5. a student constructed the following table of values for a third - degree polynomial\nthe equation of the…

Explanation:

Step1: Recall zero - product property

A polynomial (y = a(x - r_1)(x - r_2)(x - r_3)) has roots (r_1), (r_2), (r_3) when (y = 0). From the table, the roots of the third - degree polynomial are (x=-4), (x = 0), and (x = 5) since (y = 0) at these (x) - values. So the polynomial can be written in the form (y=a(x + 4)(x-0)(x - 5)=ax(x + 4)(x - 5)).

Step2: Find the value of (a)

We can use another point from the table, say ((1,5)). Substitute (x = 1) and (y = 5) into (y=ax(x + 4)(x - 5)). Then (5=a\times1\times(1 + 4)\times(1 - 5)). First, simplify the right - hand side: ((1 + 4)\times(1 - 5)=5\times(-4)=-20). So the equation becomes (5=a\times(-20)). Solving for (a), we get (a=-\frac{1}{4}).

Step3: Write the polynomial equation

The polynomial equation is (y =-\frac{1}{4}x(x + 4)(x - 5)), which is equivalent to (y=-\frac{1}{4}x(x - 5)(x + 4)).

Answer:

C. (y =-\frac{1}{4}x(x - 5)(x + 4))