Multiplying & Dividing Algebraic Fractions

Exam code: 4024

Written by: Ashika|Reviewed by: Caroline Carroll|Updated 2 July 2026

Multiplying & Dividing Algebraic Fractions

Multiplying & dividing algebraic fractions

How do I multiply algebraic fractions?

STEP 1
Simplify both fractions first by fully factorising
- E.g. $\frac{x}{3 x + 6} \times \frac{2 x + 4}{x + 7} = \frac{x}{3 (x + 2)} \times \frac{2 (x + 2)}{x + 7}$
STEP 2
Cancel any common factors on top and bottom (from either fraction)
- E.g. $\frac{x}{3 (x + 2)} \times \frac{2 (x + 2)}{x + 7} = \frac{x}{3} \times \frac{2}{x + 7}$
STEP 3
Multiply the tops together
Multiply the bottoms together
- E.g. $\frac{2 x}{3 (x + 7)}$
STEP 4
Check for any further factorising and cancelling
- E.g. $\frac{2 x}{3 (x + 7)}$ has no common factors so is in its simplest form

Worked example

Write the following as a single fraction in its simplest form

$\frac{2 x^{2} - 11 x + 12}{x^{2} + 3 x - 28} \times \frac{x^{2} - 5 x - 14}{9 - 4 x^{2}}$

Answer:

Factorise the four quadratics

$2 x^{2} - 11 x + 12$

Two numbers that multiply to give 2×12=24 and add to give -11
-3 and -8

$2 x^{2} - 3 x - 8 x + 12 x (2 x - 3) - 4 (2 x - 3) (2 x - 3) (x - 4)$

$x^{2} + 3 x - 28$

Two numbers that multiply to give -28 and add to give 3
7 and -4

$(x + 7) (x - 4)$

$x^{2} - 5 x - 14$

Two numbers that multiply to give -14 and add to give -5
2 and -7

$(x + 2) (x - 7)$

$9 - 4 x^{2}$

Difference of two squares
$3^{2} - {(2 x)}^{2}$

$(3 - 2 x) (3 + 2 x)$

Write the expression using the factors

$\frac{(2 x - 3) (x - 4)}{(x + 7) (x - 4)} \times \frac{(x + 2) (x - 7)}{(3 - 2 x) (3 + 2 x)}$

Rewrite $2 x - 3$ as $- (3 - 2 x)$ and cancel common factors

$\frac{- (3 - 2 x) (x - 4)}{(x + 7) (x - 4)} \times \frac{(x + 2) (x - 7)}{(3 - 2 x) (3 + 2 x)} \frac{- 1}{(x + 7)} \times \frac{(x + 2) (x - 7)}{(3 + 2 x)}$

Write as a single fraction

$(x + 7)$ and $(x - 7)$ are not common factors
You can multiply $(x - 7)$ by -1 to get rid of the negative sign in front of the front
Leave the expressions factorised

$\frac{- (x + 2) (x - 7)}{(x + 7) (3 + 2 x)} or \frac{(x + 2) (7 - x)}{(x + 7) (3 + 2 x)}$

How do I divide algebraic fractions?

Flip (find the reciprocal of) the second fraction and replace ÷ with ×
- So $\div \frac{a}{b}$ becomes $\times \frac{b}{a}$
- E.g. $\frac{3 x - 12}{x} \div \frac{2 x + 8}{x + 3} = \frac{3 x - 12}{x} \times \frac{x + 3}{2 x + 8}$

Then follow the same rules for multiplying two fractions

Worked example

Divide $\frac{x + 3}{x - 4}$ by $\frac{2 x + 6}{x^{2} - 16}$ , giving your answer as a simplified fraction.

Answer:

Division is the same as multiplying by the reciprocal (the fraction flipped)

$\frac{x + 3}{x - 4} \div \frac{2 x + 6}{x^{2} - 16} = \frac{x + 3}{x - 4} \times \frac{x^{2} - 16}{2 x + 6}$

Factorise all numerators and denominators to see which factors cancel out
You need to use the difference of two squares, $x^{2} - 4^{2} = (x - 4) (x + 4)$

$\frac{x + 3}{x - 4} \times \frac{x^{2} - 16}{2 x + 6} = \frac{x + 3}{x - 4} \times \frac{(x - 4) (x + 4)}{2 (x + 3)}$

Multiply the remaining numerators and denominators together

$\frac{1 \times (x + 4)}{1 \times 2} = \frac{x + 4}{2}$

Check to see if you missed any terms that are the same on the top and bottom that could be cancelled

$\frac{x + 4}{2}$ is already in its simplest form

$\frac{x + 4}{2}$

Multiplying & Dividing Algebraic Fractions

Multiplying & dividing algebraic fractions

How do I multiply algebraic fractions?

How do I divide algebraic fractions?

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