Problem 32 - College Algebra

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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 \(x^4 - 2x^2 + x - 1\).