# Thread: 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: $(x^3)(x^2)- (x^3)(x)+ (x^3)(3)- (2x)(x^2)+ (2x)(x)- (2x)(3)- (1)(x^2)- (1)(x)+ (1)(3)$

I shouldn't use the simple method because we need a faster way on doing this
(for example, when the calculations are made by a computer
it is too slow to calculate them for a wide range of values).
I think that a way to evaluate the polynomials makes use of matrices,
the results are correct but I don't know if this is an efficient way.

This is how I have done it (see attachment)

I have done the same hor B(x) and again the results are correct.
I now have to do the multiplication and interpolation...