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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. 1 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)? 2 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] 3