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Patterns in Polynomials
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Patterns in Polynomials

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

Graph of several cubic polynomials

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

        addition formula for cosine.  

So if x equals cosine theta,   then  cosine three theta equals 4 x cubed minus three x and we define the third Chebyshev polynomial to be T sub 3 of x equals four x cubed minus three x.  

The enth Chebyshev polynomial is defined to be the polynomial, T sub n of x  such that  cosine of n theta equals T sub n of cosine theta.

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  T sub 0 of x, T sub 1 of x, T sub 2 of x,  and T sub 4 of x ?
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 T sub n +1 of x, T sub n of x, and T sub n -1 of x,  where n greater than or equal to one, which you can use this identity to derive:
        cosine of n plus one theta plus cosine of n minus one theta equals two cosine theta times cosine n theta.
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 the sequence of Chebyshev polynomials dash defined only with the recursive formula that you derived in Question 2.  How might you identify the trigonometric connection with this sequence?

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