Questions: Identify which of the graphs could be the graph of a polynomial function. f(x)=-x^2(x-2)(x+1)

Solution Steps

The given polynomial function is \( f(x) = -x^2(x - 2)(x + 1) \).

The polynomial \( f(x) = -x^2(x - 2)(x + 1) \) can be expanded to find its degree: \[ f(x) = -x^2(x^2 - x - 2) = -x^4 + x^3 + 2x^2 \] The degree of the polynomial is 4.

Since the leading term is \(-x^4\), the end behavior of the polynomial is:

• As \( x \to \infty \), \( f(x) \to -\infty \)
• As \( x \to -\infty \), \( f(x) \to -\infty \)

The roots of the polynomial are found by setting \( f(x) = 0 \): \[ -x^2(x - 2)(x + 1) = 0 \] The roots are \( x = 0 \) (with multiplicity 2), \( x = 2 \), and \( x = -1 \).

• The graph should have roots at \( x = 0 \), \( x = 2 \), and \( x = -1 \).
• The graph should have end behavior where both ends go to \(-\infty\).

Final Answer