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

Verified step by step guidance

1. Start by identifying the degree of the polynomial function ƒ(x) = (x-2)^2(x-5). Since (x-2)^2 is squared, it contributes degree 2, and (x-5) contributes degree 1, so the total degree is 2 + 1 = 3.
2. Determine the leading term by multiplying the highest degree terms from each factor: (x)^2 * (x) = x^3. This tells us the polynomial is cubic with a positive leading coefficient, so the end behavior will be: as x → ∞, ƒ(x) → ∞ and as x → -∞, ƒ(x) → -∞.
3. Identify the zeros of the function from the factors: x = 2 (with multiplicity 2) and x = 5 (with multiplicity 1). The multiplicity affects the graph's behavior at these points.
4. Analyze the behavior at each zero: at x = 2, since the multiplicity is even (2), the graph will touch the x-axis and turn around (bounce off) at this point; at x = 5, with multiplicity 1 (odd), the graph will cross the x-axis.
5. Use this information to match the graph: look for a cubic graph that crosses the x-axis at 5, touches and bounces at 2, and has the end behavior of going down to the left and up to the right.

Key Concepts

Polynomial Functions and Their Graphs

A polynomial function is an expression involving variables raised to whole-number exponents combined using addition, subtraction, and multiplication. Its graph is a smooth curve with no breaks, and the degree of the polynomial determines the general shape and number of turning points.

Zeros and Their Multiplicities

The zeros of a polynomial are the values of x that make the function equal to zero. The multiplicity of a zero indicates how many times that root is repeated, affecting the graph's behavior at that point: even multiplicities cause the graph to touch and turn around, while odd multiplicities cause it to cross the x-axis.

End Behavior of Polynomial Functions

End behavior describes how the graph behaves as x approaches positive or negative infinity. It depends on the leading term's degree and coefficient: for example, an odd-degree polynomial with a positive leading coefficient falls to the left and rises to the right, guiding the overall shape of the graph.