Chapter 8.5, Problem 33WE

To determine

To show that r is a double root of $P\left(x\right)=0$ by verifying that $x-r$ is a factor of both $P\left(x\right)\text{ and }P\left(x\right)÷\left(x-r\right)$

Explanation of Solution

Given information:

The given function is $P\left(x\right)=x^4+2x^2+8x+5,r=-1$

Proof:

Here $r=-1$ , use synthetic division to show that $x-r$ is a factor of $P\left(x\right)$

  $\begin{array}{l}-\text{1 }\overline{)\begin{array}{l}1\text{ 0 2 8 5}\\ -\text{1 1 }-\text{3 }-\text{5}\end{array}}\\ \text{ 1}-\text{1 3 5 0}\end{array}$

The reminder is 0 therefore $x-r$ is a factor of $P\left(x\right)$ .

From above division you got that $P\left(x\right)÷\left(x-r\right)=x^3-x^2+3x+5$

Again use synthetic division to show that $x-r$ is a factor of $P\left(x\right)÷\left(x-r\right)$ .

  $\begin{array}{l}-\text{1 }\overline{)\begin{array}{l}1-\text{1 3 5}\\ -\text{1 2 }-\text{5}\end{array}}\\ \text{ 1 }-2\text{ 5 0}\end{array}$

The reminder is 0 therefore $x-r$ is a factor of $P\left(x\right)÷\left(x-r\right)$ .