Reduce the number of multiplications required for the product of two polynomials

This was a problem done in my CIS 315 Algorithms class in Spring 2006.

The product of two linear polynomials ax+b and cx+d

[tex]\LARGE (ax+b)(cx+d)[/tex]

is

[tex]\LARGE acx^2 + (ad+bc)x + bd[/tex]

Note that this requires 4 different multiplications of the coefficients:
ac, ad, bc and bd.

We want to find a way to determine this product with only 3 multiplications.

let:
L = (a+b)(c+d)

note what happens when we FOIL L:
L = ac + ad + bc + bd

We think that maybe we can combine the middle term (ad+bc) with only one multiplication. What we want is ad + bc, so subtract ac and bd from L:

J = ac
K = bd

L – J – K = (ac + ad + bc + bd) – ac – bd

Note at this point we can rewrite the original FOILed product as:
[tex]\LARGE Jx^2 + (L – J – K)x + K[/tex]

We now have solved the problem with only 3 multiplications. They are:
J, K and L, or ac, bd, and (a+b)(c+d)