8.4 Building rational expressions and the lcd

Overview

• The Process
• The Reason For Building Rational Expressions
• The Least Common Denominator (LCD)

The process

Recall, from Section [link] , the equality property of fractions.

Equality property of fractions

Using the fact that $1=\frac{b}{b},b\ne 0$ , and that 1 is the multiplicative identity, it follows that if $\frac{P}{Q}$ is a rational expression, then

$\begin{array}{cc}\frac{P}{Q}·\frac{b}{b}=\frac{Pb}{Qb},& b\ne 0\end{array}$

This equation asserts that a rational expression can be transformed into an equivalent rational expression by multiplying both the numerator and denominator by the same nonzero number.

Process of building rational expressions

Suppose we're given a rational expression $\frac{P}{Q}$ and wish to build it into a rational expression with denominator $Q{b}^2$ , that is,

$\frac{P}{Q}\to \frac{?}{Q{b}^2}$

Since we changed the denominator, we must certainly change the numerator in the same way. To determine how to change the numerator we need to know how the denominator was changed. Since one rational expression is built into another equivalent expression by multiplication by 1, the first denominator must have been multiplied by some quantity. Observation of

$\frac{P}{Q}\to \frac{?}{Q{b}^2}$

tells us that $Q$ was multiplied by $b^2$ . Hence, we must multiply the numerator $P$ by $b^2$ . Thus,

$\frac{P}{Q}=\frac{P{b}^2}{Q{b}^2}$

Quite often a simple comparison of the original denominator with the new denominator will tell us the factor being used. However, there will be times when the factor is unclear by simple observation. We need a method for finding the factor.