Rational expressionsEdexcel A-Level Maths: Revision notes

Section 1

Simplifying by factorising and cancelling

A rational expression is a fraction whose numerator and denominator are polynomials. To simplify one, factorise the numerator and denominator fully, then cancel common factors (not common terms). Example: $\frac{x^2-9}{x^2+x-6}=\frac{(x-3)(x+3)}{(x+3)(x-2)}=\frac{x-3}{x-2}$ . Look for: common factors, the difference of two squares $x^2-a^2=(x-a)(x+a)$ , and quadratics $px^2+qx+r$ . Denominators here are linear, e.g. $\frac{1}{ax+b}$ , or quadratic, e.g. $\frac{ax+b}{px^2+qx+r}$ . The simplified form is equal to the original everywhere the original is defined: $x=-3$ is still excluded in the example above, because it made the original denominator zero.

Key termsrational expressioncommon factor

Common mistake

Cancelling terms rather than factors: $\frac{x+3}{x+5}$ is not $\frac35$ . Only a whole factor can be cancelled.

Section 2

Factorising cubics in a fraction

A cubic in a numerator or denominator, such as $x^3+8$ , is factorised using the factor theorem: $f(-2)=-8+8=0$ , so $(x+2)$ is a factor. Dividing gives $x^3+8=(x+2)(x^2-2x+4)$ . In general $x^3+a^3=(x+a)(x^2-ax+a^2)$ and $x^2-a^2=(x-a)(x+a)$ , so $\frac{x^3+a^3}{x^2-a^2}=\frac{x^2-ax+a^2}{x-a}$ . Example: $\frac{x^3+8}{x^2-4}=\frac{(x+2)(x^2-2x+4)}{(x-2)(x+2)}=\frac{x^2-2x+4}{x-2}$ . The quadratic $x^2-2x+4$ has no real roots, so nothing more cancels.

Key termsfactor theorem

Exam tip

If the numerator is a cubic and the denominator a quadratic, look for a shared linear factor with the factor theorem.

Section 3

Multiplying, dividing, adding and subtracting

Multiply: factorise everything, cancel across numerators and denominators, then multiply what remains. Example: $\frac{x-3}{x+2}\times\frac{x+2}{x+1}=\frac{x-3}{x+1}$ . Divide: multiply by the reciprocal of the second fraction, then proceed as for multiplication. Add or subtract: write each fraction over a common denominator (use the lowest one), combine the numerators, then factorise and cancel if possible. Example: $\frac{1}{x+1}+\frac{2}{x-1}=\frac{(x-1)+2(x+1)}{(x+1)(x-1)}=\frac{3x+1}{(x+1)(x-1)}$ .

Key termsreciprocalcommon denominator

Common mistake

Adding fractions by adding denominators: $\frac1a+\frac1b\ne\frac{2}{a+b}$ .

Section 4

Algebraic division: quotient and remainder

If the numerator has degree at least the denominator, divide to write the expression as a polynomial plus a proper fraction. Example: $\frac{x-3}{x-2}=\frac{(x-2)-1}{x-2}=1-\frac{1}{x-2}$ . Example: $\frac{x^2-2x+4}{x-2}$ . Dividing, $x^2-2x+4=x(x-2)+4$ , so the expression is $x+\frac{4}{x-2}$ . For $\frac{x^2+3x+5}{x+1}$ long division gives quotient $x+2$ and remainder $3$ , so the expression is $x+2+\frac{3}{x+1}$ . This form shows how the value behaves: in $1-\frac{4}{x+1}$ the fraction is never $0$ , so the expression can never equal $1$ .

Key termsquotientremainderproper fraction

Exam tip

For a denominator $(x+k)$ , you can often rewrite the numerator directly: $x-3=(x+1)-4$ .

Section 5

Solving and checking

To solve an equation containing a rational expression, simplify first, then multiply through by the denominator. Example: $\frac{3x-2}{2x-3}=2\Rightarrow3x-2=4x-6\Rightarrow x=4$ . Always check that your answer does not make the original denominator zero. Substituting $x=4$ into $\frac{3x^2+x-2}{2x^2-x-3}$ gives $\frac{50}{25}=2$ , as expected.

Key termsrestriction

Common mistake

Accepting a solution such as $x=-1$ when $(x+1)$ was a cancelled factor of the original denominator.

That's the notes covered.

Carry on to the next subtopic.

Exam questions on Rational expressions

The expression $\frac{x^2-9}{x^2+x-6}$ is defined for all real $x$ except where the denominator is zero.
Hence write $\frac{x^2-9}{x^2+x-6}$ in the form $A+\frac{B}{x-2}$ , where $A$ and $B$ are constants.2 marks
The expression $\frac{x^3+8}{x^2-4}$ is considered for values of $x$ where it is defined.
Hence write $\frac{x^3+8}{x^2-4}$ in the form $x+\frac{k}{x-2}$ , where $k$ is a constant.2 marks
The function $f(x)=\frac{3x^2+x-2}{2x^2-x-3}$ is defined for all real $x$ except where its denominator is zero.
Simplify $f(x)$ .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).