Simplifying algebraic fractions

Covers AS91261

What an algebraic fraction is

An algebraic fraction is a fraction with algebra on the top (numerator) and/or bottom (denominator), like $\dfrac{x^2 - 9}{x^2 + 5x + 6}$ .
Simplifying means writing it in its lowest terms by cancelling anything common to top and bottom.
Cancelling is really dividing the top and bottom by the same factor — which is allowed because it does not change the value (you are dividing by 1).

Simplifying by factorising and cancelling

Factorise the top and the bottom fully into brackets.
Cancel any factor (a whole bracket, or a whole term) that appears on both top and bottom.
Golden rule: you can only cancel factors (things multiplied), never individual terms inside a sum.

$\frac{x^2 - 9}{x^2 + 5x + 6} = \frac{(x - 3)(x + 3)}{(x + 2)(x + 3)} = \frac{x - 3}{x + 2}$

Multiplying and dividing algebraic fractions

Multiply: factorise everything, cancel any common factors between the tops and bottoms, then multiply the tops and the bottoms.
Divide: flip the second fraction (its reciprocal) and multiply — flip only the fraction after the $\div$ sign.

Worked ExampleSimplifying a single fraction

Simplify $\dfrac{x^2 - 16}{x^2 + 3x - 4}$ .

Step 1 — Factorise the top

The top is a difference of two squares:

$x^2 - 16 = (x - 4)(x + 4)$

Step 2 — Factorise the bottom

Two numbers multiplying to $-4$ and adding to $+3$ are $+4$ and $-1$ :

$x^2 + 3x - 4 = (x + 4)(x - 1)$

Step 3 — Cancel the common bracket

$\frac{(x - 4)(x + 4)}{(x + 4)(x - 1)} = \frac{x - 4}{x - 1}$

Worked ExampleSimplifying a quotient of fractions

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

Step 1 — Factorise every quadratic

$\frac{(x - 2)(x + 2)}{(x - 3)(x + 2)} \div \frac{x + 2}{x - 3}$

Step 2 — Flip the second fraction and multiply

$\frac{(x - 2)(x + 2)}{(x - 3)(x + 2)} \times \frac{x - 3}{x + 2}$

Step 3 — Cancel common brackets

Cancel $(x + 2)$ and $(x - 3)$ where they appear top and bottom:

$= \frac{x - 2}{x + 2}$

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

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

Merit

Simplify $\dfrac{x^2 - 16}{x^2 + 3x - 4}$ .

Excellence

Simplify $\dfrac{2x^2 + 5x - 3}{x^2 - 9} \times \dfrac{x + 3}{2x - 1}$ .

Simplifying algebraic fractions

Covers AS91261

What an algebraic fraction is

An algebraic fraction is a fraction with algebra on the top (numerator) and/or bottom (denominator), like $\dfrac{x^2 - 9}{x^2 + 5x + 6}$ .
Simplifying means writing it in its lowest terms by cancelling anything common to top and bottom.
Cancelling is really dividing the top and bottom by the same factor — which is allowed because it does not change the value (you are dividing by 1).

Simplifying by factorising and cancelling

Factorise the top and the bottom fully into brackets.
Cancel any factor (a whole bracket, or a whole term) that appears on both top and bottom.
Golden rule: you can only cancel factors (things multiplied), never individual terms inside a sum.

$\frac{x^2 - 9}{x^2 + 5x + 6} = \frac{(x - 3)(x + 3)}{(x + 2)(x + 3)} = \frac{x - 3}{x + 2}$

Multiplying and dividing algebraic fractions

Multiply: factorise everything, cancel any common factors between the tops and bottoms, then multiply the tops and the bottoms.
Divide: flip the second fraction (its reciprocal) and multiply — flip only the fraction after the $\div$ sign.

Worked ExampleSimplifying a single fraction

Simplify $\dfrac{x^2 - 16}{x^2 + 3x - 4}$ .

Step 1 — Factorise the top

The top is a difference of two squares:

$x^2 - 16 = (x - 4)(x + 4)$

Step 2 — Factorise the bottom

Two numbers multiplying to $-4$ and adding to $+3$ are $+4$ and $-1$ :

$x^2 + 3x - 4 = (x + 4)(x - 1)$

Step 3 — Cancel the common bracket

$\frac{(x - 4)(x + 4)}{(x + 4)(x - 1)} = \frac{x - 4}{x - 1}$

Worked ExampleSimplifying a quotient of fractions

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

Step 1 — Factorise every quadratic

$\frac{(x - 2)(x + 2)}{(x - 3)(x + 2)} \div \frac{x + 2}{x - 3}$

Step 2 — Flip the second fraction and multiply

$\frac{(x - 2)(x + 2)}{(x - 3)(x + 2)} \times \frac{x - 3}{x + 2}$

Step 3 — Cancel common brackets

Cancel $(x + 2)$ and $(x - 3)$ where they appear top and bottom:

$= \frac{x - 2}{x + 2}$

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

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

Merit

Simplify $\dfrac{x^2 - 16}{x^2 + 3x - 4}$ .

Excellence

Simplify $\dfrac{2x^2 + 5x - 3}{x^2 - 9} \times \dfrac{x + 3}{2x - 1}$ .