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Chapter 2: Problem 32

Use synthetic division to divide. $$\left(9 x^{3}-16 x-18 x^{2}+32\right) \div(x-2)$$

Short Answer

Expert verified
The quotient polynomial from the division is \(9x^{2} - 0x + 8\) and the remainder is 16.

Step by step solution

01

Arrange the Polynomial and the Binomial

First, rearrange the terms in the polynomial in descending order of their degree, then write down the coefficients. The polynomial becomes \(9x^{3} -18x^{2} -16x + 32\) and the coefficients are 9, -18, -16 and 32. For binomial, \(x-2\), we only need the number -2, which we will place on the right side outside of our synthetic division setup.
02

Apply Synthetic Division

Begin synthetic division operation by bringing down the first coefficient (9) as it is. Then, multiply this number by -2 (which is the constant from the binomial \(x-2\)), and write the result under the second coefficient and add, get the new second number. Repeat the process: multiply this new number by -2, and write the result under the next coefficient, add, get the third number. Do this until you reach the last number.
03

Interpret the Result

The coefficients resulting from the synthetic division represent the coefficients of the quotient polynomial. The last number we obtain during the synthetic division process represents the remainder of the division.
04

Write the Quotient Polynomial

Start from power 2 (since the dividend was a third degree polynomial, the quotient will be one degree less), and write the coefficients that resulted from the synthetic division in front of each corresponding power of x.

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