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Chapter 2: Problem 9

(a) use a graphing utility to graph the two equations in the same viewing window, (b) use the graphs to verify that the expressions are equivalent, and (c) use long division to verify the results algebraically. $$y_{1}=\frac{x^{2}+2 x-1}{x+3}, \quad y_{2}=x-1+\frac{2}{x+3}$$

Short Answer

Expert verified
Based on the identical graphs obtained from plotting and the algebraic result obtained from long division. It can be concluded that both the expressions \(y_{1}=\frac{x^{2}+2 x-1}{x+3}\) and \(y_{2}=x-1+\frac{2}{x+3}\) are equivalent.

Step by step solution

01

Graph the equations

Firstly, use a graphing utility to plot the equations \(y_{1}=\frac{x^{2}+2 x-1}{x+3}\) and \( y_{2}=x-1+\frac{2}{x+3}\). Note the positions of their points and lines within the viewing window.
02

Compare the graphs

For verification of their equivalence, compare the graphs of the two mathematical expressions \(y_{1}\) and \(y_{2}\). If they are equivalent, means both would produce identical graphs i.e they would overlap on each other.
03

Long Division

Finally, to confirm the results algebraically, apply long division on the equation \(y_{1}\). Starting by the division of \(x^{2}\) by \(x\), then carry that process forward till end. This should result in the equation \( y_{2}=x-1+\frac{2}{x+3}\). This would be an algebraic verification of their equivalence.

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