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Station  T
Wild Exploration 

Here are some wild explorations you might want to explore, or just think about, perhaps using the app to help you out. These are “big question” ideas that might be fun to mull on.

Question 1

EXPLORATION : CAN WE USE POLYNOMIALS TO LEARN ABOUT NUMBERS?

Use a 1x1 \leftarrow x machine to compute each of the following

a. x21x1\dfrac{x^{2}-1}{x-1}
b. x31x1\dfrac{x^{3}-1}{x-1}
c. x61x1\dfrac{x^{6}-1}{x-1}
d. x101x1\dfrac{x^{10}-1}{x-1}

Can you now see that xnumber1x1\dfrac{x^{\text{number}}-1}{x-1} will always have a nice answer without a remainder?

Another way of saying this is that xnumber1=(x1)×(something)x^{\text{number}}-1=\left(x-1\right) \times \left( \text{something}\right).

For example, you might have seen from part c) that x61=(x1)×(x5+x4+x3+x2+x+1)x^{6}-1=\left(x-1\right) \times \left( x{5}+x{4}+x{3}+x{2}+x+1\right).

This means we can say, for example, that 176117^{6}-1 is sure to be a multiple of 1616!
How? Just choose x=17x=17 in this formula to get 1761=(171)×(something)=(16)×(something)17^{6}-1=\left(17-1\right) \times \left( \text{something}\right)=\left(16\right) \times \left( \text{something}\right).

e. Explain why 9991001999^{100}-1 must be a multiple of 998998.
f. Can you explain why 210012^{100}-1 must be a multiple of 33, and a multiple of 1515, and a multiple of 3131 and a multiple of 10231023? (Hint: 2100=(22)50=4502^{100}= \left( 2^2 \right){50}=4{50}, and so on.)
g. Is xnumber1x^{\text{number}}-1 always a multiple of x+1x+1? Sometimes, at least?
h. The number 2100+12^{100}+1 is not prime. It is a multiple of 1717? Can you see how to prove this?

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Question 2

EXPLORATION: AN INFINITE ANSWER?

Here is a picture of the very simple polynomial 11 and the polynomial 1x1-x.
I6ST-image256

Can you compute 11x\dfrac {1}{1-x}? Can you interpret the answer?

(We’ll explore this example, and more like it, in the island of infinite sums!)

You can either play with some of the optional stations below or go to the next island!

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