Algebraic fractions

Simplify, add, subtract, multiply and divide algebraic fractions, and simplify compound fractions, by factoring first.

Cancelling: factor first

Algebraic fractions follow the same rules as number fractions. The key rule is that you can only cancel factors, never terms.

Example 1●

Simplify $\dfrac{x^2 - 9}{x^2 + x - 6}$.

$= \dfrac{(x - 3)(x + 3)}{(x + 3)(x - 2)}$

Factor the top (difference of squares) and the bottom (monic quadratic).

$= \dfrac{x - 3}{x - 2}$

(x + 3) is a common factor of top and bottom, so it cancels.

Is the simplified fraction really the same?

Orange: the original fraction. Dashed blue: the simplified one. They match everywhere, except at x = −3, where the original is undefined because its denominator is zero.

Try this

Substitute $x = -3$ into the original fraction and you get $\tfrac{0}{0}$, which is undefined. The graph has a single missing point (a hole) there, too small to see. That is why some questions ask you to state $x \ne -3$.

Adding and subtracting

Use a common denominator, just as with $\tfrac{1}{3} + \tfrac{1}{4} = \tfrac{4 + 3}{12}$. Factor the denominators first so you can find the lowest common denominator.

Example 2●●

Simplify $\dfrac{3}{x - 1} - \dfrac{2}{x + 2}$.

$= \dfrac{3(x + 2) - 2(x - 1)}{(x - 1)(x + 2)}$

The lowest common denominator is the product (x − 1)(x + 2).

$= \dfrac{3x + 6 - 2x + 2}{(x - 1)(x + 2)}$

Expand the numerator carefully: −2 × −1 = +2.

$= \dfrac{x + 8}{(x - 1)(x + 2)}$

Multiplying and dividing

Factor everything, cancel common factors, then multiply. To divide, multiply by the reciprocal of the second fraction.

Example 3●●

Simplify $\dfrac{x^2 - 4}{3x} \div \dfrac{x + 2}{6x^2}$.

$= \dfrac{x^2 - 4}{3x} \times \dfrac{6x^2}{x + 2}$

Dividing by a fraction means multiplying by its reciprocal.

$= \dfrac{(x - 2)(x + 2)}{3x} \times \dfrac{6x^2}{x + 2}$

$= 2x(x - 2)$

Cancel (x + 2), and 6x² ÷ 3x = 2x.

Compound fractions

To clear a fraction inside a fraction, multiply the top and bottom by the lowest common denominator of all the small fractions.

Example 4●●●

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

$= \dfrac{x^2 + x}{x^2 - 1}$

The small denominators are x and x², so multiply top and bottom by x².

$= \dfrac{x(x + 1)}{(x - 1)(x + 1)}$

$= \dfrac{x}{x - 1}$

Flashcards

What can you cancel?

Common factors only, never terms. Factor first.

Dividing by a fraction

Multiply by its reciprocal.

Compound fractions

Multiply top and bottom by the LCD of the small fractions.

Practice

Simplify $\dfrac{2x + 4}{x^2 - 4}$.
$\dfrac{1}{x} + \dfrac{1}{x + 1} =$
Which cancellation is valid?
Simplify $\dfrac{x^2 - 4x}{x^2 - 16}$.
$\dfrac{x}{3} + \dfrac{x}{6} = \dfrac{x}{k}$. Find $k$.
$\dfrac{2}{x} \times \dfrac{x^2}{6} =$
$\dfrac{1}{x - 2} - \dfrac{1}{x + 2} =$
Simplify the compound fraction $\dfrac{x/2}{x/4}$ (for $x \ne 0$).