Problem 27 - College Algebra

Question

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

Step-by-Step Solution

Answer

The result of the division \(\frac{x^{5}+x^{3}-2}{x-1}\) via synthetic division is \(x^{4}+x^{3}+x^{2}+x+1-\frac{3}{x-1}\).

Step 1: Setup the synthetic division

To initiate the synthetic division, write down all coefficients of the polynomial in descending power order, which in this case are [1, 0, 1, 0, 0, -2] - zeroes were filled in for the missing powers of x. The root from the divisor, \(x - 1\), is 1, so place this outside of the synthetic division bracket.

Step 2: Carry out the synthetic division

The synthetic division process comprises a sequence of multiply, add, and drop down steps. Begin by dropping down the first coefficient, 1, from the polynomial. Then multiply this by the root, 1, and write the result under the second coefficient. This results in a new coefficient row [1, 1, 1, 1, 1, -1]. Now, sum the corresponding values in each column, i.e., [1+0, 1+0, 1+0, 1+0, 1+0, -2+(-1)], to derive a new row of [1, 1, 1, 1, 1, -3].

Step 3: Interpret the synthetic division result

The last value, -3, is the remainder and all the other values [1, 1, 1, 1, 1] are the coefficients of the result polynomial. So, the polynomial resulting from the synthetic division is \(x^{4}+x^{3}+x^{2}+x+1-\frac{3}{x-1}\).