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 Real Zeros of Polynomials

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 .

• Real Zero: A value where .

• Polynomial Function: Polynomials are continuous everywhere, so IVT applies on any interval.

Step-by-Step Guidance

1. First, identify the function and the interval: on .

2. Evaluate at the endpoints of the interval: Calculate and .

3. 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 $1$.

4. Write out the calculations for and , but do not simplify to the final values yet.

Try solving on your own before revealing the answer!