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. First, identify the polynomial function given: \(f(x) = 3x^3 - 8x^2 + x + 2\).
2. Evaluate the function at the endpoints of the interval given, which are \(x = -1\) and \(x = 0\). Calculate \(f(-1)\) and \(f(0)\) separately by substituting these values into the function.
3. Check the signs of \(f(-1)\) and \(f(0)\). If \(f(-1)\) and \(f(0)\) have opposite signs (one positive and one negative), then by the Intermediate Value Theorem, there must be at least one real zero of \(f(x)\) between \(-1\) and \$0$.
4. Recall that the Intermediate Value Theorem states: If a function \(f\) is continuous on a closed interval \([a, b]\) and \(f(a)\) and \(f(b)\) have opposite signs, then there exists at least one \(c\) in \((a, b)\) such that \(f(c) = 0\).
5. Since polynomial functions are continuous everywhere, this theorem applies directly to \(f(x)\) on the interval \([-1, 0]\), confirming the existence of a real zero between \(-1\) and \$0$ if the sign change is observed.

Key Concepts

Polynomial Functions

A polynomial function is an expression consisting of variables and coefficients combined using addition, subtraction, multiplication, and non-negative integer exponents. Understanding the degree and behavior of polynomial functions helps in analyzing their roots and graph shapes.

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 essential for proving the existence of real zeros between two points.

Evaluating Functions at Specific Points

Evaluating a polynomial at specific values helps determine the sign of the function at those points. By checking values at the endpoints of an interval, one can apply the Intermediate Value Theorem to confirm the presence of a root within that interval.