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 zeros of the polynomial by setting each factor equal to zero: from \((x-2)^2\) we get \(x=2\), and from \((x-5)^2\) we get \(x=5\).
2. Determine the multiplicity of each zero: both zeros have multiplicity 2, which means the graph will touch the x-axis at these points and turn around (no crossing).
3. Note the leading term behavior: since the polynomial is \(f(x) = -(x-2)^2 (x-5)^2\), when expanded, the highest degree term will be \(-x^4\), indicating the graph behaves like \(-x^4\) for large \(|x|\) (both ends go down).
4. Use the sign of the leading coefficient (negative) to confirm that as \(x \to \pm \infty\), \(f(x) \to -\infty\), so the graph falls on both ends.
5. Combine all this information: the graph touches the x-axis at \(x=2\) and \(x=5\) without crossing, and both ends go down. Use these characteristics to match the correct graph from choices A–F.

Key Concepts

Polynomial Function and Degree

A polynomial function is an expression involving variables raised to whole-number exponents combined using addition, subtraction, and multiplication. The degree of the polynomial is the highest exponent sum in any term, which influences the general shape and end behavior of its graph.

Zeros and Their Multiplicities

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; even multiplicities cause the graph to touch the x-axis and turn around, while odd multiplicities cause the graph to cross the axis.

End Behavior of Polynomial Graphs

The end behavior describes how the graph behaves as x approaches positive or negative infinity. It depends on the leading term's degree and coefficient sign: an even degree with a negative leading coefficient means both ends of the graph point downward.