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Chapter 7: Problem 41

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{y^{2}-3 y+2}{y^{2}+7 y-18}$$

Short Answer

Expert verified
The simplified form of the rational expression is \(\frac{(y-1)}{(y+9)}\).

Step by step solution

01

Factorize the expression

The first thing to do is to factorize the polynomial in the numerator (\(y^{2}-3 y+2\)) and the polynomial in the denominator (\(y^{2}+7 y-18\)). The factored form of the numerator is \((y-1)(y-2)\) and the factored form of the denominator is \((y-2)(y+9)\). Thus, the original expression becomes \(\frac{(y-1)(y-2)}{(y-2)(y+9)}\).
02

Cancel out common factors

Now, cancel out the common factors in numerator and denominator. Here, (y-2) is a common factor in both the numerator and denominator. After cancelling out this common factor, the expression becomes \(\frac{(y-1)}{(y+9)}\).
03

Write the simplified expression

The simplified form of the given rational expression is \(\frac{(y-1)}{(y+9)}\). Therefore, \( \frac{y^{2}-3 y+2}{y^{2}+7 y-18}\) in its simplest form is \(\frac{(y-1)}{(y+9)}\).

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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. $$\frac{y}{y-1}-\frac{1}{1-y}$$

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

perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{1}{2}+\frac{2}{3}$$

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

perform the indicated operations. Simplify the result, if possible. $$\left(\frac{3 x^{2}-4 x+4}{3 x^{2}+7 x+2}-\frac{10 x+9}{3 x^{2}+7 x+2}\right) \div \frac{x-5}{x^{2}-4}$$
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