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Chapter 3: Problem 27

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

Short Answer

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

Step by step solution

01

Preparing for Synthetic Division

For Synthetic Division, list down the coefficients of the dividend polynomial in the order of decreasing power of x. If any term is missing, then a zero should be used as place holder for that coefficient. In our case, the coefficients would be 1 (coefficient of \(x^{5}\)), 0 (coefficient of \(x^{4}\)), 1 (coefficient of \(x^{3}\)), 0 (coefficient of \(x^{2}\)), 0 (coefficient of \(x^{1}\)), and -2 (coefficient of \(x^{0}\)). The number you will use for division is the opposite number of what you see in the binomial (x-c), which is 1 in this case.
02

Performing Synthetic Division

Drop the first coefficient (1) down, multiply this by 1 (our divisor from the binomial) and write the result under the second coefficient. You add down the column and repeat this process. So the new row will be 1 (from dropping down the first coefficient), 1 (multiply 1 and 1), 1 (add 0 and 1), 1 (multiply 1 and 1), 1 (add 0 and 1), 1 (multiply 1 and 1), -1 (add 0 and -1), -1 (multiply -1 and 1) and -2 (add -1 and -1).
03

Writing the Result

The numbers at the bottom line of the synthetic division form the coefficients of the quotient polynomial. Starting from left, each number is the coefficient of the descending powers of x. So your answer will be \(x^{4}+x^{3}+x^{2}+x-1\) with the reminder of -2.

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