Rewriting Rational Expressions

Maths: Edexcel A Level Pure Maths, topic 2.4.5

Rewriting Rational Expressions

We can use everything we have learned about dividing polynomials to rewrite rational expressions.

Rational Expression

Rational Expression

A rational expression is an expression of the form $\frac{p(x)}{q(x)}$ , where $p(x)$ and $q(x)$ are polynomials and $q(x)\ne0$ .

Division

Division

We've already looked at the decomposition of a polynomial in terms of its quotient $q(x)$ , divisor $(x-c)$ , and remainder $r$ :
$f(x)=q(x)(x-c)+r$
We can generalize the decomposition for any divisor $d(x)$ that gives remainder $r(x)$ :
$f(x)=q(x)d(x)+r(x)$
Where $r(x)$ is a polynomial of a degree less than that of $d(x)$ .

Rewriting rational expressions

Rewriting rational expressions

Dividing this equation by $d(x)$ on both sides gives the following:
$\frac{f(x)}{d(x)} = q(x) + \frac{r(x)}{d(x)}$
Where $d(x)\ne0$ .
We can use this to rewrite rational expressions.

Example

Example

For example, $\frac{x^2+5x+6}{x+2}$ can be rewritten as $x+3$ .
This is because $x+2$ is a factor of the numerator, and so in this case $r(x) = 0$ .
Here $x\ne-2$ , as this would make the denominator equal to zero and the rational expression would be undefined.

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