Question

use the intermediate value theorem to show that the polynomial has a real zero between the given integers. f(x)=x³−7x+9; between −4 and −1 select the correct choice below and, if necessary, fill in the answer box(es) within your choice. (simplify your answers.) a. because f(x) is a polynomial with f(−4)= <0 and f(−1)= <0, the function has a real zero between −4 and −1. b. because f(x) is a polynomial with f(−4)= <0 and f(−1)= >0, the function has a real zero between −4 and −1. c. because f(x) is a polynomial with f(−4)= >0 and f(−1)= >0, the function has a real zero between −4 and −1. d. because f(x) is a polynomial with f(−4)= >0 and f(−1)= <0, the function has a real zero between −4 and −1.

Explanation:

Step1: Calculate \(f(-4)\)

Substitute \(x = -4\) into \(f(x)=x^{3}-7x + 9\).

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Step2: Calculate \(f(-1)\)

Substitute \(x=-1\) into \(f(x)=x^{3}-7x + 9\).

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Answer:

B. Because \(f(x)\) is a polynomial with \(f(-4)=-27<0\) and \(f(-1)=15>0\), the function has a real zero between \(-4\) and \(-1\).