Problem 12 - College Algebra Essentials

Question

Simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. $$\frac{y^{2}-4 y-5}{y^{2}+5 y+4}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{y-5}{y+4}\). And the value (-4) must be excluded from the domain of the expression.

Step 1: Factorize the numerator

The numerator \(y^{2}-4 y-5\) is a quadratic expression, which can be factorized into two binomial expressions. We look for two numbers that add up to -4 (the coefficient of \(y\)), and multiply to -5 (the constant term). These numbers are -5 and 1. Hence, the factorized form of the numerator is \((y-5)(y+1)\).

Step 2: Factorize the denominator

The denominator \(y^{2}+5 y+4\) is also a quadratic expression. We look for two numbers that add up to 5 (the coefficient of \(y\)), and multiply to 4 (the constant term). These numbers are 4 and 1. Hence, the factorized form of the denominator is \((y+4)(y+1)\).

Step 3: Simplify the expression

Next, we simplify the rational function by canceling out the common factor \((y+1)\) from the numerator and the denominator. So, the simplified form of the expression is \(\frac{y-5}{y+4}\) .

Step 4: Find the numbers excluded from the domain

The numbers that must be excluded from the domain of a rational expression are the roots of the denominator, as these values would make the denominator zero and cause the expression to be undefined. Here, the excluded value is \(y = -4\) as it makes the denominator \(y+4 = 0\)