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Chapter 0: Problem 31

Multiply or divide as indicated. $$\frac{x^{2}+x-12}{x^{2}+x-30} \cdot \frac{x^{2}+5 x+6}{x^{2}-2 x-3} \div \frac{x+3}{x^{2}+7 x+6}$$

Short Answer

Expert verified
\(\frac{x^{3}+7x^{2}+14x+8}{x+5}\)

Step by step solution

01

Factorize the given expressions

Factorize each numerator and denominator: \\[\frac{x^{2}+x-12}{x^{2}+x-30} = \frac{(x+4)(x-3)}{(x+6)(x-5)}, \]\[\frac{x^{2}+5x+6}{x^{2}-2x-3} = \frac{(x+2)(x+3)}{(x+1)(x-3)}, \]\[\frac{x+3}{x^{2}+7x+6} = \frac{x+3}{(x+1)(x+6)}\]
02

Rewrite the division operation as multiplication

Rewrite the division operation as multiplication by taking the reciprocal of the dividing term: \[\frac{(x+4)(x-3)}{(x+6)(x-5)} \cdot \frac{(x+2)(x+3)}{(x+1)(x-3)} \cdot \frac{(x+1)(x+6)}{x+3}\]
03

Cancel out common terms

In the multiplication operation of several fractions, cancel out same terms appearing both in numerators and denominators: \[\frac{(x+4)(x+2)(x+1)}{(x+5)}\]
04

Expand the product of the terms

Square out the terms in brackets and put the product together: \[\frac{x^{3}+7x^{2}+14x+8}{x+5}\]

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Most popular questions from this chapter

Place the correct symbol, \(>\) or \(<\), in the shaded area between the given numbers. Do not use a calculator. Then check your result with a calculator. \(a, 3^{\frac{1}{2}}\square 3^{1}\) b. \(\sqrt{7}+\sqrt{18} \square \sqrt{7+18}\)

Explain how to determine which numbers must be excluded from the domain of a rational expression.

In Exercises \(133-136\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$8^{-\frac{1}{4}}=-2$$

Perform the indicated operations. Simplify the result, if possible. $$\left(2-\frac{6}{x+1}\right)\left(1+\frac{3}{x-2}\right)$$

This will help you prepare for the material covered in the next section. Multiply and simplify: \(12\left(\frac{x+2}{4}-\frac{x-1}{3}\right)\).
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