Problem 32 Divide as indicated. Check each ... [FREE SOLUTION]

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Introductory Algebra for College Students

Robert Blitzer

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 5: Problem 32

Divide as indicated. Check each answer by showing that the product of the divisor and the quotient, plus the remainder, is the dividend. $$\frac{8 y^{3}+y^{4}+16+32 y+24 y^{2}}{y+2}$$

Short Answer

Expert verified

The quotient obtained after the polynomial division is $(y^{3}+4y^{2}+8y+16)$. The result has been confirmed to be correct since multiplying the quotient and the divisor, and adding the remainder gives us the original dividend.

Step by step solution

Arrange Polynomials

First, we need to write down the dividend and the divisor. Make sure that both the dividend $8 y^{3}+y^{4}+16+32 y+24 y^{2}$ and the divisor $y+2$ are written in descending order of powers of $y$. In this case, the dividend should be written as $y^{4}+8 y^{3}+24 y^{2}+32 y+16$. Next, you set up the problem as you would a long division problem, with the dividend inside the long division symbol and the divisor on the outside.

Perform Polynomial Division

The first step in polynomial division is to divide the first term in the dividend by the first term in the divisor, i.e., $y^{4} / y = y^{3}$. The result $y^{3}$ is the first term of the quotient. Multiply the divisor by the quotient obtained, i.e., $y y^{3}=y^{4}$, $-2*y^{3}=-2*y^{3}$. Write these results under the dividend and subtract them from the dividend. Repeat this method with the new polynomial $0+6*y^{3}+24*y^{2}+32*y+16$, until the degree of the remainder is less than the degree of the divisor.

Verification

Multiply the quotient derived $(y^3+4y^2+8y+16)$ with the divisor $(y+2)$ and add the remainder to check whether it equals the given dividend. In this case, you should get back the original dividend, verifying that your division is correct.

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91影视

Divide as indicated. Check each answer by showing that the product of the divisor and the quotient, plus the remainder, is the dividend. $$\frac{8 y^{3}+y^{4}+16+32 y+24 y^{2}}{y+2}$$

Short Answer

Step by step solution

Arrange Polynomials

Perform Polynomial Division

Verification

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