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Chapter 3: Problem 32

Divide using synthetic division. $$\frac{x^{5}-2 x^{4}-x^{3}+3 x^{2}-x+1}{x-2}$$

Short Answer

Expert verified
The result of the synthetic division is \(x^{4} - x^{2} + x + 1 + \frac{3}{x-2}\)

Step by step solution

01

Setup the Synthetic Division Table

List the coefficients of the polynomial in descending order of their degrees: 1, -2, -1, 3, -1, 1. The divisor is \(x - 2\), so use 2 (i.e., -(-2)) in the procedure.
02

Perform Synthetic Division

Start by dropping down the leading coefficient (1 in this case). Then, multiply 1 by 2 to get 2. Add this to the next coefficient of -2, resulting in 0. Continue this procedure: multiply 0 by 2 (gets 0) and add to the next coefficient -1 (gets -1). Again, multiply -1 by 2 to get -2, and add to 3 to get 1. Then, multiply 1 by 2, getting 2 and add this to -1 to give 1. Lastly, multiply 1 by 2 to get 2, and add this to 1 to get 3.
03

Write Down the Resulting Polynomial

The numbers present at the bottom of the synthetic division are the coefficients of the quotient, starting from the highest power of \(x\), down to the constant. Therefore, the resulting polynomial is \(x^{4} - x^{2} + x + 1\). The last number on the bottom line is the remainder which is 3.

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