Problem 71 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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 1: 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 \(\frac{(y-4)(y+3)}{4-y}\).

Step 2: 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 \(\frac{(y-4)(y+3)}{-1 * (y-4)}\).

Step 3: Simplify the Expression

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