polynomial multiplication as convolution

Multiplication of polynomials can be view as the discrete convolution of its coefficients ¹

We can visualize this process using the diagonal sum visualization of convolution.

More formally, if we view a polynomial as a sum of coefficients multiplied by corresponding powers of :

Suppose we have two polynomials, and . When we multiply them together, the resulting polynomial is where for each term :

And the entire polynomial is:

Footnotes

1. But what is a convolution? - YouTube ↩