When is a rational expression undefined?

When its original denominator is zero.

State the factor theorem.

If f(k)=0 then (x-k) is a factor of f(x).

Can you cancel the x^2 terms in \frac{x^2-9}{x^2+x-12}?

No. Only common factors cancel, not common terms.

Simplify \frac{x^2-9}{x^2+x-12}.

\frac{x+3}{x+4}, with x\neq3,-4

In algebraic division, what is the quotient?

The polynomial part of the answer; the remainder has lower degree than the divisor.

Simplify \frac{x^3+1}{x^2-1}.

\frac{x^2-x+1}{x-1}, with x\neq\pm1

Why check solutions to a rational equation?

A solution equal to an excluded value is not valid.

Simplifying rational expressionsEdexcel International A Level Maths: Flashcards

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What is a rational expression?

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What is a rational expression?: A fraction with polynomials in the numerator and denominator.
When is a rational expression undefined?: When its original denominator is zero.
Factorise $x^2-a^2$ .: $(x-a)(x+a)$
Factorise $x^3+a^3$ .: $(x+a)(x^2-ax+a^2)$
Factorise $x^3-a^3$ .: $(x-a)(x^2+ax+a^2)$
State the factor theorem.: If $f(k)=0$ then $(x-k)$ is a factor of $f(x)$ .
Can you cancel the $x^2$ terms in $\frac{x^2-9}{x^2+x-12}$ ?: No. Only common factors cancel, not common terms.
Simplify $\frac{x^2-9}{x^2+x-12}$ .: $\frac{x+3}{x+4}$ , with $x\neq3,-4$
In algebraic division, what is the quotient?: The polynomial part of the answer; the remainder has lower degree than the divisor.
Write $\frac{x^2-x+1}{x-1}$ in quotient-remainder form.: $x+\frac{1}{x-1}$
Simplify $\frac{x^3+1}{x^2-1}$ .: $\frac{x^2-x+1}{x-1}$ , with $x\neq\pm1$
Why check solutions to a rational equation?: A solution equal to an excluded value is not valid.
What does a cancelled factor $(x-k)$ leave on the graph?: A missing point (hole) at $x=k$ .

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).