Question

use the intermediate value theorem to show that the polynomial ( f(x)=x^{3}+x^{2}-2x + 11 ) has a real zero between -3 and -2. select the correct choice below and fill in the answer boxes to complete your choice. a. because ( f(x) ) is a polynomial with ( f(-3)=square<0 ) and ( f(-2)=square>0 ), the function has a real zero between -3 and -2. b. because ( f(x) ) is a polynomial with ( f(-3)=square>0 ) and ( f(-2)=square>0 ), the function has a real zero between -3 and -2. c. because ( f(x) ) is a polynomial with ( f(-3)=square>0 ) and ( f(-2)=square<0 ), the function has a real zero between -3 and -2. d. because ( f(x) ) is a polynomial with ( f(-3)=square<0 ) and ( f(-2)=square<0 ), the function has a real zero between -3 and -2.

Explanation:

Step1: Calculate \(f(-3)\)

Substitute \(x = - 3\) into \(f(x)=x^{3}+x^{2}-2x + 11\).

$$
LATEXBLOCK0
$$

Step2: Calculate \(f(-2)\)

Substitute \(x=-2\) into \(f(x)=x^{3}+x^{2}-2x + 11\).

$$
LATEXBLOCK1
$$

Answer:

A. Because \(f(x)\) is a polynomial with \(f(-3)=-1<0\) and \(f(-2)=11>0\), the function has a real zero between \(-3\) and \(-2\).