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Chapter 2: Problem 7

Use long division to verify that \(y_{1}=y_{2}\). $$y_{1}=\frac{x^{2}}{x+2}, \quad y_{2}=x-2+\frac{4}{x+2}$$

Short Answer

Expert verified
The output from the long division, \(x - 2 + \frac{4}{(x + 2)}\), coincides exactly with the given function \(y_{2}=x-2+\frac{4}{x+2}\). Thus, it has been verified that \(y_{1}=y_{2}\).

Step by step solution

01

Perform Long Division

Start by dividing \(x^{2}\) by \(x+2\) using long division. This results in a quotient of \(x-2\) with a remainder of 4.
02

Reformulate the Division Result

Reformulate the result of the long division by adding the remainder divided by the divisor to the quotient. Therefore, the expression becomes \(x - 2 + \frac{4}{(x + 2)}\).
03

Compare the Results

Compare this new expression \(x-2+\frac{4}{(x + 2)}\) with the given function \(y_{2}=x-2+\frac{4}{x+2}\). Both expressions are identical, hence this confirms \(y_{1}=y_{2}\).

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