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Chapter 7: Problem 79

Simplify each rational expression. $$\frac{9-y^{2}}{y^{2}-3(2 y-3)}$$

Short Answer

Expert verified
The simplified rational expression is \( \frac{-(y+3)}{(y-3)} \).

Step by step solution

01

Rewrite the expression

Rewrite the expression \( \frac{9-y^{2}}{y^{2}-3(2 y-3)} \) as \( \frac{9-y^{2}}{y^{2}-6y+9} \) by distributing the -3 in the denominator.
02

Rearrange the numerator

Refactor \(9 - y^{2}\) in the denominator and it becomes \(-(y^{2} - 9)\).
03

Factor the expressions

Apply the difference of squares to both the numerator and denominator. The rational expression is then: \( \frac{-(y+3)(y-3)}{(y-3)(y-3)} \).
04

Cancel common terms

Cancel the common term in the numerator and the denominator (y-3). The simplified rational expression is: \( \frac{-(y+3)}{(y-3)} \).

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Most popular questions from this chapter

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{x-2}{x^{2}-25}-\frac{x-2}{25-x^{2}}$$

If the ratio of the corresponding sides of two similar triangles is 1 to 1 ( \(\frac{1}{1}\) ), what must be true about the triangles?

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{x^{2}}{x-3}+\frac{9}{3-x}$$

Factor: \(25 x^{2}-81 .\) (Section 6.4, Example 1)

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{6 b^{2}-10 b}{16 b^{2}-48 b+27}+\frac{7 b^{2}-20 b}{16 b^{2}-48 b+27}-\frac{6 b-3 b^{2}}{16 b^{2}-48 b+27}$$
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