Section 3.5 Rational Expressions
A rational expression is a fraction involving polynomials, such as
\begin{equation*}
\frac{4x^2-3x+2}{(4-x)^2}.
\end{equation*}
Being able to perform algebraic operations on these and similar expressions is essential to success in calculus. Because division is involved, the domain of the expression is restricted all values of the variable that do not result in division by zero. In the case of the example above, we must restrict the domain to \(x\neq 4\text{,}\) or in interval notation \((-\infty,4)\cup(4,+\infty)\text{.}\)
Example 3.15. Cancellation.
You may cancel common factorsin a fraction
\begin{equation*}
\frac{8x^2-4xy}{4x^2} = \frac{4x(2x-y)}{4x^2} = \frac{4x(2x-y)}{4x\cdot x} = \frac{2x-y}{x}.
\end{equation*}
But you must never cancel common terms
\begin{equation*}
\frac{y+4x}{4x} \quad \text{is not} \quad y + \frac{4x}{4x} = y+1.
\end{equation*}
Although the following is correct
\begin{equation*}
\frac{y+4x}{4x} = \frac{y}{4x} + \frac{4x}{4x} = \frac{y}{4x} + 1.
\end{equation*}
Example 3.16. Finding Domains of Rational Expressions.
Find the domain of each rational expression. Video solutions follow.
\(\displaystyle \frac{4x^3-2x^2+1}{2x^2+5}\)
\(\displaystyle \frac{2x^3+8x^2}{2x^3+6x^2-8x}\)
Example 3.17. Multiplication.
To multiply rational expressions, simply multiply their numerators and denominators together.
\begin{equation*}
\frac{2x}{3y} \cdot \frac{z^2}{y} = \frac{(2x)(z^2)}{(3y)(y)} = \frac{2xz^2}{3y^2}.
\end{equation*}
Example 3.18. Multiplication.
If the numerator or denominator includes a sum or difference, then be sure to distribute carefully.
\begin{equation*}
\frac{-3}{2x} \cdot \frac{x-2}{x+2} = \frac{-3(x-2)}{2x(x+2)} = \frac{-3x+6}{2x^2+4x}.
\end{equation*}
Example 3.19. Addition.
To add two rational expressions, you must first find a common denominator by providing missing factors to each term.
\begin{align*}
\frac{2}{x} + \frac{3}{x-1}
&= \frac{2}{x} \cdot \frac{x-1}{x-1} + \frac{3}{x-1}\cdot \frac{x}{x}\\
&= \frac{2(x-1)}{x(x-1)} + \frac{3x}{(x-1)x}\\
&= \frac{2(x-1)+3x}{x(x-1)}\\
&= \frac{2x-2+3x}{x(x-1)}\\
&= \frac{5x-2}{x(x-1)}\\
&= \frac{5x-2}{x^2-x}
\end{align*}
Example 3.20. Subtraction.
Subtraction is similarly to addition, but remember to distribute the sign.
\begin{align*}
\frac{4}{x^2-1}-\frac{2}{x+1}
&= \frac{4}{(x-1)(x+1)} - \frac{2}{x+1}\\
&= \frac{4}{(x-1)(x+1)} - \frac{2}{x+1}\cdot \frac{x-1}{x-1}\\
&= \frac{4}{(x-1)(x+1)} - \frac{2(x-1)}{(x-1)(x+1)}\\
&= \frac{4-2(x-1)}{(x-1)(x+1)}\\
&= \frac{4-2x+2}{(x-1)(x+1)}\\
&= \frac{6-2x}{(x-1)(x+1)}\\
&= \frac{-2x+6}{x^2-1}
\end{align*}
Example 3.21. Simplifying Rational Expressions.
Perform the indicated operation and simplify. Video solutions follow.
Multiply \(\displaystyle \frac{-3}{x+2}\cdot \frac{4x+1}{x-2}\text{.}\)
Subtract \(\displaystyle \frac{-3}{x+2}-\frac{4x+1}{x-2}\text{.}\)
Add \(\displaystyle \frac{2x}{x^2-4}+\frac{3x}{2x+4}\text{.}\)
Example 3.22. Dividing Rational Expressions.
To divide fractions, you could instead multiply by the reciprocal of the denominator. For example, let’s perform the division
\begin{equation*}
\frac{3}{x-1} \div \frac{1}{x+1}.
\end{equation*}
We have
\begin{align*}
\frac{3}{x-1} \div \frac{1}{x+1}
= \frac{\frac{3}{x-1}}{\frac{1}{x+1}}
&= \frac{3}{x-1}\cdot \frac{x+1}{1}\\
&= \frac{3(x+1)}{x-1} \\
&= \frac{3x+3}{x-1}
\end{align*}
A compound rational expression is a rational expression involving multiple divisions within one ratio. We always want to simplify these into a simple rational expression.
Example 3.23. Dividing Rational Expressions.
You may multiply top and bottom by the appropriate expression and distribute it through to clear the fractions. This is particularly useful when simplifying a compound rational expressions such as
\begin{equation*}
\frac{1}{1+\left(\frac{x}{y}\right)^2}.
\end{equation*}
We have
\begin{align*}
\frac{1}{1+\left(\frac{x}{y}\right)^2} &= \frac{1}{1+\frac{x^2}{y^2}}\\
&= \frac{1}{1+\frac{x^2}{y^2}} \cdot \frac{y^2}{y^2}\\
&= \frac{y^2}{(1+\frac{x^2}{y^2})y^2}\\
&= \frac{y^2}{y^2+\frac{x^2}{y^2}\cdot y^2}\\
&= \frac{y^2}{y^2+x^2}
\end{align*}
Example 3.24. Simplifying Rational Expressions.
Perform the indicated operation and simplify. Video solutions follow.
\(\displaystyle \displaystyle \frac{x}{x-3}\div \frac{2}{3x+2}\)
\(\displaystyle \displaystyle \frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{xy}}\)
\(\displaystyle \displaystyle \frac{\frac{2}{x+1}-\frac{3}{x+2}}{x-1}\)
