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Chapter 1: Problem 14

Which of the following is equivalent to \(\frac{z^2+7 z-3}{z+2}\) ? A) \(z+5-\frac{13}{z+2}\) B) \(z+5-\frac{7}{z+2}\) C) \(z+9-\frac{21}{z-2}\) D) \(z+9-\frac{15}{z-2}\)

Short Answer

Expert verified
The equivalent form of the given rational expression \(\frac{z^2+7z-3}{z+2}\) is \(\mathbf{z+5-\frac{13}{z+2}}\).

Step by step solution

01

Setup Polynomial Long Division

We will perform polynomial long division of the given expression: \(\frac{z^2+7z-3}{z+2}\). We will divide \(z^2+7z-3\) by \(z+2\) and try to obtain a result that matches one of the given choices.
02

Perform Polynomial Long Division

Divide \(z^2+7z-3\) by \(z+2\): 1. Divide \(\frac{z^2}{z}\) and write down the result: \(z\). 2. Multiply \((z+2)\) by \(z\) and subtract the result from the dividend: \((z^2+7z-3) - (z^2+2z) = 5z-3\). 3. Divide \(\frac{5z}{z}\) and write down the result: \(+5\). 4. Multiply \((z+2)\) by \(5\) and subtract the result from the current dividend: \((5z-3) - (5z+10) = -13\). 5. Since the degree of the remaining dividend \(-13\) is less than the degree of the divisor \((z+2)\), we stop. Thus, the result of our polynomial long division is \(z+5-\frac{13}{z+2}\).
03

Compare Result with Choices

We obtained the result \(z+5-\frac{13}{z+2}\). Now let's compare it with the given choices: A) \(z+5-\frac{13}{z+2} \) (Match!) B) \(z+5-\frac{7}{z+2}\) C) \(z+9-\frac{21}{z-2}\) D) \(z+9-\frac{15}{z-2}\)
04

Solution

Since our result matches choice (A), the equivalent form of the given rational expression \(\frac{z^2+7z-3}{z+2}\) is \(\mathbf{z+5-\frac{13}{z+2}}\).

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Most popular questions from this chapter

Josephine purchases a computer for \(\$ 4,590\). The computer decreases in value at a constant rate for 9 years, after which it is considered not to have any monetary value. How much is the computer worth 6 years after it is purchased? A) \(\$ 1,530\) B) \(\$ 2,295\) C) \(\$ 3,060\) D) \(\$ 4,080\)

The economy of Argentina as measured by its Gross Domestic Product (GDP) is shrinking at a rate of \(2.6 \%\) per year. In 2015 , the GDP of Argentina was \(\$ 630\) billion. Which of the following functions represents Argentina's GDP, \(A\), in billions of dollars, \(y\) years since \(2015 ?\) A) \(A(y)=630-(1-0.26) y\) B) \(A(y)=630(1-0.26)^y\) C) \(A(y)=630-(1-0.026) y\) D) \(A(y)=630(1-0.026)^y\)

The ordered pair \((3,-1)\) satisfies which of the following inequalities? I. \(x+3 y>0\) II. \(2 x+3 y>2\) III. \(x+y<0\) A) I only B) II only C) I and III only D) II and III only

A) NO CHANGE B) leave; C) leave, D) leave

If \(15-3 b=21\), what is the value of \(5-b\) ?
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