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 given: \(f(x) = 3x^3 - 8x^2 + x + 2\).
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. 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. Conclude that since polynomial functions are continuous everywhere, and if \(f(2)\) and \(f(3)\) have opposite signs, the function \(f(x)\) must have 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. Finding zeros is essential for understanding the behavior and graph of the polynomial.

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 used to show the existence of a real zero between two points where the function values have opposite signs.

Evaluating Polynomial Functions at Specific Points

To apply the Intermediate Value Theorem, you evaluate the polynomial at given points to check the sign of the function values. If the function values at two points have opposite signs, it confirms a zero exists between those points.