p(a) = 0$ .
For a polynomial with integer coefficients and leading coefficient 1, any integer root divides the constant term. Try those first.
For the factor $x + 3$ , test $x = -3$ .

Remainder theorem

p(x) = (x - a)\,q(x) + p(a)

$q(x)$ the quotient, one degree lower than $p(x)$
$p(a)$ the remainder, a number

Worked example

Factorise

p(x) = x^3 - 2x^2 - 5x + 6

completely and find its roots.

Practice this conceptself-check · 4 quick reps

For the factor x + a, test x = minus a

x + a

is zero when

x = -a

, so that is the value to substitute. Substituting

x = a

answers a different question, and its result is always among the options.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

Expanding brackets and the special products
Special products
$(a \pm b)^2 = a^2 \pm 2ab + b^2 \qquad (a+b)(a-b) = a^2 - b^2$
Factorising: common factors, grouping and trinomials
Monic trinomial
$x^2 + sx + p = (x + m)(x + n) \quad\text{where}\quad m + n = s,\; mn = p$
Simplifying and combining algebraic fractions
Adding fractions
$\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}$
Polynomial division and the factor theorem
Remainder theorem
$p(x) = (x - a)\,q(x) + p(a)$

Watch out for (4)

The square of a sum is not the sum of the squares
→ Expanding brackets and the special products
A sum of two squares does not factorise
→ Factorising: common factors, grouping and trinomials
Cancel factors, never terms
→ Simplifying and combining algebraic fractions
For the factor x + a, test x = minus a
→ Polynomial division and the factor theorem

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