4.10+Binomial+Theorem+(1-4)

Minds In Action
To start out the day we read aloud the minds in action dialogue that is found in media type="custom" key="12010017"4.10. This dialogue primarily talks about why the coefficients in the Binomial theorem are choose notation. If we want to look at media type="custom" key="12010035" then the coefficient of media type="custom" key="12010039" will be media type="custom" key="12010043" because we are choosing which three factors of media type="custom" key="12010045" are giving us a factor of b. The remaining factors all provide a as a factor.

Different Forms for the Binomial Theorem:
The form of the binomial theorem that is found in the book and that is most closely linked to the discussion in Minds in Action is: media type="custom" key="12010151"

However, there are other forms that we sometimes see when we look at the binomial theorem. For one thing the form above looks at choosing which of the media type="custom" key="12010131" factors are providing the b's. We could just as easily look at which of the factors provides the a's and then the binomial theorem would look like this:

media type="custom" key="12010243"

Both of the forms presented above are kind of ugly as we don't know what n is and we have to use ellipses in the formula. Sometimes it is more convenient to use summation notation to condense the presentation. The first version of the binomial theorem can be condensed as:

media type="custom" key="12010195"

while the second can be condensed once you reverse it's order:

media type="custom" key="12019805"

Or if you prefer not to reverse the order it can be condensed as:

media type="custom" key="12010255"

Examples:
Once we looked at these various presentation of the binomial theorem. We then looked at some examples of how to use the Binomial theorem in some different ways.


 * FIRST** We expanded some binomials using the theorem above. It got a little trickier when the coefficients weren't 1

media type="custom" key="12019853"

Many students initially didn't put the media type="custom" key="12019043" in the parenthesis. The next day we looked at why the parentheses are necessary. Each factor of media type="custom" key="12019065" contributes either an media type="custom" key="12019071" or a media type="custom" key="12019075". So the term in the answer that has a media type="custom" key="12019083" must have had 2 factors that contributed an media type="custom" key="12019093" and three factors that contributed a media type="custom" key="12019099" which means that you really have media type="custom" key="12019123". It's annoying that the rules of exponents keep coming into play, but they do matter. When you write media type="custom" key="12019147" which is not the same thing. (This additional discussion actually happened the next day on the 5th).


 * SECOND** We also looked at some questions about identifying the coefficient on a particular term in a large expansion.


 * Finally** we looked at recoginizing the binomial theorem in the other direction.

If I am asked to evaluate media type="custom" key="12019309" then with an aha moment I recognize that it almost looks like the first condensed form above. However, I have media type="custom" key="12019289" and I don't have a factor with an exponent of media type="custom" key="12019303". If I want to create that second factor, I will have to use the base 1 so that I don't change the value of the sum. So my first step is to realize that: media type="custom" key="12019327" and that second sum matches the binomial theorem so now I can say that

media type="custom" key="12019331" Surprisingly this also tells me that if I take any row of Pascal's triangle and alternate adding and subtracting across the row that the final answer will be zero. Neat.