17.05 Simplifying algebraic fractions (Extended)
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.
Worked example
Simplify $\frac{4x^2-16x}{6x-24}$4x2−16x6x−24.
Think: To simplify the fraction we want to cancel any common factors in the numerator and denominator. To find these common factors, we first want to factorise the expressions in the numerator and denominator.
Do: In $4x^2-16x$4x2−16x there is a common factor of $2x$2x. In $6x-24$6x−24 there is a common factor of $3$3.
| $\frac{4x^2-16x}{6x-24}$4x2−16x6x−24 | $=$= | $\frac{2x\left(2x-8\right)}{3\left(2x-8\right)}$2x(2x−8)3(2x−8) |
Factorising the numerator and denominator |
| $=$= | $\frac{2x}{3}$2x3 |
Cancelling $\left(2x-8\right)$(2x−8) from the numerator and denominator |
Since there are no more common factors between $2x$2x and $3$3, we conclude that $\frac{4x^2-16x}{6x-24}=\frac{2x}{3}$4x2−16x6x−24=2x3.
Reflect: 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.
An algebraic fraction is a fraction where the numerator and denominator are algebraic expressions.
To simplify an algebraic fraction, first factorise the expressions in the numerator and denominator and then cancel any common factors in the numerator and denominator.
Practice questions
Question 1
Simplify: $\frac{64mn^2}{40m^2n}$64mn240m2n
Question 2
Factorise and simplify: $\frac{5u}{5uv-30uw}$5u5uv−30uw
Question 3
Factorise and simplify: $\frac{m^2-16}{m-4}$m2−16m−4