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Chapter 0: Problem 20

Multiply or divide as indicated. $$\frac{x^{2}+5 x+6}{x^{2}+x-6} \cdot \frac{x^{2}-9}{x^{2}-x-6}$$

Short Answer

Expert verified
So the result of \(\frac{x^{2}+5 x+6}{x^{2}+x-6} \cdot \frac{x^{2}-9}{x^{2}-x-6}\) is \(x^{2}-x-6\).

Step by step solution

01

Factorize the Polynomials

Factorize each polynomial. The numerator \(x^{2}+5 x+6\) can be written as \((x+2)(x+3)\), and the denominator \(x^{2}+x-6\) as \((x+3)(x-2)\). Then, the numerator \(x^{2}-9\) can be written as \((x+3)(x-3)\), and similarly, the denominator \(x^{2}-x-6\) can be written as \((x+3)(x-2)\).
02

Insert the Factored Expressions into the Original Problem

The original problem can be rewritten using the factored expressions as follows: \(\frac{(x+2)(x+3)}{(x+3)(x-2)} \cdot \frac{(x+3)(x-3)}{(x+3)(x-2)}\).
03

Cancel out Common Factors

Cancellation of terms takes place when the same term appears in both the numerator and denominator. Here, we can cancel out \((x+3)\) and \((x-2)\) from the numerator and denominator. This leaves us with \((x+2) \cdot (x-3)\).
04

Multiply The Remaining Terms

Finally, the multiplication of the remaining terms gives \(x^{2}-x-6\).

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \(a x^{2}+c=0, a \neq 0,\) cannot be solved by the quadratic formula.

Describe the solution set of \(|x|>-4\)

Perform the indicated operations. Simplify the result, if possible. $$\frac{a b}{a^{2}+a b+b^{2}}+\left(\frac{a c-a d-b c+b d}{a c-a d+b c-b d} \div \frac{a^{3}-b^{3}}{a^{3}+b^{3}}\right)$$

This will help you prepare for the material covered in the next section. Evaluate $$\frac{-b-\sqrt{b^{2}-4 a c}}{2 a}$$ for \(a=2, b=9,\) and \(c=-5\)

Explain how to determine the restrictions on the variable for the equation $$ \frac{3}{x+5}+\frac{4}{x-2}=\frac{7}{x^{2}+3 x-6} $$
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