Is there a simpler method of polynomial multiplication?

Question

I've been brushing up on my algebra lately and recently began to refresh my memory on polynomial multiplication. In the case of $monomial \cdot binomial$ this is very straight forward and I can see the usefulness of the expansion; for example: $2x(3 + 2x) = 6x + 4x^2$. This makes it easier to read in my opinion and thus is useful. However, given $polynomial \cdot polynomial$ where there are more than two terms on each it begins to easily confuse me:

$$(a + b + c)(d + e + f)$$

In this case we know that the multiplication is the equivalent of:

$$ad + ae + af + bd + be + bf + cd + ce + cf$$

This is a very simple case and is easy to understand but I can definitely see this growing out of control quickly:

$$(a + b + c)(d + e + f)(g + h + i)$$

Now in this case it expands very quickly and seems like it would be the cleanest answer to a more complex problem; plus it gets confusing to keep track of if you multiply the terms out:

$$adg + adh + adi + aeg + aeh + aei + afg + afh + afi + bdg + bdh + bdi + beg + beh + bei + bfg + bfh + bfi + cdg + cdh + cdi + ceg + ceh + cei + cfg + cfh + cfi$$

I understand that multiplication is a foundational operation and thus as a whole is typically the simplest answer (aside from addition); however, is there a simpler (or less confusing) way to perform the multiplication on larger polynomials; or a way to reduce the number of steps needed to reach the answer?

As a side question (if you feel like answering); is there a practical application for polynomial multiplication with more than two terms or is it typically reserved for educational purposes? A single example will do.

Answer 1

While there is no simpler way to multiply larger polynomials, there is a method that may make performing the operation easier to visualize. It is commonly known as the box method.

If you line up each polynomial on adjacent sides of a box like the picture shows and multiply each term by every other term, you get boxes that contains every term. It makes the process neater and it makes it easier to see which terms cancel or combine.

I hope this helps.

Answer 2

Multiplication of polynomials (and reduction modulo yet another polynomial) is one of the major tools in many standard encryption algorithms - so important that even a specific CPU instruction PCLMULQDQ was introduced for the purpose.