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  $-5=f(0)=b(0-a)^{3}=b(-a) ^{3}=-ba ^{3}$

$-5=-ba ^{3}$

$5=ba ^{3}$

$b= \frac{5}{ a^{3} }$

5. the function is $f(x)=\frac{5}{ a^{3} }(x-a)^{3}$ where a can be any real number except 0