Fun With Polynomials

If you came here you're either surprised what a polynomial is or what the fun with it is, so let me explain...
There are these things, you probably heard the name before: Binomials , they're Polynomials but very specific ones. Bi just standing for 2 because Binomials are (a+b)², Polynomials on the other hand are (a+b)^x.
Lets just start easy: Pascal's Triangle
- - - - - - - 1 - - - - - - - (11^0)
- - - - - - -1 1- - - - - - -(11^1)
- - - - - - 1 2 1 - - - - - -(11^2)
- - - - --1 3 3 1 - - - - - (11^3)
- - - - -1 4 6 4 1 - - - - (11^4)
and so on... the pattern is really easy, as you can see in the first row (from now on referred to as 0th for.. reasons) there is the number 1 which is any polynomial with the power 0 (a+b)^0 = 1 .
The 1st row is any polynomial with the power 1 (a+b)^1 = 1a +1b
Now it gets to school levels, Binomials (power 2) (a+b)² = 1a²b^0 + 2a^1*b^1 + b²a^0, notice how complicated I put it to clarify that thw powers go in reverse order, in this example a is counting down and b is counting up.
The 3rd power is something we have to use either the method I will use soon or Pascal's Triangle. As you can see in Pascals Triange there is 1 3 3 1, meaning that there will be 1a³ + 3a²b^1 + 3a^1*b² + 1b³ , you might see the pattern by now. A fun thing I've noticed that you can "remember" the first 5 rows of it just by simply doing 11^x, x being the number of row you want to have, this is sort of coincidental and will not work with the 5th row just as simple. But now that the introduction is over let me come to my point:

Given a polynomial with the power x and a and b being inside the brackets you will have 2^x possibilities.
That means that a polynomial with the power 3 (a+b)³ , which has a³+3a²b+3ab²+b³ has 8 possible ways
(1+3+3+1=8). This works for more objects inside the brackets, too. Which means, that a polynomial with the power x and a objects inside the brackets has a^x results. Now that you have the basic knowledge, I am coming to my point: This not only works for abstract mathematical concepts like Polynomials but also for Probabilities (with putting back). That means you can display Polynomials in a branch-like diagram and list all the possibilities in an ab form.

If you have the options a and b and the amount of attempts x then listing all possible events is easily done by saying (a+b)^x . Example: You are at a raffle. You will have to put back the balls / cards / chips / whatever you draw.
You will draw twice. There are 2 possible events that can happen; Failure (F) and Prize (P). The whole amount of options is (F+P)² = F²+2FP+P². Given that the probability for F to happen is 90% and P is the rest, the probabilites are 0.81+0.18+0.01 (or 81% + 18% + 1%) . The (a+b)^x will always be 1 (or 100%).
If given the task, the probability of at least 1 F you just have to look: F²+2FP = 0.99 (or 99%).

Now moving on to what I was originally trying to say: you can basically say that e.g. (a+b)² = a(a+b) + b(a+b)
= aa + ab + ba + bb . Now it gets interesting; this listing can and will now be done by me, but at a much larger scale and we'll see what happens.

(a+b)^5 =
a(a(a(a(a+b) = aaaaa+aaaab
a(a(a(b(a+b) = aaaab+aaabb
a(a(b(a(a+b) = aaaab+aaabb
a(a(b(b(a+b) = aaabb+aabbb
a(b(a(a(a+b) = aaaab+aaabb
a(b(a(b(a+b) = aaabb+aabbb
a(b(b(a(a+b) = aaabb+aabbb
a(b(b(b(a+b) = aabbb+abbbb
b(a(a(a(a+b) = aaaab+aaabb
b(a(a(b(a+b) = aaabb+aabbb
b(a(b(a(a+b) = aaabb+aabbb
b(a(b(b(a+b) = aabbb+abbbb
b(b(a(a(a+b) = aaabb+aabbb
b(b(a(b(a+b) = aabbb+abbbb
b(b(b(a(a+b) = aabbb+abbbb
b(b(b(b(a+b) = abbbb+bbbbb

now, this looks really weird, doesn't it? But now let me color-code it:
a(a(a(a(a+b) = aaaaa+aaaab
a(a(a(b(a+b) = aaaab+aaabb
a(a(b(a(a+b) = aaaab+aaabb
a(a(b(b(a+b) = aaabb+aabbb
a(b(a(a(a+b) = aaaab+aaabb
a(b(a(b(a+b) = aaabb+aabbb
a(b(b(a(a+b) = aaabb+aabbb
a(b(b(b(a+b) = aabbb+abbbb
b(a(a(a(a+b) = aaaab+aaabb
b(a(a(b(a+b) = aaabb+aabbb
b(a(b(a(a+b) = aaabb+aabbb
b(a(b(b(a+b) = aabbb+abbbb
b(b(a(a(a+b) = aaabb+aabbb
b(b(a(b(a+b) = aabbb+abbbb
b(b(b(a(a+b) = aabbb+abbbb
b(b(b(b(a+b) = abbbb+bbbbb

Now you may ask yourself what that means. And that is a very good question. If you recognized the pattern in which it was color-coded congratulations! For those that want entertainment and education, the coding is about how many a's anb b's there are. If there are exactly 5 of any its blue, with exactly 1 or 4 its green and with exactly 2 or 3 its red. Now I could have colored it differently depending on which (a or b) has 1;2;3;4;5 ,
if I did it that way I'd have exactly 1 of the color 5 times a listed (=a^5) , and 1 of the color b^5, 5 of each, ba^4 and ab^4 and 10 of each a²b³ and a³b². Which means I would have a^5 + a^4*b + a³b² + a²b³ + ab^4 + b^5 which by the way is a total amount of 32 outcomes (2^5) and the Polynomial of (a+b)^5.

But now I'll go on and tell you something trippy, if we list them in a row like this:
(For that purpose I'll give Blue the number 1, green 2 and red 3)

12232333233333322333333233323221

(listed going from right to left and then down)
But what is so funny about that number you ask? Let me show you:
If we only use every 2nd number ( or any pattern that I colored)
12232333233333322333333233323221
12232333233333322333333233323221
12232333233333322333333233323221
12232333233333322333333233323221
12232333233333322333333233323221
1223233323333332
You will always end up with
a) half the amount of numbers
b) the same number
c) the first half of the original number / the last half backwards

I can not explain any of it but the last part of c. C comes from the symmetry that lies within the math.

Thank you for reading this post, if you find any typos please let me know about them :).

 

thats a lot of numbers