An algebraic fraction is a fraction where the numerator and denominator are algebraic expressions. To simplify a numeric fraction we cancel any common factors in the numerator and denominator. We use the same method to simplify algebraic fractions.

Examples

Example 1

Simplify \dfrac{15x-10}{3x^3-2x^2}.

Worked Solution

Create a strategy

Factorise the numerator and denominator and cancel out the common factors.

Apply the idea

\displaystyle \dfrac{15x-10}{3x^3-2x^2}	\displaystyle =	\displaystyle \frac{5\left(3x-2\right)}{x^2\left(3x-2\right)}	Factorise the numerator and denominator
	\displaystyle =	\displaystyle \frac{5}{x^2}	Cancel out 3x-2

Reflect and check

It's worth fully factorising the numerator and denominator first in case there are any common factors. In some cases we might have to use other factorisation techniques to do this.

Example 2

Factorise and simplify: \dfrac{50m^2+70mn}{80m^2}

Worked Solution

Create a strategy

Factorise the numerator and simplify.

Apply the idea

\displaystyle \dfrac{50m^2+70mn}{80m^2}	\displaystyle =	\displaystyle \frac{10m(5m+7n)}{80m^2}	Factorise the numerator
	\displaystyle =	\displaystyle \frac{10m(5m+7n)}{10m \times 8m}	Write the denominator as a multiple of 10m
	\displaystyle =	\displaystyle \frac{5m+7n}{8m}	Cancel out 10m

Example 3

Factorise and simplify: \dfrac{a^2-81}{9-a}

Worked Solution

Create a strategy

Factorise the numerator using difference of two squares.

Apply the idea

\displaystyle \dfrac{a^2-81}{9-a}

\displaystyle =

\displaystyle \dfrac{\left(a-9\right)\left(a+9\right)}{9-a}

Factorise the numerator

By factorising -1 out of the denominator, we can write the denominator in terms of a common factor of (a-9) which we can then cancel out.

\displaystyle \dfrac{a^2-81}{9-a}	\displaystyle =	\displaystyle \dfrac{\left(a-9\right)\left(a+9\right)}{-1(a-9)}	Factorise -1 out of the denominator
	\displaystyle =	\displaystyle \dfrac{\left(a+9\right)}{-1}	Cancel out (a-9)
	\displaystyle =	\displaystyle -a-9	Divide both terms by -1

Idea summary

An algebraic fraction is a fraction where the numerator and denominator are algebraic expressions.

To simplify a numeric fraction we cancel any common factors in the numerator and denominator. We use the same method to simplify algebraic fractions.

Outcomes

MA5.2-6NA

simplifies algebraic fractions, and expands and factorises quadratic expressions

MA5.3-5NA

selects and applies appropriate algebraic techniques to operate with algebraic expressions

1.09 Simplifying algebraic fractions

Ideas

Simplify algebraic fractions

Examples

Example 1

Example 2

Example 3

Outcomes

MA5.2-6NA

MA5.3-5NA

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