Algebraic Fractions

Lessons

Senior 3 · Mathematics

Algebraic Fractions

5 lessons

Lesson 1 of 5

Algebraic fractions and when they exist

22 min

Sharing by an unknown number

A savings group in Ngoma shares a profit of 600,000 Frw equally among its members. With $n$ members, each receives

\frac{600\,000}{n} \ \text{Frw}.

This is an algebraic fraction: a fraction with a variable in it. With 8 members each gets 75,000 Frw; with 12 members, 50,000 Frw. But the expression makes no sense for $n = 0$ : nobody can share among zero people, and no number equals $600\,000 \div 0$ . Every algebraic fraction comes with such forbidden values, and finding them is always the first step. In S2 you learned to factorise polynomials; in this unit you use factorising on fractions.

Definition

Algebraic fraction (also called a rational expression): a fraction $\frac{P}{Q}$ whose numerator $P$ and denominator $Q$ are polynomials, where $Q$ is not the zero polynomial.

Notation: written with a fraction bar, for example $\frac{x + 1}{x^2 - 4}$ , or as $(x + 1) \div (x^2 - 4)$ in a line of text.

Example: $\frac{3}{x}$ , $\frac{2a}{5}$ and $\frac{x + 1}{x^2 - 4}$ are algebraic fractions. $\frac{\sqrt{x}}{2}$ is not, because $\sqrt{x}$ is not a polynomial.

Definition

Excluded value: a value of the variable that makes the denominator equal to zero. The fraction does not exist (is undefined) there. Stating the excluded values is called stating the conditions of existence.

Notation: " $x \ne 2$ ", or the domain written as $\mathbb{R} \setminus \{2\}$ (all real numbers except 2).

Example: $\frac{5}{x - 2}$ has the excluded value $x = 2$ , because $2 - 2 = 0$ .

Important

Rule. An algebraic fraction exists for every value of the variable except those that make its denominator zero. The numerator may be zero; then the fraction is simply 0.

Why division by zero is impossible: if $\frac{5}{0}$ were some number $k$ , then $5 = 0 \times k = 0$ , which is false. And $\frac{0}{0}$ cannot be given one value, because $0 = 0 \times k$ is true for every $k$ . So a zero denominator is never allowed, whatever the numerator.

Finding the excluded values

Set the denominator equal to zero.
Solve that equation, factorising if it is quadratic (the methods of the last unit).
Its solutions are the excluded values. If it has no real solution, the fraction exists for every real number.

The graph of $y = \frac{1}{x - 2}$ shows what an excluded value looks like: the curve breaks into two pieces and never touches the line $x = 2$ . Close to 2, the denominator is tiny, so the fraction is very large, positive on one side and negative on the other.

$y = \frac{1}{x - 2}$ has no point on the dashed line $x = 2$ .

Worked example 1: conditions of existence

State the excluded values of (a) $\frac{5}{x - 3}$ (b) $\frac{x + 2}{x^2 - 9}$ (c) $\frac{4x}{x^2 - 5x + 6}$ (d) $\frac{x}{x^2 + 4}$ (e) $\frac{1}{2x^2 + 3x - 2}$ .

Solution

(a) $x - 3 = 0$ when $x = 3$ . Condition: $x \ne 3$ .

(b) $x^{2} - 9 = (x - 3) (x + 3) = 0$ when $x = 3$ or $x = - 3$ . Condition: $x \ne \pm 3$ .

(d) $x^2 \ge 0$ , so $x^2 + 4 \ge 4$ and is never zero. No excluded values: the domain is $\mathbb{R}$ .

(e) $2 x^{2} + 3 x - 2 = (2 x - 1) (x + 2)$ . Conditions: $x \ne \frac{1}{2}$ and $x \ne -2$ .

Worked example 2: evaluating an algebraic fraction

Find the value of $\frac{x^2 - 1}{x + 3}$ when (a) $x = 2$ (b) $x = 1$ (c) $x = - 3$ .

Solution The excluded value is $x = - 3$ .

(a) $\frac{4 - 1}{2 + 3} = \frac{3}{5}$ . (b) $\frac{1 - 1}{1 + 3} = \frac{0}{4} = 0$ : a zero numerator is allowed. (c) The denominator is $- 3 + 3 = 0$ , so the fraction is undefined at $x = - 3$ .

Worked example 3: when the situation adds conditions

A bus travels 150 km at an average speed of $v$ kilometres per hour. Write the journey time as an algebraic fraction and state the values of $v$ it may take.

Solution Time $= \frac{\text{distance}}{\text{speed}} = \frac{150}{v}$ hours. Algebra alone excludes only $v = 0$ . The situation excludes more: a speed must be positive, so $v \gt 0$ .

Warning

Look at the denominator only, and look before you simplify. The value that makes the numerator zero is not excluded; it makes the fraction 0. And find the excluded values from the fraction exactly as it is given: in the next lesson you will cancel factors, and a cancelled factor still leaves its excluded value behind.

Tip

Write the conditions straight away. As soon as you see an algebraic fraction, write its conditions next to it, such as " $x \ne \pm 3$ ". You will need them at the end of every simplification and every equation.

Mathispower4u: Ex: Find Values Where a Rational Expression is Undefined (4 min). Three examples: watch how the denominator is set equal to zero, factorised and solved, while the numerator is ignored.

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Practice

The last three questions are harder.

State the excluded value of $\frac{7}{x + 5}$ .
State the excluded values of $\frac{x - 1}{x^2 - 16}$ .
State the excluded values of $\frac{3x}{x^2 + x - 12}$ .
Does $\frac{2}{x^2 + 9}$ have any excluded values? Explain.
State the excluded values of $\frac{x + 1}{3x^2 - 7x + 2}$ .
Evaluate $\frac{x^2 - 4}{x + 1}$ at $x = 3$ and at $x = - 2$ . Why can you not evaluate it at $x = - 1$ ?
For which value of $x$ is $\frac{x - 4}{x + 2}$ equal to zero?
Find $k$ if $\frac{5}{x^2 - 6x + k}$ has exactly one excluded value, and state that value.

Show answersHide answers

$x \ne -5$
$x^{2} - 16 = (x - 4) (x + 4)$ : $x \ne \pm 4$ .
$x^{2} + x - 12 = (x + 4) (x - 3)$ : $x \ne -4$ and $x \ne 3$ .
No: $x^2 + 9 \ge 9$ for every real $x$ , so the denominator is never zero.
$3 x^{2} - 7 x + 2 = (3 x - 1) (x - 2)$ : $x \ne \frac{1}{3}$ and $x \ne 2$ .
$\frac{9 - 4}{4} = \frac{5}{4}$ ; $\frac{0}{-1} = 0$ ; at $x = - 1$ the denominator is 0.
$x = 4$ : the numerator is 0 and the denominator is $6 \ne 0$ .
One excluded value means $x^{2} - 6 x + k = 0$ has a repeated root: $\Delta = 36 - 4k = 0$ , so $k = 9$ , and the excluded value is $x = 3$ .

Sources and credits

8.1: Simplify Rational Expressions (values for which an expression is undefined), from Elementary Algebra by Lynn Marecek, MaryAnne Anthony-Smith and Andrea Honeycutt Mathis, OpenStax, provided by LibreTexts, licensed under CC BY 4.0 (licence read on the page, 30 September 2026). Adapted by Ganzaa.
The reason why division by zero is impossible, the graph of y = 1 ÷ (x − 2) and the worked examples. Written by Ganzaa.
Video: Ex: Find Values Where a Rational Expression is Undefined, Mathispower4u, embedded from YouTube.

The Rwandan worked examples, the diagram, the practice items and the quiz questions were written by Ganzaa.

1Sharing by an unknown number

2Finding the excluded values

Sharing by an unknown number

Finding the excluded values