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Chapter 0: Problem 12

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

Short Answer

Expert verified
The simplified rational expression is \((y-5)/(y+4)\) and the domain is all real numbers except -4.

Step by step solution

01

Simplify the Rational Expression

Simplify the rational expression by factoring the numerator and the denominator.\n The numerator \(y^{2}-4 y-5\) factors into \((y-5)(y+1)\) and the denominator \(y^{2}+5 y+4\) factors into \((y+4)(y+1)\). Therefore, the rational expression can be written as:\n \[\frac{(y-5)(y+1)}{(y+4)(y+1)}\]
02

Cancel Common Factors

In the rational expression \((y-5)(y+1)/(y+4)(y+1)\), cancel the common factor of \((y+1)\):\n Result: \[\frac{y-5}{y+4}\]
03

Determine the Domain

Now, to find the domain of the rational expression, we need to find all values of 'y' that make the denominator equal to zero. Therefore, we solve for 'y' in the equation \(y+4 = 0\). Solving this equation gives \(y = -4\). This value cannot be included in the domain because it will make the denominator zero and the expression undefined. Therefore, the domain is all real numbers except -4.

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