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)=3x^3-8x^2+x+2 between 2 and 3

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1. Identify the polynomial function: \( f(x) = 3x^3 - 8x^2 + x + 2 \).
2. Use the Intermediate Value Theorem, which states that if a continuous function changes sign over an interval \([a, b]\), then it has at least one real zero in that interval.
3. Evaluate \( f(x) \) at the endpoints of the interval: calculate \( f(2) \) and \( f(3) \).
4. Check the signs of \( f(2) \) and \( f(3) \). If they are different, then there is at least one real zero between 2 and 3.
5. Conclude that since \( f(x) \) is a polynomial (and thus continuous), and the signs of \( f(2) \) and \( f(3) \) are different, there is a real zero between 2 and 3.

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form of a polynomial in one variable is f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n are constants and n is a non-negative integer. Understanding the behavior of polynomial functions, including their continuity and differentiability, is crucial for analyzing their roots.

Intermediate Value Theorem

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b] and takes on different values at the endpoints, then it must take on every value between f(a) and f(b) at least once. This theorem is essential for proving the existence of real zeros in polynomial functions, as it guarantees that if the function changes signs over an interval, there is at least one root in that interval.

Finding Real Zeros

Finding real zeros of a polynomial function involves determining the values of x for which f(x) = 0. Techniques for finding these zeros include factoring, using the Rational Root Theorem, or applying numerical methods such as the Newton-Raphson method. In the context of the given polynomial, evaluating the function at specific points between 2 and 3 will help confirm the existence of a real zero in that interval.