If the numerator and/or the denominator of a fraction is an algebraic expression, then the fraction is called an **algebraic fraction**.

**Examples**: $\dfrac{x+1}{x+2},\;\;\;\dfrac{y^2+xy-2}{x^2+xy-1},\;\;\;\dfrac{3x+3}{6},\;\;\;\dfrac{x^2+2x+1}{x^2+3x+2}$

Algebraic fractions do not change when we multiply both the numerator and denominator by the same non-zero expression.

**Examples**:

\begin{align} \dfrac{y}{y^2+1}&=\dfrac{y \times (y^2+1)}{(y^2+1)\times (y^2+1)}=\dfrac{y(y^2+1)}{(y^2+1)^2}\\ \dfrac{x}{x+1}&=\dfrac{x \times (x-1)}{(x+1)\times (x-1)}=\frac{x^2-x}{x^2-1}\\ \end{align} In the last example we require that $x \neq \pm 1$ as otherwise the fraction is **undefined**.

We simplify algebraic fractions in the same way that we simplify arithmetic fractions. However when simplifying algebraic fractions we must always be aware of the possibility of dividing by zero.

Simplify $\dfrac{x^2+3x+2}{x^2+4x+3}$.

\begin{align} \dfrac{x^2+3x+2}{x^2+4x+3}&=\dfrac{(x+2)(x+1)}{(x+3)(x+1)}\\ &=\dfrac{x+2}{x+3},\;\;x \neq -1 \end{align} On cancelling the common factor $x+1$. Note that we are careful to point out the value of $x$ at which the expression is not defined.

We add and subtract algebraic fractions in the same way we do with arithmetic fractions.

**Example**: \begin{align} \frac{y+z-xy}{y+z+1}+\frac{1+xy}{y+z+1}&=\dfrac{(y+z-xy)+(1+xy)}{y+z+1}\\ &=\dfrac{y+z+1}{y+z+1}\\ &=1,\;\;y+z+1 \neq 0 \end{align}

Write $\dfrac{x+1}{y}+\dfrac{y+1}{x}$ as a single algebraic fraction.

\begin{align} \dfrac{x+1}{y}+\dfrac{y+1}{x}&=\dfrac{(x+1)\times x+(y+1)\times y}{xy}\\ &=\dfrac{x^2+x+y^2+y}{xy} \end{align} The numerator and denominator have no common factors so the fraction is in its simplest form.

Write $\dfrac{x+y}{xy-x}-\dfrac{xy-1}{xy+x}$ as a single algebraic fraction.

\begin{align} \dfrac{x+y}{xy-x}-\dfrac{xy-1}{y-1}&=\dfrac{(x+y)(xy+x)}{(xy-x)(xy+x)}-\dfrac{(xy-1)(xy-x)}{(xy+x)(xy-x)}\\ &=\dfrac{(x+y)(xy+x)-(xy-1)(xy-x)}{(xy-x)(xy+x)}\\ &=\dfrac{x^2y+x^2+xy^2+xy-x^2y^2+x^2y+xy-x}{(xy-x)(xy+x)}\;\;\;\text{ expanding brackets in the numerator}\\ &=\dfrac{x(xy+x+y^2+y-xy^2+xy+y-1)}{x^2(y-1)(y+1)}\\ &=\dfrac{x(2xy+x+y^2+2y-xy^2-1)}{x^2(y-1)(y+1)}\;\;\;\text{ collecting like terms in the numerator}\\ &=\dfrac{2xy+x+y^2+2y-xy^2-1}{x(y-1)(y+1)}\;\;\;\text{ cancelling the common factor of }x\\ &=\dfrac{2xy+x+y^2+2y-xy^2-1}{x(y^2-1)}\;\;\;\text{ expanding brackets in the denominator} \end{align} The numerator and denominator have no common factors so the fraction is in its simplest form.

The rule for multiplying algebraic fractions is the same as that for multiplying arithmetic fractions: \[\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\]

Calculate $\dfrac{3x}{y} \times \dfrac{2xy^2}{z}$.

\begin{align} \frac{3x}{y} \times \frac{2xy^2}{z} &= \frac{3x \times 2xy^2}{y \times z}\\ \\ &=\frac{6x^2y^2}{yz}\\ \\ &=\frac{6x^2y}{z}\;\;\;\text{ (on cancelling the common factor of }y)\\ \\ \end{align} The numerator and denominator now have no common factors so the fraction is in its simplest form.

The rule for dividing algebraic fractions is the same as that for dividing arithmetic fractions: \[\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}.\]

Express $\dfrac{4}{1-2x} - 3$ as a single fraction.

First, we put everything over the same denominator $1-2x$, then expand the brackets in the numerator and simplify.

\begin{align} \frac{4}{1-2x} - 3 & = \frac{4}{1-2x} - \frac{3(1-2x)}{1-2x}\\\\ &=\frac{4-3(1-2x)}{1-2x}\\\\ &=\frac{4-3+6x}{1-2x}\\\\ &=\frac{1+6x}{1-2x} \end{align} $1+6x$ and $1-2x$ have no factors in common so the fraction is in its simplest form.

Prof. Robin Johnson calculates $\dfrac{7}{60}+\dfrac{2}{5}-\dfrac{5}{12}$ and simplifies $\dfrac{1+2x}{(x+1)(x-2)}+\dfrac{3x-2}{x^2-1}$.

Prof. Robin Johnson simplifies the combines the fractions $\dfrac{2}{1+x} - \dfrac{3}{2-x}$, into a single fraction.

Prof. Robin Johnson combines the fractions $\dfrac{x^3y^4}{z^5}\times\dfrac{z^3}{x^2y^7}$ and $\dfrac{x^2-1}{x+2}\times\dfrac{2x^2+3x-2}{x+1}$.

This workbook produced by HELM is a good revision aid, containing key points for revision and many worked examples.

Test yourself: Numbas test on adding and subtracting algebraic fractions

Test yourself: Numbas test on fractions

- An Algebra Refresher workbook at mathcentre.
- Simplifying Algebraic Expressions workbook at mathcentre.
- Fractions in Algebra at Maths is Fun.
- Fractions videos, worksheets and quizzes at BBC Skillswise.