PreCalcP7­ 10.5 ­ Binomial Theorem Proof
Find the Next Row...
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Find the Next Row...Another Way!
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C2 PreCalcP7­ 10.5 ­ Binomial Theorem Proof
A Real, Rigorous, and Important Proof (or Two)
The Binomial Theorem is an important theorem in algebra, and uses many of the things that we have covered so far.
You will not be tested on the proof...but this is a good chance for you to get a sense of what a mathematical proof is really like (and it's not as much like the ones you may remember from geometry class).
In this proof, we will use induction, combinatorics, sigma notation, and more to prove this nice formula:
In our next class, we will see how we can apply it, but for now, let's just enjoy this proof!
Step 1: Prove Pascal's Identity
No, not who he was...this thing here:
Start by writing the equation with factorials:
Now, give the left side the same denominator (multiply left by r, and right by k ­ r + 1:
Add the two fractions on the left:
Simplify the factorials:
Factor the top and simplify:
More simplification of factorials:
PreCalcP7­ 10.5 ­ Binomial Theorem Proof
Step 2: Show It's True for n = 1 and n = 2
when n = 1:
when n = 2:
Step 3: Assume It's True for n ­ What about n + 1?