Question

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

Accepted Answer

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. 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) → -∞. 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. 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. 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.

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

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Key Concepts

Polynomial Functions and Their Graphs

Zeros and Their Multiplicities

End Behavior of Polynomial Functions