# Simplifying algebraic fractions. Learn how to simplify a fraction which contains algebra (math help)

Updated on March 14, 2013 ## How To Simplify A Fraction Video

An algebraic can be simplified by factorising the numerator and denominator of the algebraic fraction. You will need to identify whether the expressions on the numerator and denominator can be factorised into a single bracket or a double bracket. If the expression is of the form ax² + bx + c then the expression can be factorised into a double bracket. With most algebraic fractions, the two expressions that have to be factorised will be two double brackets expressions, or a single and a double bracket expression.

Once the numerator and denominator have been factorised you will find that one of the factors will be the same on the numerator and denominator. These two factors can be cancelled out to leave your final answer. Before you start these examples make sure that you are confident with factorising expressions into single and double brackets.

Example 1

Simplify this algebraic fraction:

(x² + 3x)/(x² + 2x + 3)

First factorise x² + 3x into a single bracket x(x+3)

Secondly, factorise x² + 2x + 3 into a double bracket as (x+3)(x-1)

So the algebraic fraction can be written as:

[x(x+3)]/[(x-1)(x+3)]

Since you have x+3 on the numerator and denominator of the fraction then these can be cancelled out to give:

= x/(x-1)

Example 2

Simplify this algebraic fraction:

(x² - 3x – 10)/(x² - 8x + 15)

This time both expressions are double bracket expressions:

x² - 3x – 10 can be factorised to (x+2)(x-5)

x² - 8x + 15 can be factorised to (x-3)(x-5)

So the algebraic fraction can be written as:

[(x+2)(x-5)]/[(x-3)(x-5)]

Since you have x-5 on the numerator and denominator of the fraction then these can be cancelled out to give:

= (x+2)/(x-3)

Example 3

Simplify this algebraic fraction:

(x² - 49)/(x² +6x -7)

This time both expressions are double bracket expressions:

x² - 49 can be factorised to (x+7)(x-7). This expression is known as the difference between two square.

x² + 6x - 7 can be factorised to (x+ 7)(x-1)

So the algebraic fraction can be written as:

[(x+7)(x-7)]/[(x + 7)(x-1)]

Since you have x+ 7 on the numerator and denominator of the fraction then these can be cancelled out to give:

= (x-7)/(x-1)