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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
The simplified result of the given operation is \( \frac{(x+4)(x+2)(x+6)}{x+10} \).

Step by step solution

01

Factorize the Numerators and Denominators

Begin by factorizing each of the polynomials to find their roots. The factorized form of the polynomials are: \[ \frac{(x-3)(x+4)}{(x-3)(x+10)} \cdot \frac{(x+2)(x+3)}{(x-3)(x+1)} \div \frac{x+3}{(x+1)(x+6)}\] Each term in the original expression has been replaced with a factorized equivalent.
02

Multiply and Divide

Perform the multiplication and division as indicated: \[ \frac{(x-3)(x+4)(x+2)(x+3)}{(x-3)(x+10)(x-3)(x+1)} \div \frac{(x+3)}{(x+1)(x+6)} \] We can treat the division as a multiplication with the reciprocal, and then merge the multiplication: \[ \frac{(x-3)(x+4)(x+2)(x+3)(x+1)(x+6)}{(x-3)(x+10)(x-3)(x+1)(x+3)} \]
03

Simplify the Expression

Now eliminate the common factors from the numerator and the denominator. The factors \(x-3\), \(x+3\) and \(x+1\) can be eliminated: \[ \frac{(x+4)(x+2)(x+6)}{x+10} \] This is the simplified result.

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Most popular questions from this chapter

Simplify by reducing the index of the radical. $$\sqrt[6]{x^{4}}$$

Simplify using properties of exponents. $$\frac{\left(2 y^{\frac{1}{5}}\right)^{4}}{y^{\frac{3}{10}}}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ x^{2}+36-(x+6)^{2} $$

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. $$\text { a. } 3^{\frac{1}{2}} \square 3^{\frac{1}{3}}$$ $$\text { b. } \sqrt{7}+\sqrt{18} \square \sqrt{7}+18$$

Factor completely. $$ -x^{2}-4 x+5 $$
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