- An algebraic fraction is a fraction with an algebraic expression on the top (numerator) and/or the bottom (denominator)

**Factorise fully**top and bottom**Cancel**common factors (including common brackets)

- If you are asked to simplify an algebraic fraction and have to factorise the top or bottom, it is very likely that one of the factors will be the same on the top and the bottom – you can use this to help you factorise difficult quadratics!

Simplify

Factorise the top, by using 2 as a common factor

Factorise the bottom using your preferred method

Using the fact that the top factorised to may help!

The common factors on the top and bottom reduce to 1 (cancel out)

- The rules are the same as fractions with numbers:

- Find the
**lowest common denominator**(LCD)

- The LCD of
*x -*2 and*x*+ 5 is found by multiplying them together: LCD = (*x*- 2)(*x*+ 5)- this is the same as with numbers, where the LCD of 2 and 9 is 2 × 9 = 18

- The LCD of
*x*and 2*x*is not found by multiplying them together, as 2*x*already includes an*x*^{ }, so the LCD is just 2*x*- this is the same as with numbers, where the LCD of 2 and 4 is just 4, not 2 × 4 = 8

- The LCD of
*x*+ 2 and (*x*+ 2)(*x*- 1) is just (*x*+ 2)(*x*- 1), as this already includes an (*x*+ 2) - The LCD of
*x*+ 1 and (*x*+ 1)^{2}is just (*x*+ 1)^{2}, as this already includes an (*x*+ 1) - The LCD of (
*x*+ 3)(*x*- 1) and (*x*+ 4)(*x*- 1) is three brackets: (*x*+ 3)(*x*- 1)(*x*+ 4), without repeating the (*x*- 1)

- The LCD of
- Write each fraction over this lowest common denominator
**Multiply the numerators**of each fraction by the**same amount**as the denominators- Write as a
**single fraction**over the lowest common denominator (by**adding**or**subtracting**the**numerators**, taking care to use brackets when subtracting) **Check**at the end to see if the top factorises and cancels

- Leaving the top and bottom of the fraction in factorised form will help you see if anything cancels at the end.

(a) Express as a single fraction

The lowest common denominator is

Write each fraction over this common denominator, remember to multiply the top of the fractions too

Simplify the numerators

Combine the fractions, as they have the same denominator

Factorise the top

There are no terms which would cancel here, so this is the final answer

(b) Express as a single fraction

The lowest common denominator is (You could also use but this wouldn't be the *lowest* common denominator)

Write each fraction over this common denominator, remember to multiply the top of the fractions too

Simplify the numerators

Combine the fractions, as they have the same denominator

There is nothing else that can be factorised on the numerator, so this is the final answer

**Simplify**both fractions first by**fully factorising**, then**cancelling**any common brackets on top or bottom (from either fraction)- Multiply the
**tops**together - Multiply the
**bottoms**together **Check**for any further factorising and cancelling

**Flip**("reciprocate") the**second**fraction and replace ÷ with ×- So becomes

- Then follow the same rules for
**multiplying**two fractions

Divide by , giving your answer as a simplified fraction

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

It can often help to factorise first, as there may be factors that cancel out

Multiply the numerators and denominators, and cancel any terms that are the same on the top and bottom

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