BackUsing the Intermediate Value Theorem to Determine Real Zeros of a Polynomial
Study Guide - Smart Notes
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Q2. Determine whether the given polynomial has a real zero between and by using the Intermediate Value Theorem.
Background
Topic: Intermediate Value Theorem (IVT) and Polynomial Functions
This question tests your understanding of how to use the Intermediate Value Theorem to determine if a continuous function (in this case, a polynomial) has a real zero within a given interval.
Key Terms and Formulas
Intermediate Value Theorem (IVT): If is continuous on and is any number between and , then there exists at least one in such that .
For this problem, we are interested in (i.e., a zero of the function).
Polynomial Function: is continuous everywhere.
Step-by-Step Guidance
Evaluate at the endpoints of the interval: and .
Calculate by substituting into the polynomial: .
Calculate by substituting into the polynomial: .
Check the signs of and . If one is positive and the other is negative, then by the IVT, there must be a zero between and $1$.
Set up the comparison: Are and of opposite signs?
Try solving on your own before revealing the answer!
Final Answer: Yes, there is a real zero between and .
Calculating the values:
Since (positive) and (negative), the function changes sign over the interval. By the Intermediate Value Theorem, there must be at least one real zero between and .
