Question

Determine which consecutive integers do not have a real zero of f(x) = x^3+5x^2 -x -6 between them.

(-6,-5)
(-2,-1)
(-5,-4)
(1,2)

Answer 1

(-6, -5).

Explanation:

If there is a zero between the integers there will be a change of sign of f(x).

x^3 + 5x^2 - x - 6

f(1) = 1 + 5 - 1 - 6 = -1

f(-1) = -1 + 5 + 1 - 6 = -1

f(2) = 8 + 20 - 2 -6 = 20

There' a change of sign between f(1) and f(2) so there is a zero between (1, 2).

f(-2) = -8 + 20 + 2 - 6= 6

So there is also a zero between -1 and -2. (-2, -1)

F(-4) = 14, f(-5) = -1 = change of sign so there is also a zero in (-5,-4)

f(-6) = -36 - no change in sign between -5 and -6.