Question

Divide. $\left(17 z^{2}-10-12 z^{3}+32 z\right) \div(4 z+5)$

   Divide.
$\left(17 z^{2}-10-12 z^{3}+32 z\right) \div(4 z+5)$

Beginning and Intermediate Algebra

Beginning and Intermediate Algebra

Sherri Messersmith 2nd Edition

Chapter 6, Problem 32

Instant Answer

Solved by Expert Maurizia De Palma

07/06/2020

Step 1

So, we rewrite the given expression as $(-12z^{3}+17z^{2}+32z-10) \div (4z+5)$. Show more…

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Divide. $\left(17 z^{2}-10-12 z^{3}+32 z\right) \div(4 z+5)$

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Key Concepts

Polynomial Long Division

Polynomial long division is a systematic method for dividing a polynomial by another polynomial of lower or equal degree. The process involves aligning terms by order, dividing the leading term of the dividend by the leading term of the divisor, multiplying the entire divisor by that quotient term, subtracting the result from the dividend, and repeating the process with the remainder until a term of lower degree than the divisor is reached.

Standard Form (Ordering of Terms)

Before beginning polynomial division, it is essential to write both the dividend and divisor in standard form, meaning that their terms are arranged in descending order of degree. This ordering ensures that the division process is applied correctly, allowing for proper alignment of like terms during subtraction.

Determining the Remainder

In polynomial division, the process is repeated until the degree of the remaining polynomial (the remainder) is less than the degree of the divisor. At this point, the result of the division is expressed as the sum of the quotient and a remainder over the divisor, analogous to the division algorithm for integers.

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