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Chapter 2: Problem 8

Use long division to verify that \(y_{1}=y_{2}\). $$y_{1}=\frac{x^{4}-3 x^{2}-1}{x^{2}+5}, \quad y_{2}=x^{2}-8+\frac{39}{x^{2}+5}$$

Short Answer

Expert verified
\(y_{1}\) is indeed equal to \(y_{2}\) as verified by long division of polynomials.

Step by step solution

01

Apply Long Division

To reshape each function into a comparable form, Long Division of Polynomials is applied. For long division, one must divide \(x^{4}-3 x^{2}-1\) by \(x^{2}+5\) following the rule of 'Divide, Multiply, Subtract, Bring down' iteratively until you can't bring down.
02

Carry out the Division

The calculation leads to \(y_{1}= x^{2}-8+\frac{39}{x^{2}+5}\)
03

Compare functions y1 and y2

After long division, the function \(y_{1}\) is indeed equal to the function \(y_{2}\), as they are now written in exactly the same form. So, the division indeed verifies \(y_{1}=y_{2}\).

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