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 -1 and 0

Verified step by step guidance

1. insert step 1: Understand that the problem is asking to show that the polynomial function \( f(x) = 3x^3 - 8x^2 + x + 2 \) has a real zero between \(-1\) and \(0\).
2. insert step 2: Use the Intermediate Value Theorem, which states that if a continuous function changes sign over an interval, then it has at least one real zero in that interval.
3. insert step 3: Evaluate the function at the endpoints of the interval. Calculate \( f(-1) \) and \( f(0) \).
4. insert step 4: Check the signs of \( f(-1) \) and \( f(0) \). If \( f(-1) \) and \( f(0) \) have opposite signs, then by the Intermediate Value Theorem, there is at least one real zero between \(-1\) and \(0\).
5. insert step 5: Conclude that since the function changes sign between \(-1\) and \(0\), there is a real zero in that interval.

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, a_(n-1), ..., a_0 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 signs at the endpoints, then there exists at least one c in (a, b) such that f(c) = 0. This theorem is essential for proving the existence of real zeros in polynomial functions, as it guarantees that a root must exist between points where the function changes sign.

Finding Real Zeros

Finding real zeros of a polynomial function involves determining the values of x for which f(x) = 0. This can be done through various methods, including factoring, using the Rational Root Theorem, or applying numerical methods like the Newton-Raphson method. In the context of the given polynomial, evaluating the function at specific points within the interval can help identify where the function crosses the x-axis, indicating the presence of real zeros.