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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
After performing the synthetic division, the quotient is \(x^4 - 2x^2 + x - 1\) and the remainder is 3. Thus, \(\frac{x^{5}-2 x^{4}-x^{3}+3 x^{2}-x+1}{x-2} = x^4 - 2x^2 + x - 1 + \frac{3}{x-2}\).

Step by step solution

01

Setting Up the Synthetic Division

Write the coefficients of the polynomial and the constant term in a row: 1 -2 -1 3 -1 1. Write the '2' (from 'x-2') to the left of this row.
02

Begin the Iterative Synthetic Division Process

Bring the first coefficient (1) straight down and multiply it by 2. Write the result under the second coefficient (-2) and add to get 0. Repeat the process for the remaining coefficients: Each number obtained is multiplied by 2 and added to the next coefficient. The final sequence should be 1, 0, -2, 1, -1, 3.
03

Confirm Interpretation of the Final Sequence

The last entry, 3, is the remainder. The coefficient sequence before the remainder: 1, 0, -2, 1, -1, is the result of the division and can be written as a polynomial. So, the quotient is \(x^4 - 2x^2 + x - 1\).

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Most popular questions from this chapter

In Exercises \(27-34,\) find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zero. $$f(x)=-3\left(x+\frac{1}{2}\right)(x-4)^{3}$$

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