# Math Help - Polynomial (evaluation - multiplication - interpolation)

1. ## Polynomial (evaluation - multiplication - interpolation)

Hello, I have no idea on how to solve this problem:
Given the polynomials $A=x^3 +2x -1 , B=x^2-x+3$ evaluate them for $x=-2:1:3$
and calculate their product and its coefficients.
I need another way on doing this, instead of calculating A and B one by one
(I shouldn't use the simple method ie. $x=-1: A_{-1}=(-1)^3 +2(-1) -1= -4$ ....).
How can this be done? Thank you.

2. Why should you not use the "simple" method?

If you really want to do it "the hard way", you can use "synthetic division":

-2) 1 0 -2 -1 ("1 0 -2 -1" are the coefficients of [tex]x^3- 2x- 1)
-2 4 -4
1 -2 2 -5 so A(x)= -5

where I brought down the first "1" to the third row then multiplied it by -2 to get the -2 in the second row and added it to 0 to get -2 in the third row, multiplied that by -2 to get the 4 in the second row, add that to -2, etc.

To multiply A and B, use the "distributive law":
$(x^3- 2x- 1)(x^2- x+ 3)= x^3(x^2- x+ 3)- 2x(x^2- x+ 3)- 1(x^2- x+ 3)$ and distribute again: $(x^3)(x^2)- (x^3)(x)+ (x^3)(3)- (2x)(x^2)+ (2x)(x)- (2x)(3)- (1)(x^2)- (1)(x)+ (1)(3)$