Rational expressions

A rational expression is an expression of the form \( \frac{P}{Q} \), where \( P \) and \( Q \) are similar expressions. In this lesson both \(P\) and \(Q\) will be polynomials.

Simplifying rational expressions

This lesson is essentially an application of factoring polynomials. When the polynomials in the numerator and denominator of an expression share factors, you can cancel these factors out.

It is important to note that x cannot be equal to 3 or 2 because this would put 0 in the denominator, and it is not possible to divide by zero. For this reason a potential COMPASS Test problem could be phrased like this: For \( x\neq5 \), simplify the following expression: \( \frac{x^{2}-3x-10}{x-5} \).

You simplify by factoring the numerator, and the context of the problems hints that \( (x-5) \) is a factor.

Beware of the tricky case of the ‘difference of squares’:

\( x^{2}-a^{2} \) will always factor to \( (x-a)(x+a) \). It can get a little trickier if we switch the order up, but just remember you can factor out a negative sign.

Practice questions

Simplify the rational expressions

1. \( \frac{x^{2}+12x+35}{x+5} \)
2. \( \frac{x^{2}+6x-16}{x-2} \)
3. \( \frac{x^{2}+7x+6}{x^{2}+2x-24} \)
4. \( \frac{x^{2}-6x+9}{x^{2}-9} \)
5. \( \frac{81-x^{2}}{x^{2}-8x-9} \)
6. \( \frac{2x^{2}-7x-4}{x^{2}-x-12} \)
7. \( \frac{3x^{2}+12x-36}{x^{2}-2x} \)

Solutions

1. \( \frac{x^{2}+12x+35}{x+5}=\frac{(x+5)(x+7)}{x+5}=x+7 \)
2. \( \frac{x^{2}+6x-16}{x-2}=\frac{(x-2)(x+8)}{x-2}=x+8 \)
3. \( \frac{x^{2}+7x+6}{x^{2}+2x-24}=\frac{(x+1)(x+6)}{(x-4)(x+6)}=\frac{x+1}{x-4} \)
4. \( \frac{x^{2}-6x+9}{x^{2}-9}=\frac{(x-3)(x-3)}{(x+3)(x-3)}=\frac{x-3}{x+3} \)
5. \( \frac{81-x^{2}}{x^{2}-8x-9}=\frac{-(x-9)(x+9)}{(x-9)(x+1)}=\frac{-9-x}{x+1} \)
6. \( \frac{2x^{2}-7x-4}{x^{2}-x-12}=\frac{(2x+1)(x-4)}{(x+3)(x-4)}=\frac{2x+1}{x+3} \)
7. Solution: $$ \begin{align*} & \frac{3x^{2}+12x-36}{x^{2}-2x}=\frac{3(x^{2}+4x-12)}{x^{2}-2x}= \\[0.5em] & \frac{3(x+6)(x-2)}{x(x-2)}=\frac{3(x+6)}{x} \end{align*} $$