Multiplying General Polynomials

Here's what the multiplication looks like when it's done horizontally:

That was painful! Now I'll do it vertically:

That was a lot easier! But, by either method, the answer is the same:

I'm just going to do this one vertically; horizontally is too much trouble.

Note that, since order doesn't matter for multiplication, I can still put the "x + 2" polynomial on the bottom for the vertical multiplication, just as I always put the smaller number on the bottom when I was doing regular vertical multiplication with just plain numbers back in grammar school.

x⁴ + 5x³ + 10x² − 9x − 34

I'll take my time, and do my work neatly:

6x⁴ − 6x³ − 47x² + 83x − 35

First off, I notice that terms of these polynomials have some power (that is, degree) "gaps".

The first polynomial has an x³ term, an x² term, and a constant term, but no x term; and the second polynomial has an x³ term, an x term, and a constant term, but no x² term. When I do the vertical multiplication, I will need to leave spaces in my set-up, corresponding to the "gaps" in the degrees of the polynomials' terms, because I will almost certainly need the space.

(This is similar to using zeroes as "place holders" in regular numbers. You might have a thousands digit of 3, a hundreds digit of 2, and a units digit of 5, so you'd put a 0 in for the tens digits, creating the number 3,205.)

Here's what that looks like:

See how I needed the gaps? See how it helped that I had everything lined up according to the term's degree? If I hadn't left gaps when writing out my original factors, my terms could easily have become misaligned in the rows below. By taking the time to write things out explicitly neatly, I saved myself from many needless difficulties.

My answer is: