HomeAbout Our ProjectContact UsSite Web Map
Mathematics ProjectsSupport for StudentsSupport for TeachersSupport for MentorsSupport for ParentsHard Math Cafe

Home > Mathematics Projects > Patterns in Polynomials > Warm Up Problems


Patterns in Polynomials Project Description Prerequisites Warm up Problems Hints Resources Teaching Notes Extension Problems Results

Warm-up Problems for Patterns in Polynomials

1.   For each of value of y below, identify whether it is a polynomial in the indicated value of x.  For example, if you were given y equals t squared minus one and x equal t minus one,  
     y equal x squared plus 2 x
therefore, t squared minus one is a polynomial in t minus one.
    a.     y equal t squared minus four   and  x equal t plus one
    b.    y equal to the sum of eight to the t plus two, four to the t plus one, two the t, and one   and x equal two to the t
    c.     y equal three to the t plus two to the t   and x equal two to the t
    d.    y equal cosing of 2 theta  and x equal cosine theta
    e.      y equal sine of two theta  and x equal sine of theta

2.  For each figure, find the indicated lengths in terms of the trigonometric functions of alpha and beta.
    a.  Find x.

figure for warmup problem 2 a

     b.  Find x and y.

figure for warmup problem 2 b

3.  There are countless "proofs with no words" of the identity identity for the cosine of a sum.  Here is one of them.  Convince yourself that this proof is correct.  Can you explain the reasoning to someone else?

proof with no words of the cosine of a sum identity

4.   You will be asked to derive a recursive formula for the Chebyshev polynomials from the identity cosine of n plus one theta plus cosine of n minus one theta equals two cosine of theta time cosine of n theta.  The formula will look a little like one of the sequences below.  Can you describe these sequences of polynomials without using recursion?

    a.  P sub 0 equals one and for positive n, P sub n of x equals P sub n minus one of x plus x times P sub n minus one of x.

     b.  Q sub zero of x equals 1 and for positive n, Q sub n of x equals Q sub n minus one of x times the factor one minus x over n.

     c.  R sub zero of x equals one, R sub one of x equals x, and if n is greater than one, R sub n of x equals R sub n minus one of x plus R sub n minus 2 of x.

Note!
Solutions to these warm up problem are available in the results section of this project.

Back to Top


Making Mathematics Home |
Mathematics Projects | Students | Teachers |
Mentors | Parents | Hard Math Café |
Patterns in Polynomials Project Description |
Prerequisites | Warm Up Problems | Hints |
Resources | Teaching Notes | Extensions |
Results |


Translations of mathematical formulas for web display were created by tex4ht.

© Copyright 2003 Education Development Center, Inc. (EDC)

EDC Logo