Chapter 4.4, Problem 70E

To determine

Whether the conclusion that $f\left(x\right)$ cannot be factored is correct or not

Answer to Problem 70E

The conclusion is not correct.

Explanation of Solution

Given information:

After divide $f\left(x\right)$ by $\left(x-a\right)$ the remainder does not equal 0.

Calculation:

If dividing $f\left(x\right)$ by $x-a$ doesn’t gives remainder zero, it means that $x-a$ is not a factor of $f\left(x\right)$ .

But it doesn’t means that the polynomial can’t be factored.

For example, $f\left(x\right)=x\left(x+4\right)\left(x-3\right)$ can’t be factored by $x=6$ but it can be factored by $x=0,3$ and $x=-4$ .

So, the conclusion is not correct.