Problem 8 Use long division to verify that... [FREE SOLUTION]

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

By performing long division of polynomials, it is verified that \(y_{1}\) is indeed equal to \(y_{2}\).

Step by step solution

Setting Up the Long Division

Set up the long division with polynomial \(x^{4}-3 x^{2}-1\) in the dividend (numerator) place and \(x^{2}+5\) in the divisor (denominator) place.

Performing the Long Division

The first term of the dividend \(x^{4}\) is divided by the first term of the divisor \(x^{2}\), which gives the first term of the quotient \(x^{2}\). This quotient is then multiplied with the divisor, and the result is subtracted from the dividend to form the new dividend \(x^{2}-8\). Repeat the same processes until the degree of the new dividend is less than the degree of the divisor. You will get a final quotient of \(x^{2}-8\) and a remainder of \(39\).

Expressing \(y_{1}\) in terms of Quotient and Remainder

Express \(y_{1}\) as the quotient from the division plus the remainder over the divisor. Therefore, rewriting \(y_{1}\) gives us \(x^{2}-8+\frac{39}{x^{2}+5}\).

Verifying that \(y_{1}\) Equals \(y_{2}\)

Now, compare the new expression of \(y_{1}\) with \(y_{2}\). Since both are the same, it can be concluded that \(y_{1}\) equals \(y_{2}\).

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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

Step by step solution

Setting Up the Long Division

Performing the Long Division

Expressing \(y_{1}\) in terms of Quotient and Remainder

Verifying that \(y_{1}\) Equals \(y_{2}\)

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