Problem 41 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \(\frac{(y-1)(y-2)}{(y-2)(y+9)}\).

Step 2: 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 \(\frac{(y-1)}{(y+9)}\).

Step 3: 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 \(\frac{(y-1)}{(y+9)}\).