Question

Suppose the graph of a cubic polynomial function has the same zeroes and passes through the coordinate (0, –5). Describe the steps for writing the equation of this cubic polynomial function.

Answer 1

Use the zeroes to determine the roots.

Write the polynomial as a product of the leading coefficient, a, and the factors, where each factor is x minus a root.

Use the y-intercept (0, –5) to solve for the leading coefficient.

Substitute the leading coefficient into the polynomial function for a and simplify.

Answer 2

1. "the graph has the same zeros" : so let a be the "triple" root of the cubic polynomial function.

2. So f(x)=
`(x-a)^(3)`

3. Don't forget that the expression might have a coefficient b as well, and still maintain the conditions:


`f(x)=b(x-a)^(3)`

4. Now, f(0)=-5 so
`-5=f(0)=b(0-a)^(3)=b(-a) ^(3)=-ba ^(3)`


`-5=-ba ^(3)`


`5=ba ^(3)`


`b= (5)/( a^(3) )`

5. the function is
`f(x)=(5)/( a^(3) )(x-a)^(3)` where a can be any real number except 0