Formulas for the product of sums

The product of sums formula generalizes to multiplying any two polynomials. If $A(x) = a_1 + a_2 + ext{...} + a_n$ and $B(x) = b_1 + b_2 + ext{...} + b_m$, then:

$A(x)B(x) = (a_1 + a_2 + ... + a_n)(b_1 + b_2 + ... + b_m) = \sum_{i=1}^n \sum_{j=1}^m a_i b_j$

This means you multiply every term from the first sum by every term from the second sum, and add all the products.

Important properties

• The distributive property allows us to expand products of sums.

• The number of terms in the expanded product is the product of the number of terms in each sum.

• This process is the basis for multiplying polynomials.