Simplifying rational expressionsEdexcel International A Level Maths: Revision notes

Section 1

Rational expressions and excluded values

A rational expression is a fraction in which the numerator and denominator are polynomials, for example $\frac{1}{ax+b}$ or $\frac{ax+b}{px^2+qx+r}$ . In this course the denominator is linear or quadratic. The expression is undefined wherever the denominator equals zero, so those values of $x$ are excluded. For $\frac{x^2-9}{x^2+x-12}$ the denominator is $(x+4)(x-3)$ , so $x\neq-4$ and $x\neq3$ . Always state the exclusions from the original denominator, not the simplified one.

Key termsrational expressionexcluded value

Common mistake

Reading the exclusions from the simplified fraction. If $(x-3)$ cancels, $x=3$ is still excluded.

Section 2

Factorising before cancelling

To simplify, factorise numerator and denominator completely, then cancel common factors (never common terms). Useful patterns: $x^2-a^2=(x-a)(x+a)$ ; $x^3-a^3=(x-a)(x^2+ax+a^2)$ ; $x^3+a^3=(x+a)(x^2-ax+a^2)$ ; quadratics $ax^2+bx+c$ by inspection or the quadratic formula. For a cubic, use the factor theorem: if $f(k)=0$ then $(x-k)$ is a factor. Example: $\frac{x^3+1}{x^2-1}=\frac{(x+1)(x^2-x+1)}{(x-1)(x+1)}=\frac{x^2-x+1}{x-1}$ , for $x\neq\pm1$ .

Key termscommon factorfactor theorem

Common mistake

Cancelling terms such as the $x^2$ in $\frac{x^2-9}{x^2+x-12}$ . Only whole factors can cancel.

Exam tip

A quadratic that will not factorise nicely is a sign the numerator and denominator may share no common factor.

Section 3

Algebraic division

When the numerator has degree equal to or greater than the denominator, divide using algebraic (long) division. Divide the leading term of the numerator by the leading term of the divisor, multiply back, subtract, and repeat until the remainder has lower degree than the divisor. $\frac{N(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)}$ where $Q$ is the quotient and $R$ the remainder. Example: $x^2-x+1=x(x-1)+1$ , so $\frac{x^2-x+1}{x-1}=x+\frac{1}{x-1}$ . An alternative is to match coefficients: write $N(x)\equiv Q(x)D(x)+R(x)$ and compare.

Key termsquotientremainderalgebraic division

Exam tip

Write every power of $x$ in the numerator, inserting $0x^2$ for any missing term, so the columns line up.

Section 4

Worked example: cancel then divide

Simplify $g(x)=\frac{x^3-8}{x^2-4}$ .

Factorise: $x^3-8=(x-2)(x^2+2x+4)$ and $x^2-4=(x-2)(x+2)$ .
Cancel $(x-2)$ : $g(x)=\frac{x^2+2x+4}{x+2}$ , $x\neq\pm2$ .
Divide: $x^2+2x+4=x(x+2)+4$ , so $g(x)=x+\frac{4}{x+2}$ . Check at $x=0$ : original $\frac{-8}{-4}=2$ ; simplified $0+\frac42=2$ . The graph has a missing point where $x=2$ , at $(2,3)$ .

Key termsequivalent expression

Exam tip

Substitute a simple value such as $x=0$ or $x=1$ into the original and the answer to check your simplification.

Section 5

Exam technique

Use the command word. Show that or simplify needs the factorisation written out. Express in the form asks for division. State any excluded values when the question asks where an expression is undefined. For a solved equation such as $\frac{x+3}{x+4}=2$ , multiply through by the denominator, solve, then check the answer is not an excluded value.

Key termsshow that

Common mistake

Multiplying through by an expression and forgetting that its zero values must be rejected as solutions.

That's the notes covered.

Carry on to the next subtopic.

Exam questions on Simplifying rational expressions

Let $f(x)=\dfrac{x^2-9}{x^2+x-12}$ .
Solve $f(x)=2$ .2 marks
Let $h(x)=\dfrac{x^3+1}{x^2-1}$ .
Hence find constants $a$ , $b$ and $c$ such that $h(x)\equiv ax+b+\dfrac{c}{x-1}$ for $x\neq\pm1$ .2 marks
Let $F(x)=\dfrac{2x^3-3x^2-11x+6}{x^2-x-6}$ , where $x\in\mathbb{R}$ , $x\neq 3$ , $x\neq -2$ .
Simplify $F(x)$ fully, showing your factorisation of the numerator.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).