# 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.

Station T

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.

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

a. $\dfrac{x^{2}-1}{x-1}$

b. $\dfrac{x^{3}-1}{x-1}$

c. $\dfrac{x^{6}-1}{x-1}$

d. $\dfrac{x^{10}-1}{x-1}$

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

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

For example, you might have seen from part c) that $x^{6}-1=\left(x-1\right) \times \left( x{5}+x{4}+x{3}+x{2}+x+1\right)$.

This means we can say, for example, that $17^{6}-1$ is sure to be a multiple of $16$!

How? Just choose $x=17$ in this formula to get $17^{6}-1=\left(17-1\right) \times \left( \text{something}\right)=\left(16\right) \times \left( \text{something}\right)$.

e. Explain why $999^{100}-1$ must be a multiple of $998$.

f. Can you explain why $2^{100}-1$ must be a multiple of $3$, and a multiple of $15$, and a multiple of $31$ and a multiple of $1023$? (Hint: $2^{100}= \left( 2^2 \right){50}=4{50}$, and so on.)

g. Is $x^{\text{number}}-1$ always a multiple of $x+1$? Sometimes, at least?

h. The number $2^{100}+1$ is not prime. It is a multiple of $17$? Can you see how to prove this?

Here is a picture of the very simple polynomial $1$ and the polynomial $1-x$.

Can you compute $\dfrac {1}{1-x}$? 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!