Question

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

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

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.

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 \(z+5-\frac{13}{z+2}\).

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) \(z+9-\frac{15}{z-2}\)

Solution

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