The value 5 is an upper bound for the zeros of the function shown below. f (x) = x^4+x^3 - 11x^2-9x+18

A.) True

B.) False

Answer: True

Explanation:

According to the rational zeros theorem, if x=a is a zero of the function f (x), then f (a) = 0.

Given: f (x) = x⁴ + x³ - 11x² - 9x + 18

From the calculator, obtain

f (5) = 448

f (4) = 126

f (3) = 0

f (2) = - 20

f (1) = 0

f (0) = 18

f (-1) = 16

f (-2) = 0

f (-3) = 0

The polynomial is of degree 4, so it has at most 4 real zeros.

From the calculations, we found all 4 zeros as x = - 3, - 2, 1, and 3.

Therefore

f (x) = (x+3) (x+2) (x-1) (x-3).

For x>3, f (x) increases rapidly. Therefore there are no zeros for x>3.

The statement that x=5 is an upper bound for the zeros of f (x) is true.