Problem 42 - Introductory Algebra for College Students

Question

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{y^{2}+5 y+4}{y^{2}-4 y-5}$$

Step-by-Step Solution

Answer

The simplified form of the given rational expression is \(\frac{y + 4}{y-5}\).

Step 1: Factor the Numerator

First, we need to factor the numerator, \(y^{2}+5y+4\). Looking for two numbers that multiply to 4 and add to 5, we find 1 and 4. Therefore, the factored form is \((y + 1)(y + 4)\).

Step 2: Factor the Denominator

Now, factor the denominator, \(y^{2} - 4y -5\). We find that 5 and -1 multiply to -5 and add to -4, so the factored form is \((y -5)(y+1)\).

Step 3: Simplify the Expression

The rational expression now takes the form \(\frac{(y + 1)(y + 4)} {(y -5)(y+1)}\). Here we can see that \(y + 1\) is a common factor in the numerator and the denominator, which gives us the simplified expression \(\frac{y + 4}{y-5}\).