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.

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

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