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

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 graphing them, as it helps identify their degree, leading coefficient, and behavior at infinity.

Factoring and Roots

Factoring a polynomial involves expressing it as a product of its linear factors, which correspond to its roots or x-intercepts. For the function f(x) = 2x^2(x - 1)^3(x + 2), the roots are x = 0 (with multiplicity 2), x = 1 (with multiplicity 3), and x = -2 (with multiplicity 1). Knowing the roots is crucial for graphing, as they indicate where the graph intersects the x-axis and how it behaves near those points.

Graphing Techniques

Graphing polynomial functions requires understanding their key features, including intercepts, end behavior, and turning points. The degree of the polynomial determines the number of turning points, while the leading coefficient influences the direction of the graph as x approaches positive or negative infinity. Techniques such as plotting key points, analyzing symmetry, and using the roots help create an accurate representation of the polynomial's behavior.