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 Find the zero in part (b) to three decimal places.

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the real zeros of the polynomial function \( f(x) = 3x^3 - 8x^2 + x + 2 \).
2. Step 2: Use the Rational Root Theorem to list possible rational zeros. The possible rational zeros are the factors of the constant term (2) divided by the factors of the leading coefficient (3).
3. Step 3: Test the possible rational zeros using synthetic division or direct substitution to see if they make the polynomial equal to zero.
4. Step 4: Once a real zero is found, use it to factor the polynomial and simplify it to a quadratic equation.
5. Step 5: Solve the quadratic equation using the quadratic formula or factoring to find the remaining zeros, and ensure to find the zero in part (b) 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 'n' is a non-negative integer and 'a_n' are constants. 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 can be determined using various methods, such as the Rational Root Theorem, synthetic division, or numerical methods like the Newton-Raphson method. Identifying these zeros is crucial for understanding the function's graph and its intersections with the x-axis.

Numerical Methods for Root Finding

Numerical methods are techniques used to approximate the solutions of equations when analytical solutions are difficult or impossible to obtain. For polynomial functions, methods such as the bisection method, Newton's method, or the secant method can be employed to find real zeros to a specified degree of accuracy. These methods are particularly useful for finding zeros to three decimal places, as required in the problem.