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

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1. Identify the roots of the polynomial function \( f(x) = (x-2)^2(x-5) \). The roots are \( x = 2 \) and \( x = 5 \).
2. Determine the multiplicity of each root. The root \( x = 2 \) has a multiplicity of 2, and the root \( x = 5 \) has a multiplicity of 1.
3. Analyze the behavior of the graph at each root. Since \( x = 2 \) has an even multiplicity, the graph will touch the x-axis at this point and turn around. At \( x = 5 \), with an odd multiplicity, the graph will cross the x-axis.
4. Consider the end behavior of the polynomial. Since the leading term is positive and the degree of the polynomial is 3 (odd), the graph will start from the bottom left and end at the top right.
5. Use the information about the roots, their multiplicities, and the end behavior to match the polynomial function to its graph from the given choices.

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.