Problem 11 - College Algebra Essentials

Question

Simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. $$\frac{y^{2}+7 y-18}{y^{2}-3 y+2}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{y+9}{y-1}\) and the values excluded from the domain are \(y = 1\) and \(y = 2\).

Step 1: Factor the expression

First we start by factoring the numerator and the denominator. The factored form of the numerator \(y^{2}+7 y-18\) is \((y-2)(y+9)\) and the factored form of the denominator \(y^{2}-3 y+2\) is \((y-1)(y-2)\). So the rational expression becomes \(\frac{(y-2)(y+9)}{(y-1)(y-2)}\).

Step 2: Simplify the expression

Next, cancel out any common terms in the numerator and the denominator. The term \((y-2)\) is common to both the numerator and the denominator, thus can be cancelled out. This gives the simplified rational expression as \(\frac{y+9}{y-1}\).

Step 3: Find the excluded values

In a rational expression, any values that make the denominator equal to zero are excluded from the domain. Thus, we set the denominator equal to zero and solve for \(y\). This gives \(y-1 = 0\), thus \(y = 1\). So \(y = 1\) is excluded from the domain. Additionally, since we cancelled the term \((y-2)\) in the previous step, \(y = 2\) is also excluded from the domain, as it would have made the original denominator zero.