Question

How can I find a polynomial of the form y=ao+a1x+a2x^2+a3x^3 which passes through the points (-3,-2), (-1,2), (1,5), (2,1)?

Answer 1

Take each of the points and plug the x and y values into the equation to yield a linear equation with the constants a, b, c, and d as the "variables". Make sure your equation is consistent: use \[d+cx+bx ^{2}+ax ^{3}=y\] or something similar, and make sure your columns are properly formatted. I'll show you how it works for (-3,-2), but you should do the rest of the points on your own for practice.
\[\left[ d-3c+9b-27a=-2 \right]\]
Then, once you have a 4 X 4 matrix augmented with the y values, solve your matrix.

Answer 2

or you can use Lagrange polynomial ! http://en.wikipedia.org/wiki/Lagrange_polynomial
\[x_0=-3;y_0=-2 \]
\[x_1=-1;y_1=2\]
\[x_2=1;y_2=5\]
\[x_3=2;y_3=1\]
\[P=y_0l_0+y_1l_1+y_2l_2+y_3l_3\]