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Chapter 3: Problem 25

Divide using synthetic division. $$\left(x^{2}-5 x-5 x^{3}+x^{4}\right) \div(5+x)$$

Short Answer

Expert verified
The result of the division is \(x^{3}-6x^{2}-5x+5\).

Step by step solution

01

Organize the Polynomial and the Divisor

First, ensure that the polynomial is written in order of descending exponents. Also, if there is a missing term in the sequence, it should be included as zero. This gives us \[x^{4}-5x^{3}-5x^{2}+0x+0\]. The \[(5+x)\] should be rewritten as \[-x+5\] or \[-1x+5\] because we need the 'x' to be first and if we do not change the sign, synthetic division will not provide the correct answer.
02

Create the Synthetic Division Layout

Next is to set up the synthetic division. Write -1 on the left side and the coefficients of the polynomial on the right side, i.e. [1, -5, -5, 0, 0]. Draw a line beneath these numbers.
03

Begin Synthetic Division

Carry down the first number which is 1. Multiply -1 by the number just written below, which is also 1, and write the answer, i.e. -1, under the next number, -5. Continue this process for each number and sum up the top and bottom numbers at each step.
04

Write the Result

Write your answer in polynomial form using the results from the bottom row. The degree of the polynomial has to start from n-1, which in this case is 3. This gives the answer as \[x^{3}-6x^{2}-5x+5\] with no remainder.

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