which polynomial function has x - intercepts -1, 0, and 2 and passes through the point (1…

Explanation:

Step1: Recall x - intercepts form

If the x - intercepts of a polynomial are (a), (b), (c), the polynomial can be written as (f(x)=k(x - a)(x - b)(x - c)). Here (a=-1), (b = 0), (c = 2), so (f(x)=k(x + 1)x(x - 2)=k(x^{3}-x^{2}-2x)).

Step2: Find the value of (k)

Since the polynomial passes through the point ((1,-6)), substitute (x = 1) and (y=-6) into (f(x)=k(x^{3}-x^{2}-2x)). We get (-6=k(1^{3}-1^{2}-2\times1)), which simplifies to (-6=k(1 - 1-2)=-2k). Solving for (k) gives (k = 3).

Step3: Determine the polynomial function

Substitute (k = 3) into (f(x)=k(x^{3}-x^{2}-2x)), we get (f(x)=3x^{3}-3x^{2}-6x).

Answer:

(f(x)=3x^{3}-3x^{2}-6x) (the second option)