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Math 32 Discussion Problems
Polynomials
1. Use the quadratic formula (and factoring by grouping ) to
find all complex
solutions to the
given equations:
2. This problem explains one of the deep connections between complex numbers
and trigonometry.
You may have done it already, it was at the end of the handout from November 20.
Let's define a function from the real numbers to the complex numbers by the following formula:
(a) Using and the sum ofangles formula to show that f(x + y) = f(x) f(y).
(b) Use part (a) to show that f(3x) = f(x)^{3}. More generally, show that f(nx) =
f(x)^{n} for
any positive integer n.
(c) Expand out the formula in part (b) in terms of sines and cosines. I.e.
expand out
You should use the fact that (A + B)^{3} = A^{3} + 3A^{2}B + 3AB^{2} + B^{3}. What is i^{3}?
(d) By comparing just the real parts or just the pure imaginary parts of your
answer to
problem (c), conclude formulas for cos(3x) and sin(3x).
(e) Use the fact that
to derive formulas for cos(7x) and sin(7x).
3. Divide the following polynomials. Do these enough to get the hang of it
(maybe you already
have the hang of it from your homework.) Be sure to do the last four.
4. Find the value of k such that when x^{3} + kx + 1 is divided by x + 3, the remainder is 4.
5. For each of the following polynomials, I've listed one or more roots. Find
the rest of the roots.
(Hint: use synthetic division .)
6. Find a polynomial of degree 3 . . .
(a) . . . such that the coefficient of x ^{3} is 1 and the roots are
3, 4, and 5.
(b) . . . with integer coefficients such that the roots are 1/2, 2/5, and 3/4.
(c) . . . with a root of multiplicity two at 1 and such that x + 6 is a factor.
7. Find all possible values of b such that one root of the equation x ^{2}
+ bx + 1 = 0 is twice the
other one.
8. Let the roots of a polynomial be:
with
mulitplicity
with
multiplicity 2, 4i with
multiplicity 1, and 4i with multiplicity 1. If the leading coefficient is 1,
what is the polynomial?
9. Factor x^{4} + 64 into linear factors.
10. Define the nth quantum integer to be the polynomial
.
For example,
(The conventional notation is to write [n]
for f_{n}, but I think this might be confusing.)
(a) What is f_{n}(1)? What is f_{n}(0)? What is f_{n}(1)?
What is f_{n}(i)?
(b) What is f_{n}(q)*f_{2}(q)?
(c) Use synthetic division to show that f_{n}(q) is divisible by f_{2}(q)
if and only if n is even.
(d) More generally, show that f_{n}(q) is divisible by f_{d}(q)
if and only if n is divisible by d.
(e) If d divides n, show that f_{n}(q)=f_{d}(q) = f_{n/d}(q^{d}).
(f) Completely factor
11. (a) Show that
other terms, for
any integer n and numbers
(b) Let r_{1}, r_{2}, r_{3}, and r_{4}
be four real roots of the equation Show that
Hint: explain how to factor
into linears .
(c) Suppose that a circle intersects the parabola y = x^2
in the points (x_{1}, y_{1}), (x_{2}, y_{2}),
(x_{3}, y_{3}), and (x_{4}, y_{4}). Show that x_{1}
+ x_{2} + x_{3} + x_{4} = 0.
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