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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
The simplified result is \(x^2+6x+9\).

Step by step solution

01

Factorize the polynomials

First, four given polynomials must be factorized, if possible. \n\The factorization of the polynomial \(x^{2}+5 x+6\) is \((x+2)(x+3)\), the polynomial \(x^{2}+x-6\) is \((x-3)(x+2)\), and the polynomial \(x^{2}-9\) is a difference of squares and can be written as \((x-3)(x+3)\). The polynomial \(x^{2}-x-6\) again is \((x-3)(x+2)\).
02

Substitute the factorized polynomials into the expression

Replace the original polynomials in the given expression with their factorized counterparts: So we will get \(\frac{(x+2)(x+3)}{(x-3)(x+2)} * \frac{(x-3)(x+3)}{(x-3)(x+2)}\)
03

Simplify the Expression

Now we can cancel out the common factors from the numerator and denominator: The \(x+2\) and \(x-3\) will cancel themselves out and you'll be left with \((x+3)(x+3)\). Therefore, now the expression simplifies as \((x+3)^2\) or \(x^2+6x+9\).

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Most popular questions from this chapter

Give an example of a rational number that is not an integer.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Although \(20 x^{3}\) appears in both \(20 x^{3}+8 x^{2}\) and \(20 x^{3}+10 x\) I'll need to factor \(20 x^{3}\) in different ways to obtain each polynomial's factorization

Your local electronics store is having an end-of-the-year sale. The price on a plasma television had been reduced by \(30 \% .\) Now the sale price is reduced by another \(30 \%\). If \(x\) is the television's original price, the sale price can be modeled by $$(x-0.3 x)-0.3(x-0.3 x)$$ a. Factor out \((x-0.3 x)\) from each term. Then simplify the resulting expression. b. Use the simplified expression from part (a) to answer these questions. With a \(30 \%\) reduction followed by a \(30 \%\) reduction, is the television selling at \(40 \%\) of its original price? If not, at what percentage of the original price is it selling?

Evaluate each algebraic expression for the given value or values of the variable(s). $$\frac{2 x+y}{x y-2 x}, \text { for } x=-2 \text { and } y=4$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. . You grouped the polynomial's terms using different groupings than I did, yet we both obtained the same factorization.
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