Exercises 53–60 show incomplete graphs of given polynomial functions. a) Find all the zeros of each function. b) Without using a graphing utility, draw a complete graph of the function. f(x)=−x^3+x^2+16x−16

Verified step by step guidance

1. Identify the polynomial function: \( f(x) = -x^3 + x^2 + 16x - 16 \).
2. Use the Rational Root Theorem to list possible rational zeros, which are the factors of the constant term divided by the factors of the leading coefficient.
3. Test the possible rational zeros using synthetic division or direct substitution to find actual zeros of the polynomial.
4. Once a zero is found, factor the polynomial by dividing it by \( (x - \text{zero}) \) to simplify the polynomial and find other zeros.
5. Use the zeros and the end behavior of the polynomial (determined by the leading term \(-x^3\)) to sketch a complete graph of the function, noting that the graph will have a downward end behavior on both sides.

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 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' is not zero. Understanding the structure of polynomial functions is essential for analyzing their behavior, including finding zeros and graphing.

Finding Zeros

The zeros of a polynomial function are the values of 'x' for which f(x) = 0. These points are crucial as they indicate where the graph intersects the x-axis. To find the zeros, one can use methods such as factoring, the Rational Root Theorem, or synthetic division, depending on the degree and complexity of the polynomial.

Graphing Polynomial Functions

Graphing polynomial functions involves plotting points based on the function's values and understanding its general shape, which is influenced by the degree and leading coefficient. Key features to consider include intercepts, end behavior, and turning points. A complete graph can be sketched by identifying zeros, evaluating the function at various points, and analyzing the intervals of increase and decrease.