How many real roots does the equation have in the interval ?
A)no real roots
B)at most one real root
C)at most five real roots
D)at most three real roots
E)at most two real roots

Question

How many real roots does the equation  have in the interval  ?
A)no real roots 
B)at most one real root 
C)at most five real roots 
D)at most three real roots 
E)at most two real roots

Quizplus · Accepted Answer

Since the given equation is a polynomial of degree 10, it can have at most 10 real roots. However, the interval given is of length zero (because the two endpoints are the same), so there can be at most one real root by the Intermediate Value Theorem. That root may not even exist if the value of the polynomial at the endpoint is not opposite in sign to its value at any other point in the interval. Therefore, the answer is (B) at most one real root.

How Many Real Roots Does the Equation Have in the Interval