Problem 72 - Introductory Algebra for College Students

Question

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{y^{2}-7 y+12}{3-y}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \( -(y - 4) \)

Step 1: Factorize the Numerator

Factorize \(y^{2}-7 y+12\) into two binomial expressions. This is achieved by finding two numbers that multiply to 12 and sum to -7. The numbers -3 and -4 meet this criteria, hence \(y^{2}-7 y+12 = (y-3)(y-4)\)

Step 2: Rewrite the Expression

Rewrite the rational expression replacing the numerator with the now factorized form. So the expression becomes \(\frac{(y-3)(y-4)}{3 - y}\)

Step 3: Simplify the Expression

Next, simplify the expression by cancelling out any common factors. Here, we note a factor of (y-3) in the numerator and (3 - y) in the denominator. However, they are not exactly the same as their signs are opposite. To make them the same, factor out -1 from the denominator to get (y - 3) and then, cancel out (y-3) in both numerator and denominator. So \(\frac{(y-3)(y-4)}{3 - y} = -(y - 4)\)