Problem 12 - College Algebra

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(\frac{y - 5}{y + 4}\) and the numbers that need to be excluded from the domain are \(y = -4\) and \(y = -1\).

Step 1: Factorize Numerator and Denominator

Begin by factoring the numerator and the denominator of the fraction, which are both quadratic equations. The numerator \(y^{2}-4 y-5\) factors to \( (y - 5)(y + 1) \) and the denominator \(y^{2}+5 y+4\) factors to \( (y + 4)(y + 1) \). This gives us the new expression: \(\frac{(y - 5)(y + 1)}{(y + 4)(y + 1)}\).

Step 2: Cancel Out Common Factors

The common factor between the numerator and the denominator is \((y + 1)\). Cancel this factor out to simplify the expression. The simplified fraction now becomes \(\frac{y - 5}{y + 4}\).

Step 3: Determine Numbers Excluded from the Domain

In a rational expression, any value that makes the denominator equal to zero must be excluded from the domain, because division by zero is undefined. To find this, set the denominator equal to zero and solve for \(y\). Doing so, we get \(y + 4 = 0\), so \(y = -4\). Additionally, from the original denominator before simplification, we see that \(y + 1 = 0\) so \(y = -1\). This was cancelled during simplification but must still be included as part of the excluded values.