Rational Expression
In short: A rational expression is a fraction whose top and bottom are both polynomials, like (x + 2)/(x^2 – 9), and it only makes sense for values that keep the bottom from being zero.
Algebra 2AlgebraA-APR.D.6
What is Rational Expression?
A rational expression is the algebra version of a fraction: instead of whole numbers on the top and bottom, you have polynomials. Just as \(\frac{3}{4}\) cannot have a zero denominator, these expressions are off-limits at any input that makes the bottom equal to zero.
In symbols, it has the form \(\frac{P(x)}{Q(x)}\) where \(P\) and \(Q\) are polynomials and \(Q\) is not the zero polynomial. The values of \(x\) with \(Q(x) = 0\) are the excluded values, and they stay excluded even after common factors are cancelled.
How to spot it
- A rational expression shows up whenever a fraction contains letters and powers, such as (x + 5)/(x^2 – 4), instead of plain numbers.
- The directions say simplify, add, multiply or divide fractions that have variables in the denominator.
- A problem asks which values of x must be excluded or where the expression is undefined.
- Factoring appears first, because only factors, never separate terms, can be cancelled.
The method
- Factor the numerator and the denominator completely.
- Write down every value that makes a factor of the original denominator zero: these are the restrictions.
- Cancel factors that appear on both the top and the bottom.
- Present the simpler fraction and keep the restrictions next to it.
Rational Expression: solved example
Problem
Simplify \(\frac{x^2 – 49}{x^2 + 3x – 28}\) and state the values of \(x\) that are not allowed.
Solution
Factor the top as a difference of squares: \(x^2 – 49 = (x – 7)(x + 7)\). Factor the bottom: \(x^2 + 3x – 28 = (x + 7)(x – 4)\).
The bottom is zero when \(x = -7\) or \(x = 4\), so those two values are excluded. Cancel the shared factor \(x + 7\): the result is \(\frac{x – 7}{x – 4}\), with \(x \neq -7\) and \(x \neq 4\).
Try it
Question 1
Which values of \(x\) are excluded for \(\frac{5x}{x^2 – 6x + 8}\)?
Show the answer
The bottom factors as \((x – 2)(x – 4)\), so \(x = 2\) and \(x = 4\) are excluded.
Question 2
Simplify \(\frac{3x^2 – 12x}{6x}\).
Show the answer
The top is \(3x(x – 4)\). Cancel \(3x\) to get \(\frac{x – 4}{2}\), with \(x \neq 0\).
Rational Expression vs. Rational Function
An expression is just a piece of algebra you simplify or combine; nothing is graphed. A rational function uses that same fraction as a rule \(f(x) = \frac{P(x)}{Q(x)}\) that sends each allowed input to one output, so it has a graph, a domain, intercepts and asymptotes.
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