In Exercises 33–40, use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x^5−x^3−1; between 1 and 2

Key Concepts

Intermediate Value Theorem

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b] and takes on different signs at the endpoints, then there exists at least one c in (a, b) such that f(c) = 0. This theorem is crucial for proving the existence of real zeros in polynomials.

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In this case, f(x) = x^5 - x^3 - 1 is a polynomial of degree 5, which is continuous and differentiable everywhere, making it suitable for applying the Intermediate Value Theorem.

Sign Change

A sign change occurs when a function takes on values of opposite signs at two points. For the Intermediate Value Theorem to apply, we need to evaluate the polynomial at the endpoints of the interval (1 and 2) and check if f(1) and f(2) have different signs, indicating the presence of a real zero in that interval.