For each polynomial function, identify its graph from choices A–F. ƒ(x)=(x-2)^2(x-5)

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. Understanding the degree and leading coefficient of a polynomial helps predict its end behavior and the number of roots.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its simpler polynomial factors. For the function f(x) = (x-2)^2(x-5), it is factored into two distinct roots: x = 2 (with multiplicity 2) and x = 5. This process is crucial for identifying the x-intercepts of the graph, which are points where the function crosses the x-axis.

Graphing Polynomial Functions

Graphing polynomial functions requires understanding their key features, such as intercepts, turning points, and end behavior. The graph of f(x) = (x-2)^2(x-5) will touch the x-axis at x = 2 and cross it at x = 5. The shape of the graph is influenced by the degree of the polynomial and the nature of its roots, which helps in visualizing the function's behavior.