Find integers that are upper and lower bounds for the real zeros of the polynomial. (Be sure the lower bound is the largest possible lower bound and the upper bound is the smallest possible upper bound.)
P(x) = x3 − 3x2 + 4

Question

anonymous · Answer

wait....it should be x^3-3x^2+4

anonymous · Answer

So they want the infimum and the supremum.

anonymous · Answer

if that is the upper and lower bounds, then yes

anonymous · Answer

Least upper bound and greatest lower bound.

anonymous · Answer

yeah

anonymous · Answer

This is a calculus course, no?

anonymous · Answer

its college algebra

anonymous · Answer

Ah.

anonymous · Answer

That polynomial is factorable. If it had a rational zero it must be an integer that divides 4, so the options are$$\pm 1, \pm 2, \pm 4.$$You test these numbers and find that $$-1$$is a root of the polynomial and then use long division to get$$\frac{x^3 - 3x^2 + 4}{x -(-1)} = x^2 - 4x + 4 = (x - 2)^2$$so $$x^3 - 3x^2 + 4 = (x+1)(x-2)^2.$$Therefore, the roots are -1 and 2 (the second with multiplicity 2) and the lower and upper bounds for the real roots are -1 and 2 respectively.