Binomial Products
Before we get to the fun math, let's explore Pascal's Triangle in a fun way. First, fill in the rest of the triangle below; be careful – don’t make any addition errors. Can you see the pattern and how it should be filled out? Next, pick your 2 favorite colors – color all the odd numbers one color and all the even numbers the other color – can you stay within the lines?
I'm not going to tell you what famous pattern you just colored is right now. The first one to get me the answer to it's special name and how the pattern was originally created will get some sort of small prize from me.
(over)
Ok, so – let's get to some math. Here is some background. Your textbook is about to show you some special product patterns (look on p.339). Ask yourself, if Mr. Wittry has me memorize these right now, will I still be able to recall them 2 weeks from now. The answer is: probably not. Even if you could, you'd certainly forget them by the final – and most certainly forget them over the summer. Then, your pre-calculus/IB teacher is going to get on me next year about you forgetting this stuff – what am I to do? Ok, maybe I'm exaggerating somewhat – but in any case, here is an easy way to expand any binomial.
First, the book says that if you take (a + b)3 and expand it, you'll get a3 + 3a2b + 3ab2 + b3. In their infinite wisdom, they then suggest you memorize that – hmm?
Look back at Pascal's Triangle at the first row that contains a 3 (we're trying to expand a cubic). Notice any similarity between that row and the underlined expansion?
Write out what pattern/similarity you notice:
Ok, now – the rest of the trick is easier for me to explain/show than type here – so take good notes (because this is not explained my way in the book – it's harder in the book (and in Ch. 12) filled with more convoluted notation than is necessary. Oh, by the way, what we're doing has a fancy name – it's called the Binomial Theorem. When you're done with my way, go look at p.710 of your book and see if you can decipher what they've given you – look like fun?
Add these to your homework for next time. You must show the steps the same way I did in class. Expand on another piece of paper – write them out nicely and leave lots of room between terms:
(a + 3b)3=(3x + 2y)5 = (x2 – 3y3)6 =(3x2 + 2y)7 =
Algebra II