Thursday, September 22, 2011

Special Polynomial Products

Following up on my previous post about Polynomials in Mathematics, I wanted to dedicate a brief post to special polynomial products.  These are products that will occur very frequently throughout your mathematics studies, and so it is a very good idea to remember these products.

If you would like to refresh on multiplying together polynomials in general, follow the link to view my previous post about polynomials.  Alternately, if you would like to see how to FOIL polynomials, a special method you can use to multiply together binomials, I refer you to my previous post on this as well.

Now, on to the Special Polynomial Products.

These are not overly complicated, and can be derived using basic knowledge of multiplying together polynomials.  But, as I said, these will occur so frequently, that you might as well memorize them so you can save time and not have to worry about deriving them every time you see them.

(A - B)(A + B) = A2 - B2
This first one is the easiest to arrive at.  Notice that the signs are OPPOSITES!  Simply apply the FOIL method to the starting expression, and you will arrive at the simplified product.  I'll leave that for you to check for yourself.

(A + B)= A2 + 2AB + B2
(A - B)= A2 - 2AB + B2
This set is also quite simple to see, once you FOIL them out.  The starting terms are squared, you can easily rewrite them as (A+B)(A+B), etc, apply the FOIL method, and come up with the product above.

(A + B)= A3 + 3A2B + 3AB2 + B3
(A - B)= A3 - 3A2B + 3AB2 - B3
This set, obviously, is a bit tougher to arrive at, but by tougher I only mean 'requires more work'.  There aren't any tricks and they aren't any more difficult than the others.  I'll derive one for you at the end, after I show you the final set of special products, which is...

(A + B)(A- AB + B2) = A3 + B3
(A - B)(A+ AB + B2) = A3 - B3
This set requires the most work to derive, so you can see what I mean when I say it pays to memorize these so that you don't have to mess around every time you see them!  I won't derive these ones, but you should try to on your own so that you see how they work and how to arrive at the products IN CASE YOU FORGET THEM!  It's ALWAYS a good idea to understand how to do all these things so that you can do them WITHOUT the shortcuts.

Anyways, now I will get you started by showing you how to multiply (A + B)3 and arrive at the special product I showed you above:

(A+B)3.....rewrite this
(A+B)[(A+B)(A+B)]..... note I've isolated 2 binomials so that I can show you to FOIL easier
(A+B)(A2 + 2AB + B2)..... now, multiply the two polynomials together.  Multiply everything in the second term by A, then multiply everything in the second term by B.
(A)(A2) + (A)(2AB) + (A)(B2) + (B)(A2) + (B)(2AB) + (B)(B2)
A3 + 2A2B + AB2 + A2B + 2AB2 + B3... Now you can combine like terms.
A3 + 3A2B + 3AB2 + B3

And there you have it.  It took a little bit of work, but it's not that hard.  You just have to keep going at it.

So, those are the special products of polynomials.  Try to memorize them to make you homework easier, but MORE IMPORTANTLY, please try to work through the derivations and understand HOW to arrive at the products.

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