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Chapter 2: Problem 30

Use synthetic division to divide. \(\left(2 x^{3}+14 x^{2}-20 x+7\right) \div(x+6)\)

Short Answer

Expert verified
The result of the division is \((2 x^{2}+2 x-32)+\frac{199}{x+6}\)

Step by step solution

01

Set Up the Synthetic Division Table

The first step involves setting up the synthetic division table. Write the coefficients of the dividend polynomial \(2 x^{3}+14 x^{2}-20 x+7\) and the root of the divisor \(x+6\) in a synthetic division table. The root of the divisor is found by equating it to zero, so the root is -6.
02

Conduct the Synthetic Division

To do synthetic division, follow these steps: first, bring down the leading coefficient, which is 2, to the bottom row. Multiply this by the root, -6, and write the result, -12, under the next coefficient, 14. Add 14 and -12 to get 2. Repeat this process for the rest of the coefficients. After the final addition, we get coefficients 2, 2, -32, and 199.
03

Form the Quotient Polynomial

The result of synthetic division gives the coefficients of the quotient (answer) term by term from highest power to lowest. With the coefficients 2, 2, -32 and the remainder 199, the quotient polynomial would be \(2 x^{2}+2 x-32\) and the remainder would be 199. Therefore, the result of the division is \((2 x^{2}+2 x-32)+\frac{199}{x+6}\).

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