Step 1: Write the general form of a cubic equation having the same zeros
A cubic function having the same zeros take the form:
[tex]y=k(x-a)^3[/tex]Step 2: Find the constant, k
Since the graph passes through the point (0, -5), substitute x = 0 and y = -5 into the general equation to find the constant, k.
[tex]\begin{gathered} -5=k(0-a)^3 \\ -5=k(-a)^3 \\ -5=-ka^3 \\ k\text{ = }\frac{-5}{-a^3} \\ k\text{ = }\frac{5}{a^3} \end{gathered}[/tex]Step 3: Substitute the value of k into the general equation to find the equation for the cubic polynomial function
[tex]\begin{gathered} y\text{ = }\frac{5}{a^3}(x-a)^3 \\ y\text{ = }\frac{5(x-a)^3}{a^3} \\ y\text{ = 5(}\frac{x-a}{a})^3 \\ y\text{ = 5(}\frac{x}{a}-\frac{a}{a})^3 \\ y\text{ = 5(}\frac{x}{a}-1)^3 \end{gathered}[/tex]The equation for the cubic polynomial function is therefore:
[tex]y\text{ = }\frac{5}{a}(x-1)^3[/tex]