Algebra. Pang-Cheng, Wu. January 22, 2016

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1 Algebra Pang-Cheng, Wu January 22, 2016 Abstract For preparing competitions, one should focus on some techniques and important theorems. This time, I want to talk about a method for solving inequality which is called Schur s decomposition and facts about polynomials. 1 Polynomials 1.1 Something you should know Here are consequences without proofs because I suppose that the readers already know how to prove the following results. Proposition If A, B are two polynomials, then deg A B) max dega, degb) deg AB) = dega degb Theorem Given polynomials A, B 0, there are unique polynomials Q, R such that with degr < degb. A = BQ R Lemma If a polynomial P is divisible by a polynomial Q, then every zero of Q is also a zero of P. In particular, if P is divisible by x a, then P a) = 0. Theorem A real polynomial has a factorization of the form x r 1 ) x r k ) x 2 a 1 x b 1 ) x 2 a l x b l ) 1

2 1 POLYNOMIALS 2 Theorem Vieta s formula). If α 1,..., α n are zeros of the polynomial P = x n a 1 x n 1 a n, then a k = 1) k Σx i1 x i2...x ik with the sum being over all k-element subsets of {1,..., n}. Theorem Fundamental Theorem of Algebra). A non-constant complex polynomial has at least one root. Corollary A complex polynomial of degree n 1 has exactly n roots in C. Theorem Gauss Lemma). A non-constant integer polynomial is irreducible over Z[x] if and only if it s also irreducible over Q[x]. Theorem Eisenstein s Criterion). Suppose P is a polynomial with integer coefficients. Precisely, we have If there is a prime p satisfies 1. p divides a i for all i = 0,..., n 1 2. p isn t a divisor of a n 3. p 2 isn t a divisor of a 0 Then P is irreducible over rationals. P x) = a n x n a 0 for x Corollary Cohn s Criterion). For an integer b 2, if P = a n x n a 0 is a polynomial such that 0 a i b 1 for all i = 0,..., n. Moreover, P b) is a prime, then P is irreducible over Z[x]. Proposition If P is a polynomial with integer coefficients, then P a) P b) is divisible by a b for any distinct integers a, b. Lemma If a rational number q p is a root of an integer polynomial P = a nx n a 0, then q a 0 and p a n. Theorem For any given numbers y 0,...y n and distinct x 0,..., x n, there is a unique polynomial P of degree n such that P x i ) = y i for i = 1,...n. Furthermore, it s given by ) n x x j P x) = y i x i x j i=0 j i

3 1 POLYNOMIALS Something you maybe need to know Theorem Descarte s Law). If the coefficients and roots of the polynomial P are real, then the number of positive roots is equal to the number of sign change in the sequence of coefficients of P. Corollary If the coefficients and roots of the polynomial P are real, then the number of positive roots in the interval a, b) is equal to the number of sign change in the sequence of coefficients of P x a) minus that of P x b). For the next theorem, assume that the polynomial P has only simple roots and set P 1 to be its derivative. Then using Euclid s algorithm for P and P 1, we ll get P = P 1 Q 1 P 2 P 1 = P 2 Q 2 P 3 P 2 = P 3 Q 3 P 4 P k 1 = P k Q k In this way, we construct a sequence of polynomials with decreasing degrees,. P, P 1, P 2,..., P k, which is called Sturm chain for P. Now, choose x such that it s not a root for any of polynomials in Sturm chain. Define σ x) to be the number of sign changes in the following sequence sgn P x)), sgn P 1 x)), sgn P 2 x)),..., sgn P k x)) Theorem Sturm s Theorem). The number of real roots of P in the interval a, b) is σ a) σ b) Theorem Perron s Criterion). Suppose P = x n a n 1 x n 1 a 0 is a polynomial such that a 0 0 and Then P is irreducible. a n 1 > 1 a n 2 a 0 Theorem Schur s Theorem). If S is the set of all non-zero values of a nonconstant integer polynomial P, then the set of primes that divide some member of S is infinite. Theorem Sylvester s Theorem). If a b, then a 1) a b) has a prime factor which is greater than b. Theorem Hensel s Lemma). If P = a n x n a 0 is an integer polynomial and x is an integer such that P x) is a multiple of a prime p and P x) isn t a multiple of p. Then there exists an unique residue y such that p, p k are divisors of x y, P y), respectively.

4 1 POLYNOMIALS A piece of cake Example Find all polynomials of the form a n x n a 0 with a j { 1, 1} for all j = 0,..., n whose roots are all real. Example If a 1,..., a n are integers, show that the polynomial x a 1 ) x a n ) 1 is irreducible. Example If p is a prime number, show that the polynomial x p 1 1 is irreducible. Example Let P be a polynomial with integer coefficients. If P P P x) )) = x for some integer x, then P P x)) = x. 1.4 One eyewitness is better than ten hearsays Problem Find all polynomials P such that P 0) = 0 and P x 3 x 2 ) = P x) 3 P x) 2 Problem If 2b 2 < 5c, then the following polynomial has at least one complex root. x 5 bx 4 cx 3 dx 2 ex f = 0 Problem Let P be an integer polynomial. Suppose a, b are integers such that Prove that P a) P b) = 0. P a) P b) = a b) 2 Problem Find all real polynomials P satisfying P P x)) = P x n ) P x) 1 for all x R Problem If P is a polynomial with real coefficients with degree 2n 1 such that P k) = x for all x = n,... 1, 1,...n. Find P 0). Problem Assume P an integer polynomial such that P n) > n for n N. Also, for every integer m, there is a term of 1, P 1), P P 1)),... which is divisible by m. Show that P x) = x 1. Problem Determine the least positive integer n so that there is a real polynomial P x) = a 2n x 2n a 0 with a real root satisfying a i [2014, 2015] for all i = 0,..., 2n. Problem For a rational polynomial f of degree n whose coefficients aren t all integers, is it possible to find an integer polynomial g and S = {x 1, x 2,..., x n1 } such that f t) = g t) for all t S?

5 1 POLYNOMIALS A rolling stone gathers no moss Exercise Show that the maximum in absolute value of any monic real polynomial of degree n on [ 1, 1] is not less than 1 2 n 1. Exercise Find all real polynomials P that P x P x)) = P x) P P x)) for all x R Exercise Find all real polynomials P satisfying P x 2 1) = P x) 2 1 for all x R. Exercise If P is a real polynomial such that P x) 0 for each x 0. Prove that there are real polynomials A, B so that P 2 = A 2 xb 2. Exercise Prove that the polynomial P x) = x 2 x) is irreducible. Exercise Prove that there is no polynomial P C[x] so that the set {P z) z = 1} in complex plane forms a polygon. Exercise Find all co-prime polynomials P, Q, R with complex coefficients such that P 3 Q 4 = R 5 Exercise Let p, q R[x] such that p z) q z) R for all z C. p x) = kq x) for some constant k R or q x) = 0. Prove that Exercise A real polynomial P has the property that for every y Q, there exists x Q such that P x) = y. Prove that P is linear. Exercise If P is a complex polynomial such that P z) R for all z C with z = 1, then P is constant.

6 2 INEQUALITY 6 2 Inequality 2.1 Good medicine for health tastes bitter to the mouth Recall that for all non-negative reals a, b, c and r > 0, we have a r a b) a c) b r b a) b c) c r c a) c b) 0 The equality cases occur when a = b = c or when two of a, b, c are equal and the third is 0. Now, we re going to look its generalized form. Theorem Let a b c be reals and x, y, z be non-negative reals. Then x a b) a c) y b a) b c) z c a) c b) 0 holds if one of the following conditions is fulfilled: x z y ax cz by x,y,z are the squares of the side-lengths of a triangle. ax, by, cz are the squares of the side-lengths of a triangle. There is a convex function f such that x = f a), y = f b), z = f c) Theorem If g is an odd, monotonically increasing function such that g s) 0 for all non-negative real s. Then the inequality xg a b) g a c) yg b a) g b c) zg c a) g c b) 0 holds if one of the following conditions is fulfilled: x z y x,y,z are the squares of the side-lengths of a triangle. There is a convex function f such that x = f a), y = f b), z = f c) Theorem The inequality x a b) a c) y b a) b c) z c a) c b) 0 holds if and only if we can find a convex function f such that. x = b c) 2 f a), y = c a) 2 f b), z = a b) 2 f c)

7 2 INEQUALITY 7 Perhaps the readers know a little about SOS, and thus, this may be a useful identity: x a b) a c) = cyc cyc Finally, review the theorem about SOS method. y z x) 2 b c) 2 Theorem Consider the sum S = S a b c) 2 S b c a) 2 S c a b) 2. Then S is non-negative if one of the following conditions is fulfilled: S a, S c, S a 2S b, S c 2S b are all non-negative. S b, S a S b, S b S c are non-negative when b is the median. S b, S c, b 2 S a a 2 S b are non-negative when a b c. 2.2 Practice makes perfect Example Prove that for any three non-negative reals a, b, c, b 2 c 2) 2 c 2 a 2) 2 a 2 b 2) 2 4 b c) c a) a b) a b c) 0 Example For any triangle ABC, show the following inequality cosasin A B) sin A C)cosBsin B A) sin B C)cosCsin C A) sin C B) 0 Example If a, b, c are non-negative reals, then the following inequality holds. a 2016b) a 2016c) a b 2016a) b 2016c) b c 2016a) c 2016b) c Example If a b c and x y z are non-negative reals, then x b c) 2 b c a) y c a) 2 c a b) z a b) 2 a b c) 0 Example Suppose a, b, c are non-negatives such that a b c = 1, then 1 a2 b 2 c 2 1 a 2 b 2 c 2 1 a 2 b 2 c a 2 b 2 c 2 ) 0 Example Let a, b, c be positive reals. Prove that the following inequality. a b b c c ab bc ca a a 2 b 2 c Example Consider a, b, c > 0, show that a 2 b b2 2 c c2 8 ab bc ca) 11 2 a2 a 2 b 2 c 2

8 2 INEQUALITY Look before you leap Exercise If x, y, z are non-zero reals so that x y z = xyz, then ) x 2 2 ) 1 y 2 2 ) 1 z x y z Exercise For non-negative reals a, b, c, the following inequality holds. a b 5 c 5 ) a 2 bc b c5 a 5 ) b 2 ca c a5 b 5 ) c 2 ab Exercise Suppose that a, b, c > 0, try to show a 4 b 4 c 4 a 3 b 2 b 3 c 2 c 3 a 2) 2 3abc a 4 b 3 b 4 c 3 c 4 a 3) Exercise Let a, b, c be positives satisfying a b c = 3. Then a 5 b 5 c a 3 b 3 c 3 3 ) Exercise Consider positive reals a, b, c, we have a b a b b c c c a 9 ab bc ca) 2 a 2 b 2 c 2 ) Exercise If a, b, c are positive reals such that a b c = 1, prove that a b c) 2 b c a) 2 c a b) 2 3 Exercise Suppose a, b, c > 0. Then the following inequality holds. a b 2 c 2 ) b c2 a 2 ) c a2 b 2 ) 6 a2 b 2 c 2 ) c 2 a 2 b 2 a b c Exercise Let a, b, c, d be positives satisfying a b c d = 4. Prove 1 a 1 b 1 c 1 d 1 3 a 2 b 2 c 2 d 2) 4

Mathematical Olympiad Training Polynomials

Mathematical Olympiad Training Polynomials Definition A polynomial over a ring R(Z, Q, R, C) in x is an expression of the form p(x) = a n x n + a n 1 x n 1 + + a 1 x + a 0, a i R, for 0 i n. If a n 0,