009A Sample Midterm 2, Problem 2
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The function is a polynomial and therefore continuous everywhere.
(a) State the Intermediate Value Theorem.
(b) Use the Intermediate Value Theorem to show that has a zero in the interval
Foundations: 

What is a zero of the function 
A zero is a value such that 
Solution:
(a) 

Intermediate Value Theorem 
If is continuous on a closed interval 
and is any number between and 
then there is at least one number in the closed interval such that 
(b)
Step 1: 

First, is continuous on the interval since is continuous everywhere. 
Also, 

and
. 
Step 2: 

Since is between and 
the Intermediate Value Theorem tells us that there is at least one number 
such that 
This means that has a zero in the interval 
Final Answer: 

(a) See solution above. 
(b) See solution above. 