• Aug 9th 2007, 12:28 PM
polymerase
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.:rolleyes:
• Aug 9th 2007, 02:16 PM
galactus
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.