Given polynomials: P(x) = 4x 3 + 5x 2 -2x + 3 Q(x) = 2x 3 -3x 2 + x + 1 Calculate the product PQ(x) in the following three ways. Use standard algebraic polynomial multiplication (“big-FOIL” ) . Use...

Given polynomials:

P(x) = 4x³
+ 5x²
-2x + 3
Q(x) = 2x³
-3x²
+ x + 1

Calculate the product PQ(x) in the following three ways.

1. Use standard algebraic polynomial multiplication (“big-FOIL” ) .
   
   


2. Use Discrete Fourier Transform :

Evaluate both P(x) and Q(x) at the
eighth roots of unity.
(check your notes if you don’t remember, but basically z = cos 45 + i sin 45, and all its “powers”
in radical form!!

Multiply the values for the above to generate 8 points, (z, PQ(z) ) for all powers of z.


Use Iterpolation to find the polynomial PQ(x).