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HOMEWORK 2: COMPLEX NUMBERS AND POLYNOMIALS
1. Problem 1: complex numbers (reminder)
1.1. Write the following numbers in the form a + ib, where a, b are real numbers:
(1) i10
1
(2) 1+i
√
(3) ( 3 + i)30
(4) 1 + i + ... + i100
(5) 1+i
1−i
1.2. Recall that complex numbers can be represented as points on the plane R2 (z = a+ib
is represented by the point whose Cartesian coordinates are (a, b)). In each of the following
cases, describe the set of points on the plane for which the corresponding complex numbers
z satisfy the equation:
(1) z + z̄ = 1
(2) z · z̄ = 1
(the description should include both a description of the set in terms of Cartesian coordinates, and a geometrc description).
1.3. Let z, w be two complex numbers. Prove the inequality: |z| + |w| ≥ |z + w|.
When does equality occur?
Instruction: use the representation of complex numbers as points on the plane.
2. Problem 2: Polynomials (a reminder)
2.1. Give an example of a polynomial of degree 3 with roots 1, 2, 3 (write the coefficients
explicitly).
2.2. Let P (x), Q(x) be two polynomials. It is known that the sum of coefficients of P (x)
is 5, and the sum of coefficients of Q(x) is 7. What would be the sum of coefficients of
each of the following polynomials?
(1) P (x) + Q(x)
(2) P (x)Q(x)
Instruction: Complete the sentence: “The sum of the coefficients of a polynomial P (x)
is the value of the polynomial when we substitute x = ...”. Use this claim to solve the
problem.
3. Problem 3: Reducibility of polynomials
Recall the fundamental theorem of algebra: any polynomial with complex coefficients
decomposes into a product of linear polynomials. That is, given a polynomial P (x) =
an xn + an−1 xn−1 + ... + a1 x + a0 , where an , an−1 , ..., a1 , a0 are complex numbers, there exist
complex numbers α1 , ..., αn such that P (x) = an (x − α1 )(x − α2 )....(x − αn ).
The aim of this problem is to understand how to present a polynomial P (x) with real
coefficients (i.e. P (x) = an xn + an−1 xn−1 + ... + a1 x + a0 , where an , an−1 , ..., a1 , a0 are
Date: September 25, 2013.
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real numbers) as a product of polynomials, each having real coefficients, which have the
smallest possible degrees. That is , we want P (x) to be a product of polynomials with real
coefficients and which are indecomposable over R (i.e. themselves cannot be written
as a product of two non-constant polynomials which have real coefficients).
Example: x2 + 1 has real coefficients, but cannot be written as a product of two nonconstant polynomials which have real coefficients. Thus x2 + 1 is indecomposable over
R.
3.1. Let P (x) be a polynomial with real coefficients, and z (a complex number) be its
root. Prove that z̄ is a root of P (x) as well.
3.2. Use the above result, together with the fundamental theorem of algebra, to prove
the following statement:
Any polynomial with real coefficients can be presented as a product of polynomials
which are indecomposable over R and have each having degree at most 2.
3.3. Decompose the polynomial x8 + 1 as a product of polynomials with real coefficients
and each having degree at most 2.
4. Problem 4: Division of polynomials
4.1. Show all the steps of dividing with remainder the polynomial x7 + 1 by the polynomial x2 + 3.
4.2. Show that when dividing with remainder a polynomial P (x) by the linear polynomial
x − a (a being a complex number), we get the remainder P (a) (a constant polynomial).
Conclusion: the number a is the root of the polynomial P (x) if and only if the polynomial
P (x) is divisible by the linear polynomial x − a.
4.3. Let P (x) be a polynomial of degree 3 with leading and constant terms both equal
1, and assume P (1) = P (2) = 0. Find the polynomial P (x) (write out the coefficients
explicitly).
Instruction: Use the previous exercise.
4.4. Let P (x), Q(x) be two polynomials; let D(x) be the quotient of the division of P (x)
by Q(x) and R(x) be the remainder. Assume degP (x) = 100, degQ(x) = 43. What cold
be the degrees of D(x), R(x) (list all the possibilities)?
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5. Bonus problems
5.1. Let P (x) be a polynomial. Consider the polynomial Q(x) := P (x) · (x − 1). Could
all the coefficients of Q(x be positive?
5.2. For which positive integers n, k the polynomial xk −1 divides the polynomial xn −1?
5.3. Is there a polynomial P (x) with integer coefficients such that P (7) = 5, P (11) = 7?
5.4. (Difficult). Let P (x, y) be a polynomial in two variables, with real coefficients.
Assume P (t, sin(t)) = 0 for any real t. Prove that P (x, y) is the zero polynomial.
5.5.
• Prove the identity: (a2 + b2 )(c2 + d2 ) = (ac − bd)2 + (ad + bc)2 .
• Some positive integers satisfy an interesting property: they can be presented as a
sum of integer squares (i.e. as a2 + b2 , where a, b are integers). Prove that if m, n
satisfy this property, then so does their product.
More about the problem of detemining which numbers can be presented as a sum of
integer squares:
wikipedia.org, article: “Fermat’s theorem on sums of two squares”.
E-mail address: [email protected]
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