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Chapter 7: Problem 71

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{y^{2}-y-12}{4-y}$$

Short Answer

Expert verified
The simplified form of the rational expression \(\frac{y^{2}-y-12}{4-y}\) is \( -y + 3\). Be careful with the signs while simplifying.

Step by step solution

01

Factorization of the Numerator

Given the rational expression \(\frac{y^{2}-y-12}{4-y}\), begin by factorizing the numerator of the rational expression. The numerator \(y^{2}-y-12\) can be written in the form \(y^{2} -4y +3y -12\). This expression can be factored further by grouping, giving us \((y(y-4)+3(y-4)) = (y-4)(y+3)\). Hence the original expression becomes \(\frac{(y-4)(y+3)}{4-y}\).
02

Rewrite the Denominator

Notice that 4-y is the negative of y-4. Therefore, replace \(4-y\) with \(-1 * (y-4)\) to show this relationship. Hence, the rational expression becomes \(\frac{(y-4)(y+3)}{-1 * (y-4)}\).
03

Simplify the Expression

With the rational expression in this form, \((y-4)\) terms cancel off, and the result becomes \(-1 * y +3 = -y + 3\).

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Most popular questions from this chapter

Use a proportion to solve each problem. A person's hair length is proportional to the number of years it has been growing. After 2 years, a person's hair grows 8 inches. The longest moustache on record was grown by Kalyan Sain of India. Sain grew his moustache for 17 years. How long was each side of the moustache?

Add or subtract as indicated. Simplify the result, if possible. $$\frac{2}{x-1}+\frac{3}{x+2}$$

When two people work together to complete a job, describe one factor that can result in more or less time than the time given by the rational equations we have been using.

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

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{2 y}{y-5}-\left(\frac{2}{y-5}+\frac{y-2}{y-5}\right)$$
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