A cubic polynomial has x=-3 as one of its roots and y-intercept 9. If the graph passes through (-2,3) and (1,16), find the equation of the polynomial.
You can use the given info to build a system and then solve for the coefficients.
The y-intercept is 9, therefore, x=0:
$\displaystyle a(0)^{3}+b(0)^{2}+c(0)+d=9$
A root is -3:
$\displaystyle a(-3)^{3}+b(-3)^{2}+c(-3)+d=0$
Passes through (-2,3):
$\displaystyle a(-2)^{3}+b(-2)^{2}+c(-2)+d=3$
Passes through (1,16):
$\displaystyle a(1)^{3}+b(1)^{2}+c(1)+d=16$
Now, you have 4 equations with 4 unknowns:
$\displaystyle \left[\begin{array}{cccc|c}0&0&0&1&9\\-27&9&-3&1&0\\-8&4&-2&1&3\\1&1&1&1&16\end{array}\right]$
Solve the system and you get:
$\displaystyle a=\frac{1}{3}, \;\ b=\frac{5}{3}, \;\ c=5, \;\ d=9$
Therefore, the polynomial is:
$\displaystyle \boxed{\frac{1}{3}x^{3}+\frac{5}{3}x^{2}+5x+9}$
This is one way to approach these. Perhaps try this on your other post regarding the quartic.