Chapter 4.3, Problem 14E

To determine

To divide: $\left(x^2+20x+91\right)÷\left(x+7\right)$ using synthetic division.

Answer to Problem 14E

  $\left(x^2+20x+91\right)÷\left(x+7\right)=\left(x+13\right)$

Explanation of Solution

Given information: $\left(x^2+20x+91\right)÷\left(x+7\right)$

Calculation:

When dividing by $\left(x+a\right),$ begin with writing − a at the top left. Continue by writing down the coefficients of the polynomial to the right of this first value.

These must be in order from highest degree coefficient to lowest degree coefficient. If a certain degree is not present in the polynomial, don’t forget to write a 0.

Then, to do the synthetic division, begin with the top value in the second column and drop it down.

  $\begin{array}{l}7|1\text{ 20 91}\\ |\underset{\_}{\downarrow }\\ |\text{ 1 }|\end{array}$

Next multiply it with the top left value and write it in the empty space under the top value of the third column.

  $\begin{array}{l}7|1\text{ 20 91}\\ |\underset{\_}{\downarrow -7}\\ |\text{ 1 }|\end{array}$

Add the two values in the third column and write it in the empty space at the bottom of this column.

  $\begin{array}{l}7|1\text{ 20 91}\\ |\underset{\_}{\downarrow -7}\\ |\text{ 1 13 }|\end{array}$

Now go back to step 2 and repeat until finished.

  $\begin{array}{l}7|1\text{ 20 91}\\ |\underset{\_}{\downarrow -7-91}\\ |\text{ 1 13 }|0\end{array}$

The result can be interpreted as follows.

Look at the bottom row. The first value is the coefficient of the new polynomial, but with one degree less than what we started with. The value to the right of this value is the coefficient of one degree less and so on. The final value is the remainder.

  $\left(x^2+20x+91\right)÷\left(x+7\right)=\left(x+13\right)$