Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=4x³-37x²+50x+60 between 2 and 3

Verified step by step guidance

1. First, identify the polynomial function given: \(f(x) = 4x^3 - 37x^2 + 50x + 60\).
2. Evaluate the function at the endpoints of the interval given, which are \(x=2\) and \(x=3\). Calculate \(f(2)\) and \(f(3)\) by substituting these values into the polynomial.
3. Check the signs of \(f(2)\) and \(f(3)\). If \(f(2)\) and \(f(3)\) have opposite signs (one positive and one negative), then by the Intermediate Value Theorem, there must be at least one real zero between 2 and 3.
4. Explain the Intermediate Value Theorem: it states that if a continuous function changes sign over an interval, it must cross zero at some point within that interval.
5. Conclude that since polynomial functions are continuous everywhere, and if the sign change is confirmed, the function \(f(x)\) has at least one real zero between 2 and 3.

Key Concepts

Polynomial Functions and Their Zeros

A polynomial function is an expression involving variables raised to whole-number exponents with coefficients. The zeros of a polynomial are the values of x for which the function equals zero. Understanding how to find or estimate these zeros is essential for analyzing the behavior of the function.

Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function changes sign over an interval [a, b], then it must have at least one root in that interval. This theorem is useful for proving the existence of a zero between two points by evaluating the function's values at those points.

Evaluating Polynomial Functions at Specific Points

To apply the Intermediate Value Theorem, you need to calculate the polynomial's value at given points. By substituting x-values into the polynomial and finding the sign of the output, you can determine if a zero lies between those points based on a sign change.