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

Verified step by step guidance

1. Identify the degree of the polynomial function \( f(x) = (x-2)^2(x-5)^2 \). Since each factor is squared, the degree is \( 2 + 2 = 4 \).
2. Determine the roots of the polynomial. The roots are \( x = 2 \) and \( x = 5 \), each with multiplicity 2.
3. Understand the behavior at each root. Since both roots have even multiplicity, the graph will touch the x-axis at these points and turn around, rather than crossing it.
4. Consider the end behavior of the polynomial. Since the degree is 4 (even) and the leading coefficient is positive, the graph will rise to positive infinity on both ends.
5. Match these characteristics (degree, roots, behavior at roots, and end behavior) with the given graph choices A–F to identify the correct graph.

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 and Roots

Factoring a polynomial involves expressing it as a product of its linear factors. The roots of the polynomial are the values of x that make the function equal to zero. In the given function f(x) = (x-2)^2(x-5)^2, the roots are x = 2 and x = 5, each with a multiplicity of 2, indicating that the graph touches the x-axis at these points without crossing it.

Graph Behavior and Multiplicity

The behavior of a polynomial graph at its roots is influenced by the multiplicity of those roots. If a root has an even multiplicity, the graph will touch the x-axis and turn around at that point, while an odd multiplicity means the graph will cross the x-axis. In this case, since both roots (2 and 5) have even multiplicities, the graph will touch the x-axis at these points and not cross it.