Question

Use the Intermediate Value Theorem to show that the polynomial f(x)=x^3+x^2-2x+5 has a real zero between -3 and -1.

Answer 1

You need to note first two things.

1. The function f(x) need to be a continuous function on [-3,-1] (indeed f(x) is a polynomial function, is continuous on every real number x)

2. Take the closed interval [-3,-1]

Now, evaluate f(-3) and f(-1).

$f(-3) = (-3)^3 + (-3)^2 -2(-3) + 5 = -27 + 9 +6 + 5 = -7 < 0\\f(-1) = (-1)^3 + (-1)^2 -2 (-1) + 5 = -1 + 1 +2 + 5 = 7 >0$

So, we have f(-3) < 0  and f(-1). So,  $f(-3) < 0 < f(-1)$, then by the Intermediate Value Theorem, there exist $c \in (-3,-1)$ such that $f(c) = 0$.