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.

Verified step by step guidance

1. insert step 1> Identify the polynomial function given: $f(x)=4x^3-37x^2+50x+60$.
2. insert step 2> Use the Intermediate Value Theorem to determine if there is a real zero. Check the values of $f(x)$ at different points to see if there is a sign change.
3. insert step 3> Apply the Rational Root Theorem to list possible rational zeros. These are the factors of the constant term (60) divided by the factors of the leading coefficient (4).
4. insert step 4> Use synthetic division or polynomial division to test the possible rational zeros and find one that results in a remainder of zero.
5. insert step 5> Once a real zero is found, use numerical methods such as the Newton-Raphson method or a graphing calculator to refine the zero to three decimal places.

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 structure of polynomial functions is essential for analyzing their behavior, including finding their zeros.

Real Zeros

A real zero of a polynomial function is a value of x for which the function evaluates to zero, meaning f(x) = 0. The existence of real zeros is crucial for understanding the roots of the polynomial, which can be found using various methods such as factoring, the Rational Root Theorem, or numerical approximation techniques. Identifying these zeros helps in graphing the polynomial and understanding its intersections with the x-axis.

Numerical Methods for Finding Zeros

Numerical methods, such as the Newton-Raphson method or bisection method, are techniques used to approximate the zeros of functions when analytical solutions are difficult to obtain. These methods involve iterative calculations to hone in on a value that satisfies f(x) = 0 to a specified degree of accuracy. In the context of the given polynomial, using numerical methods allows for finding the zero to three decimal places, which is often necessary in practical applications.