Question

select the correct answer.

for the given table of values for a polynomial function, where must the zeros of the function lie?

a. between 3.0 and 3.5 and between 4.0 and 4.5
b. between 3.5 and 4.0 and between 4.0 and 4.5
c. between 3.5 and 4.0 and between 5.0 and 5.5
d. between 4.0 and 4.5 and between 4.5 and 5.0

Explanation:

Step1: Apply Intermediate Value Theorem

A polynomial function is continuous everywhere. A zero exists where \(f(x)\) changes sign (positive to negative or negative to positive).

• At \(x=3.0\), \(f(x)=4.0\) (positive); at \(x=3.5\), \(f(x)=-0.2\) (negative). Sign change occurs here.
• At \(x=4.0\), \(f(x)=-0.8\) (negative); at \(x=4.5\), \(f(x)=0.1\) (positive). Sign change occurs here.

Step2: Eliminate incorrect options

• Option B: No sign change between 3.5 and 4.0 (both negative), so no zero here.
• Option C: No sign change between 5.0 and 5.5 (both positive), so no zero here.
• Option D: No sign change between 4.5 and 5.0 (both positive), so no zero here.

Answer:

A. between 3.0 and 3.5 and between 4.0 and 4.5