"Polynomials are temperamental creatures.  If you force them to behave somewhere, they will go wild in other places." -- Oved Shisha (1932-1998)

The central "creatures" of this project are the Chebyshev polynomials.  Their definition relies on the fact that if n is a positive integer, then is a polynomial in .  For example,  it isn't too difficult to show that   using the identity

        .  

So if ,   then   and we define the Chebyshev polynomial to be .  

The Chebyshev polynomial is defined to be the polynomial,   such that  .

Several books and endless articles and research papers have been written on the Chebyshev polynomials.  Here are a few directions that you can take in exploring them, none requiring calculus.  
1   What are  ?
2.   Based on your list of the first few Chebyshev polynomials, what patterns do you see?  Can you prove that the patterns continue for the whole sequence of polynomials?  You might find it useful to use the relationship between ,  where , which you can use this identity to derive:
        .
3.   Use a graphing calculator or computer algebra system to sketch the graphs of the first few Chebyshev polynomials.  What further properties can you detect from the graphs?  Can you prove them?
4.   Do you see patterns in the coefficients of the Chebyshev polynomials?  Can you prove them?
5.   Suppose that you had been given the sequence defined only with the recursive formula that you derived in Question 2.  How might you identify the trigonometric connection with this sequence?

---