Chapter 2.6, Problem 41E

Solution

To match: all real zeros of the function.

  

The correct match is B and real zero of function $x=1,2,-1$ .

Given information: 

The function $f\left(x\right)=x^3-2x^2-x+2$ .

Formula used:

The Rational Zero Theorem:

If $f\left(x\right)=a_n{x}^n+\cdots +a_{1}x+a_0$ consists of integer coefficients, then the shape of each rational zero of $f$ is as follows:

  $\frac{p}{q}=\frac{\text{ factor of constant term }{a}_0}{\text{ factor of leading coefficient }{a}_n}$

Calculation:

Consider the function

  $f\left(x\right)=x^3-2x^2-x+2$

Rational zero theorem:

Factors of constant term $2$ :

  $p=1,2$

Leading coefficient factors $1$ :

  $q=1$

All possible combinations of $\pm \frac{p}{q}$ .

Some or none may be zeros of the function.

  $\pm \frac{p}{q}=\pm \left\{1,2\right\}$

Test the possible rational zero combinations. There are only $4$ .

  $/$ polyhornerscheme $[x=1]\left\{x^3-2x^2-x+2\right\}$

  $/$ polyhornerscheme $[x=2]\left\{x^3-2x^2-x+2\right\}$

  $/$ polyhornerscheme $[x=-1]\left\{x^3-2x^2-x+2\right\}$

  $/$ polyhornerscheme $[x=-2]\left\{x^3-2x^2-x+2\right\}$

  $x=1,2,-1$ are the three zeros. This matches graph B.