Chapter 2.4, Problem 67E

Solution

To choose: The correct statements about the function $f\left(x\right)=\left(x+2\right)\left(x^2+x-1\right)-3$

Option B is correct.

Given information:

  $f\left(x\right)=\left(x+2\right)\left(x^2+x-1\right)-3$

Calculation:

Option A. The remainder when $f\left(x\right)$ is divided by $\left(x+2\right)$ is $-3$ .

  $\frac{\left(x+2\right)\left(x^2+x-1\right)-3}{x+2}=\left(x^2+x-1\right)-\frac{3}{x+2}$ .

It is true statement.

Option B. The remainder when $f\left(x\right)$ is divided by $\left(x-2\right)$ is $-3$ .

  $\frac{\left(x+2\right)\left(x^2+x-1\right)-3}{x-2}=\left(x^2+5x+11\right)-\frac{17}{x-2}$ .

It is false statement.

Option C. The remainder when $f\left(x\right)$ is divided by $\left(x^2+x-1\right)$ is $-3$ .

  $\frac{\left(x+2\right)\left(x^2+x-1\right)-3}{x^2+x-1}=\left(x+2\right)-\frac{3}{x^2+x-1}$ .

It is true statement.

Option D. $x+2$ is not a factor of $f\left(x\right)$ .

  $\begin{array}{c}f\left(-2\right)=\left(-2+2\right)\left({\left(-2\right)}^2-2-1\right)-3\\ =0-3\\ =-3\end{array}$

It is true statement.

Option E. $f\left(x\right)$ is not evenly divisible by $x+2$

This is true statement because $x+2$ is not factor of $f\left(x\right)$ .