Chapter 11.2, Problem 8LC

To determine

Write how the multiplying rational expressions and multiplying numerical fractions are similar and how they are different.

Answer to Problem 8LC

Explanation of Solution

Given:

How are multiplying rational expressions and multiplying numerical fractions similar? How are they different?

Concept Used:

Just as you can multiply and divide fractions, you can multiply and divide rational expressions. A fraction is called that contains a polynomial as the numerator, denominator, or both.

In fact, you use the same processes for multiplying and dividing rational expressions as you use for multiplying and dividing numeric fractions.

Example:  $\frac{15}{22}·\frac{33}{50}$  You could of course multiply 15 and 33, then 22 and 50 and simplify the result.  $\frac{15}{22}·\frac{33}{50}=\frac{495}{1100}=\frac{9}{20}$

In practice it is easier to factor the numerators and denominators and "cancel" common factors.

  $\frac{15}{22}·\frac{33}{50}=\frac{5·3}{2·11}·\frac{3·11}{5·10}=\frac{5}{5}·\frac{11}{11}·\frac{3·3}{2·10}=\frac{9}{20}$

 Note that we really noticed that  $\frac{5}{5},\frac{11}{11}$  were just 1, and multiplying by 1 doesn't affect the product.

Example: The rational expression:

You could multiply the numerators and denominators and end up with a quadratic over a quadratic, and then hope to simplify the result. The easiest way is to factor first:

  $\frac{x^2+3x+2}{x^2+7x+6}=\frac{(x+1)(x+2)}{(x+1)(x+6)}$

Now rewrite so that common factors are over each other and cancel the common factors, and then simply.

  $\frac{x^2+3x+2}{x^2+7x+6}=\frac{(x+1)(x+2)}{(x+1)(x+6)}=\frac{x+1}{x+1}·\frac{x+3}{x+6}=\frac{x+3}{x+6}$ .

The same rules apply in both cases: to multiply fractions or rational expressions the numerator of the answer is the product of the numerators and the denominator of the answer is the product of the denominators. To divide you multiply by the reciprocal.