Final answer:
The zeros of the function ƒ(x) = x²(x – 100)(x – 200) are x=0 with multiplicity 2, and x=100 and x=200 each with multiplicity 1, resulting in the graph touching the x-axis at x=0 and crossing at x=100 and x=200.
Step-by-step explanation:
The zeros of the polynomial function ƒ(x) = x2(x – 100)(x – 200) are the values of x for which the function equals zero. These zeros are the x-intercepts of the graph. The given function clearly shows zeros at x=0, x=100, and x=200. Multiplicity refers to the number of times a particular factor appears in the polynomial.
The zero at x=0 has a multiplicity of 2 because the factor x appears squared in the function. The zeros at x=100 and x=200 each have a multiplicity of 1 since these factors appear only once. The multiplicity affects how the graph behaves at these intercepts. When the multiplicity is even, the graph will touch the x-axis but not cross it. When the multiplicity is odd, the graph will cross the x-axis.
Therefore, since the zero at x=0 has an even multiplicity of 2, the graph will touch and not cross the x-axis at that point. The zeros at x=100 and x=200 both have an odd multiplicity (1), so the graph will cross the x-axis at these points. The correct answer is (d): Zeros: 0, 100, 200; Multiplicity: 2, 1, 1; Touch/Cross.