Chapter 5.6, Problem 90ES

To determine

The objective is to identify the pattern in quotients and remainders and use the patterns to find $\frac{x^7-1}{x+1}$ . The result obtained can be checked using synthetic division as follows.
When we have to divide a polynomial function by a binomial of the form $(x-c)$ , then we can to use the synthetic division. Set up the synthetic division by writing the coefficients of the dividend in a box and the divisor goes on the outside of the box. When we write out the dividend make sure that we write it in descending powers and insert $0\text{'}s$ for any missing terms. Then bring down the leading coefficient to the bottom row by adding zero to it. Then, multiply $c$ by the value just written on the bottom row. Add the column created by writing the sum in the bottom row. Repeat this until done. The numbers in the last row make up our coefficients of the quotient as well as the remainder. The final value of the right is the remainder. Working right to left, the next number is our constant; the next is the coefficient for $x$ the next is the coefficient of $x^2$ and so on. From the answers obtained, we get
$Dividend=Quotient\times Divisor+Reminder$

Dividing throughout by divisor, we get the result in the format
$\frac{Dividend}{Divisor}$ = Quotient + $\frac{Reminder}{Divisor}$