# Week 14 – Precalc 11

Some of the things we learned this week in Precalc 11 is how to Multiply and Divide Rational Expressions.

#### Multiplying Rational Expressions

When multiplying rational expressions, it’s better to simplify the expression as much as possible then multiply, rather than multiplying and getting really big numbers then simplifying.

##### Example 1

$\frac{x^2+5+6}{(x+3)(x-2)} \times \frac{2(x-2)}{x+5}$
The first step would be factoring any polynomial that can be factored.
$=\frac{(x+3)(x+2)}{(x+3)(x-2)} \times \frac{2(x-2)}{x+5}$
Since it’s fractions, the denominator cannot equal to 0, to set the restrictions, determine the zeros.
$x \neq -3, 2, -5$
Then cross out any that matches in the numerator and the denominator. Note that you cannot cancel out only the x in x + 5 in the denominator, the +5 also has to be included.
$=\frac{2(x+2)}{x+5}$

##### Example 2

$\frac{x^2-36}{x^2-3x-18} \times \frac{x^2+3x}{x+1}$
$=\frac{(x+6)(x-6)}{(x+3)(x-6)} \times \frac{x(x+3)}{x+1}$
$=\frac{x(x+6)}{x+1}, x \neq -3, 6, -1$

#### Dividing Rational Expressions

Dividing rational expressions is pretty similar to multiplying rational expressions.

##### Example 1

$\frac{x^2+11x+24}{x^2-15x+56} \div \frac{x^2-x-12}{(x-7)(x-4)}$
First step is the same as multiplying, factor.
$= \frac{(x+8)(x+3)}{(x-9)(x-6)} \div \frac{(x-4)(x+3)}{(x-7)(x-4)}$
Then we can determine the zeros.
$x \neq 9, 6, 7, 4$
Next, we flip the right side of the expression to get the reciprocal because we’re dividing. When we get the reciprocal we also have to add the zeros for the new denominator.
$= \frac{(x+8)(x+3)}{(x-7)(x-8)} \times \frac{(x-7)(x-4)}{(x-4)(x+3)}$
$x \neq 9, 6, 7, 4, -3$
Then we can cancel out pairs from the numerator and the denominator.
$=\frac{x+8}{x-8}, x \neq 9, 6, 7, 4, -3$

##### Example 2

$\frac{(x+4)(x+6)}{x^2-5x+6} \div \frac{x^2+11x+28}{(x-3)^2} \times \frac{(x+7)(x-6)}{x^2+3x-18}$
$=\frac{(x+6)(x+4)}{(x-3)(x-2)} \div \frac{(x+7)(x+4)}{(x-3)(x-3)} \times \frac{(x+7)(x-6)}{(x+6)(x-3)}, x \neq 3, 2, -6$
$=\frac{(x+6)(x+4)}{(x-3)(x-2)} \times \frac{(x-3)(x-3)}{(x+7)(x+4)} \times \frac{(x+7)(x-6)}{(x+6)(x-3)}, x \neq 3, 2, -6, -7, -4$
$=\frac{x-6}{x-2}, x \neq 3, 2, -6, -7, -4$