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)=4x^3-37x^2+50x+60 Find the zero in part (b) to three decimal places.

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Identify the polynomial function given: \(f(x) = 4x^3 - 37x^2 + 50x + 60\).

Recall that a real zero of a polynomial function is a value of \(x\) for which \(f(x) = 0\). Our goal is to find such values.

Use the Intermediate Value Theorem to show the existence of a real zero: choose two values \(a\) and \(b\) such that \(f(a)\) and \(f(b)\) have opposite signs, indicating a root lies between \(a\) and \(b\).

Apply a root-finding method such as the Rational Root Theorem to test possible rational zeros or use numerical methods like the Newton-Raphson method or synthetic division to approximate the zero.

Once the zero is approximated, refine the value to three decimal places by iterating the numerical method or using a calculator with root-finding capabilities.

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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 that make the function equal to zero. Understanding how to find these zeros is essential for analyzing the behavior and roots of the polynomial.

Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function changes sign over an interval, then it must have at least one zero in that interval. This theorem helps to show the existence of real zeros for polynomial functions by evaluating the function at specific points.

Numerical Methods for Approximating Zeros

When exact zeros are difficult to find, numerical methods like the bisection method or Newton's method approximate zeros to a desired decimal accuracy. These iterative techniques refine estimates of zeros, which is necessary for finding the zero to three decimal places as requested.