Problem 11 - College Algebra

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 rational expression is \[\frac{y + 9}{y - 1}\], and the numbers that must be excluded from the domain are \(1, 2\).

Step 1: Factoring the Numerator and the Denominator

Start by factoring both the numerator and the denominator into products of binomials. The numerator \(y^{2} + 7y - 18\) can be factored into \((y - 2)(y + 9)\) , and the denominator \(y^{2} - 3y + 2\) can be factored into \((y - 2)(y - 1)\). This gives us:\[\frac{(y - 2)(y + 9)}{(y - 2)(y - 1)}\]

Step 2: Simplifying the Fraction

You will notice that the term \((y - 2)\) appears in both the numerator and the denominator. We can simplify the fraction by dividing the numerator and the denominator by this term:\[\frac{(y - 2)(y + 9)}{(y - 2)(y - 1)} = \frac{y + 9}{y - 1}\] provided \(y ≠ 2\) since \((y - 2)\) was in the denominator originally.

Step 3: Identifying the Values to Exclude from the Domain

The values to be excluded from the domain are those that render the denominator zero. As we mentioned, \(y cannot be 2\), because that would have made the original denominator zero. Similarly, \(y cannot be 1\) too, because in the simplified rational expression, \(y - 1\) is in the denominator. Hence, \(y ≠ 1, 2\) for the simplified rational expression.