in class we progressed in the powerpoint presentation up to but not
including the slide titled Palindromials. To cement your
understanding of what we did, here are some exercises:
are just intended to help you absorb the ideas from today's
class. I won't bother adding them to the assignment sheet, I
won't publish answers, and I won't include any of them in a formal
problem set later. Something similar might show up on
the midterm exam. I will take questions about these problems in
class next Tuesday.
- Make up a polynomial f(x) of
degree 6 and a polynomial p(x) of degree 3. Divide your f(x)
by p(x) and find the quotient and remainder. Then exchange your
polynomials with a classmate and double check each other's work.
Discuss any discrepancies.
- The number (1 + squareroot(5))/2
is called the golden mean or the golden ratio, and represented by the
greek letter φ (phi). It is a root of the polynomial x2-x-1.
For your polynomial f(x) from problem 1, compute an exact value of
f(φ), using the same method as we used in class when we were working on
the handout about finding local extrema of a polynomial. Double
check that your exact answer gives the same decimal approximation as
direct computation of f(φ) using a calculator.
- Do this with a partner. Each
person make up a polynomial in factored form of degree 4 or 5, and then
multiply it out into the standard descending form. Don't let your
partner know what the factored form of your polynomial is, but give
him/her the descending form. For example, if my polynomial is
(x-1)(x-2)(x-3)(x-4), then I give my partner x4 - 10x3
+ 35x2 - 50x + 24. Next, each of you is to compute,
for the polynomial given to you by your partner, the sum of the roots,
the sum of the squares of the roots, the sum of the cubes of the roots,
and the sum of the fourth powers of the roots. Do this using the
method shown in class, without finding the roots themselves.
Meanwhile, since you know the roots of the polynomial you gave your
partner, you can compute directly for that polynomial the sum of the
roots, the sum of the squares of the roots, etc. Check if your
partner found the right answers.