Tutorial: Multiplying Polynomials

And now for some algebra: How to multiply polynomials.

I’m planning on including some math tutorials in here. Once I get enough of them posted, I’ll make a place for you to easily find them all.

When students are first learning how to multiply polynomials, they usually begin with binomials (like (x+3)(x+5), for example), and then learn the F.O.I.L. mnemonic. While F.O.I.L. is useful, it tends to obscure what’s really going on: the distributive property. The F.O. part distributes the first term of the first binomial to both terms of the second, x(x+5), and the I.L. distributes the second term of the first binomial, 3(x+5). See below:

The other trouble with F.O.I.L. is that it breaks down if either of the polynomials has more than two terms. If you’re concerned about forgetting terms if you stop using F.O.I.L. and just remember the distributive property, there’s another technique you can use. I don’t know an official name for it, so let’s call it the Box Method. See below to watch the same problem as above solved with the Box Method:

In case it’s not clear from the video, the idea of the Box Method is that you’re labeling each rectangle with the area it contains. The orange rectangle has side lengths of x and x, so the area is x*x = x^2. Similarly, the pink rectangle has side lengths of x and 3, so the area is x*3 = 3x. Once you have all of the rectangle areas, you add them together. Now, let’s check out both methods with more complicated polynomials: (2x^2-3x+5)(x^3+2x^2+x-1)

As you can see, the box method ends up being a good way to keep track of all of your terms. As an added bonus, sometimes the terms you can combine (the “like terms”) line up nicely along the diagonal!

If you have good handwriting and are good at carefully following procedures, multiplying using the distributive method will probably work well for you. However, if you’re at all concerned about forgetting to multiply a pair of terms, or of not being able to see where your results came from, I recommend you get into the habit of using the box method. It might take up a bit more space on your page, but you’re much less likely to make careless errors, and if you to, it should be easier to find them.

Do you find these tutorial videos helpful? What other topics can I write about that would be useful for you? I’d love for you to let me know in the comments below!

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