Chapter 8.4, Problem 9WE

To determine

To divide : the given polynomial by the binomial using synthetic division.

Answer to Problem 9WE

  $\frac{x^5-1}{x-1}=x^4+x^3+x^2+x+1$

Explanation of Solution

Given information :

The polynomial: $x^5-1$

The binomial: $x-1$

Formula used : The divisor in a synthetic division is a binomial of the form $x-c$ .

Calculation :

As per the problem,

The dividend is $x^5-1$ . Insert the missing terms, $0x^4,0x^3,0x^2\text{, and 0}x$ .

Since the divisor is $x-1$ , use $1$ for c .

  $\begin{array}{l}\overline{)1}\text{ 1 0 }00\text{ 0 }-\text{1}\\ \underset{\_}{\text{ 1 1 1 1 1}}\\ \text{ 1 }1\text{ 1 1 1 | 0}\end{array}$

The quotient is $x^4+x^3+x^2+x+1$ , and the remainder is $0$

Therefore,

  $\frac{x^5-1}{x-1}=x^4+x^3+x^2+x+1$