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

Step by step solution

01

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 \((y-5)(y+1)\).
02

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 \((y+4)(y+1)\).
03

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 \(\frac{y-5}{y+4}\) .
04

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 \(y+4 = 0\)

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Most popular questions from this chapter

Your local electronics store is having an end-of-the-year sale. The price on a plasma television had been reduced by \(30 \% .\) Now the sale price is reduced by another \(30 \%\). If \(x\) is the television's original price, the sale price can be modeled by $$(x-0.3 x)-0.3(x-0.3 x)$$ a. Factor out \((x-0.3 x)\) from each term. Then simplify the resulting expression. b. Use the simplified expression from part (a) to answer these questions. With a \(30 \%\) reduction followed by a \(30 \%\) reduction, is the television selling at \(40 \%\) of its original price? If not, at what percentage of the original price is it selling?

Find the union of the sets. $$\\{1,3,5,7\\} \cup\\{2,4,6,8,10\\}$$

Find the intersection of the sets. $$\\{a, b, c, d\\} \cap \varnothing$$

Evaluate each algebraic expression for the given value or values of the variable(s). $$x^{2}-3(x-y), \text { for } x=8 \text { and } y=2$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The cube root of \(-8\) is not a real number.
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