Questions: This question: 1 point() possible Graph the polynomial function. Factor first if the expression is not in factored form. f(x)=x^2(x-2)(x+4)^2 Choose the correct graph below. A. B. C. D.

Solution Steps

The function f(x) = x²(x-2)(x+4)² has the following zeros and multiplicities:

• x = 0 with multiplicity 2 (even multiplicity means the graph touches the x-axis at this point)
• x = 2 with multiplicity 1 (odd multiplicity means the graph crosses the x-axis at this point)
• x = -4 with multiplicity 2 (even multiplicity means the graph touches the x-axis at this point)

The degree of the polynomial is 2 + 1 + 2 = 5, which is odd. The leading coefficient is positive (1). Therefore, the end behavior is as follows:

• As x approaches negative infinity, f(x) approaches negative infinity.
• As x approaches positive infinity, f(x) approaches positive infinity.

• Option A: The graph touches the x-axis at x = 0 and x = -4, and crosses at x = 2. This matches our analysis. The end behavior is also consistent.
• Option B: The graph has the correct roots, but the end behavior is flipped, suggesting a negative leading coefficient.
• Option C: The graph has the correct roots, but the end behavior is flipped, suggesting a negative leading coefficient.
• Option D: The graph appears to have a zero near x=-1, which our function doesn't have. Also, the graph passes through x=0 as if it were odd multiplicity.

Final Answer: The correct graph is A.