Algebraic fractions are fractions that include pronumerals in either the numerator, denominator or both. For example, the following terms are all examples of algebraic fractions:
$-\frac{2p}{3},\frac{5}{x-1},\frac{3m^2-5n^3}{mn^3+1}$−2p3,5x−1,3m2−5n3mn3+1
Factorisation and simplification
It is generally a good idea to simplify all fractions where possible before proceeding with any operation (addition, subtraction, multiplication or division). This is particularly important when we are dealing with algebraic fractions that involve binomial or quadratic expressions, as cancelling common factors can make these seemingly complicated expressions much easier to work with. We may need to use any or all of the factorising techniques from the previous lesson, so make sure you are familiar with them all. It is accepted practice to present final answers in factorised form.
Practice questions
Question 1
Simplify the rational expression $\frac{2r-8}{r^2-16}$2r−8r2−16
Sum and difference of algebraic fractions
The same rules apply to the sum and difference of numeric and algebraic fractions. A common denominator is required to add or subtract fractions. A quick way to find the common denominator is to multiply the denominators of each fraction, but just as with numbers, this may not be the lowest common denominator and hence you will probably have to do some factorising at a later stage.
Practice questions
Question 2
Simplify $\frac{4x}{7}-\frac{2x-3}{21}$4x7−2x−321.
Question 3
Simplify the following expression, giving your answer in fully factorised form:
$\frac{x}{x^2-16}-\frac{12}{x+4}$xx2−16−12x+4
Multiplication of algebraic fractions
To multiply algebraic fractions, we multiply numerators together to form the new numerator, and denominators together to form the new denominator. We also want to check for common factors that can be cancelled.
Practice questions
Question 4
Simplify the following expression:
$\frac{p+7}{5}\times\frac{5p-2}{p^2+14p+49}$p+75×5p−2p2+14p+49
Division of algebraic fractions
Dividing by an algebraic fraction is the same as multiplying by the reciprocal.
Practice questions
Question 5
Simplify $\frac{k-1}{20}\div\frac{k^2-10k+9}{4}$k−120÷k2−10k+94.
Compound fractions
A compound fraction is one where the numerator or denominator, or both, are themselves fractions. One way to deal with compound fractions is to expand the numerators or denominators and solve the problem as if it was a multi-step division problem, but there is a faster way! If we multiply each term in the overall fraction by the lowest common multiple of each term's denominator, we will simplify the expression to just a regular algebraic fraction which is usually easier to work with. Let's look at an example of this.
Worked example
Example 1
Simplify $\frac{1+\frac{a}{b}}{\frac{2b}{a}}$1+ab2ba.
Think: What is the lowest common multiple of the denominators of each term in the fraction? In this case, the denominator of the upper fraction is $b$b, and the denominator of the lower fraction is $a$a. So we can simplify this expression by multiplying the numerator and denominator of the overall fraction by $ab$ab.
Do:
Multiplying, we get:
| $\frac{1+\frac{a}{b}}{\frac{2b}{a}}$1+ab2ba | $=$= | $\frac{ab(1+\frac{a}{b})}{ab(\frac{2b}{a})}$ab(1+ab)ab(2ba) |
| $=$= | $\frac{ab+a^2}{2b^2}$ab+a22b2, |
Which we can also express as $\frac{a\left(b+a\right)}{2b^2}$a(b+a)2b2 by factorising the numerator.
Practice question
Question 7
Simplify the expression $\frac{\frac{3}{x}-\frac{1}{xy}}{\frac{1}{xy}+\frac{7}{y}}$3x−1xy1xy+7y.