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Chapter 7: Problem 27

Multiply as indicated. $$\frac{25-y^{2}}{y^{2}-2 y-35} \cdot \frac{y^{2}-8 y-20}{y^{2}-3 y-10}$$

Short Answer

Expert verified
The result of the multiplication of the fractions given in the problem statement, after simplification, is \( \frac{(5-y)(y-10)}{(y-7)(y-5)} \).

Step by step solution

01

Factorize the Polynomials

Factorize every polynomial to its simplest form. The polynomials \(25-y^{2}\), \(y^{2}-2 y-35\), \(y^{2}-8 y-20\) , and \(y^{2}-3 y-10\) should be factorized as follows: \(25-y^{2} = (5+y)(5-y)\) , \(y^{2}-2 y-35 = (y-7)(y+5)\), \(y^{2}-8 y-20 = (y-10)(y+2)\) and \(y^{2}-3 y-10 = (y-5)(y+2)\).
02

Insert the factorized polynomials into the fractions

Replace the polynomials in the original fractions with their factorized form: \(\frac{(5+y)(5-y)}{(y-7)(y+5)} \cdot \frac{(y-10)(y+2)}{(y-5)(y+2)}\). You should always make sure that you are not dividing by zero, which would make the function undefined.
03

Simplify the fractions

Now, observe both fractions, looking for common factors in the numerators and denominators across the fractions. You should find that factors \( (y+2) \) and \( (y+5) \) are common and cancel these out. The simplified equation becomes: \( \frac{(5-y)(y-10)}{(y-7)(y-5)} \)

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Most popular questions from this chapter

What is a proportion? Give an example with your description.

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{y}{a x+b x-a y-b y}-\frac{x}{a x+b x-a y-b y}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{6}{x-5}+\frac{2}{5-x}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{10}{x+3}-\frac{2}{-x-3}$$

perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\text { Simplify: } \frac{(y+2) y-2 \cdot 4}{4 y(y+4)}$$
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