The Lagrange interpolation polynomial is a sum of products.
(a) [tex]f(x)=1\times \frac{(x-2)(x-3)}{(0-2)(0-3)}+3\times \frac{(x-0)(x-3)}{(2-0)(2-3)}+0\times \frac{(x-0)(x-2)}{(3-0)(3-2)}[/tex] This can be simplified to [tex]f(x)=\frac{1}{3}(-4x^{2}+11x+3)[/tex]
(b) [tex]g(x)=0+\frac{1(x+1)(x-3)(x-5)}{(2+1)(2-3)(2-5)}+\frac{1(x+1)(x-2)(x-5)}{(3+1)(3-2)(3-5)}+\frac{2(x+1)(x-2)(x-3)}{(5+1)(5-2)(5-3)}[/tex] This can be simplified to [tex]g(x)=\frac{1}{24}(x^{3}-6x^{2}+11x+18)[/tex]
