In Exercises 25–26, graph each polynomial function. f(x) = -x^3(x + 4)^2(x-1)

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' are constants. Understanding the structure of polynomial functions is essential for analyzing their behavior, including their roots and end behavior.

Factoring and Roots

Factoring a polynomial involves expressing it as a product of its linear factors, which directly relates to finding its roots or x-intercepts. For the function f(x) = -x^3(x + 4)^2(x - 1), the roots are x = 0, x = -4, and x = 1. The multiplicity of each root affects the graph's behavior at those points, such as whether the graph crosses or touches the x-axis.

Graphing Techniques

Graphing polynomial functions requires understanding key features such as intercepts, end behavior, and turning points. The degree of the polynomial indicates the maximum number of turning points, while the leading coefficient determines the direction of the graph's ends. By plotting the roots and analyzing the function's behavior around them, one can create an accurate representation of the polynomial's graph.