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 1.1.1. If A, B are two polynomials, then deg A B) max dega, degb) deg AB) = dega degb Theorem 1.1.2. Given polynomials A, B 0, there are unique polynomials Q, R such that with degr < degb. A = BQ R Lemma 1.1.3. 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 1.1.4. 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
1 POLYNOMIALS 2 Theorem 1.1.5. 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 1.1.6. Fundamental Theorem of Algebra). A non-constant complex polynomial has at least one root. Corollary 1.1.7. A complex polynomial of degree n 1 has exactly n roots in C. Theorem 1.1.8. Gauss Lemma). A non-constant integer polynomial is irreducible over Z[x] if and only if it s also irreducible over Q[x]. Theorem 1.1.9. 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 1.1.10. 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 1.1.11. 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 1.1.12. 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 1.1.13. 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
1 POLYNOMIALS 3 1.2 Something you maybe need to know Theorem 1.2.1. 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 1.2.2. 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 1.2.3. Sturm s Theorem). The number of real roots of P in the interval a, b) is σ a) σ b) Theorem 1.2.4. 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 1.2.5. 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 1.2.6. Sylvester s Theorem). If a b, then a 1) a b) has a prime factor which is greater than b. Theorem 1.2.7. 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.
1 POLYNOMIALS 4 1.3 A piece of cake Example 1.3.1. 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 1.3.2. If a 1,..., a n are integers, show that the polynomial x a 1 ) x a n ) 1 is irreducible. Example 1.3.3. If p is a prime number, show that the polynomial x p 1 1 is irreducible. Example 1.3.4. 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 1.4.1. Find all polynomials P such that P 0) = 0 and P x 3 x 2 ) = P x) 3 P x) 2 Problem 1.4.2. 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 1.4.3. 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 1.4.4. Find all real polynomials P satisfying P P x)) = P x n ) P x) 1 for all x R Problem 1.4.5. 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 1.4.6. 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 1.4.7. 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 1.4.8. 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?
1 POLYNOMIALS 5 1.5 A rolling stone gathers no moss Exercise 1.5.1. 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 1.5.2. Find all real polynomials P that P x P x)) = P x) P P x)) for all x R Exercise 1.5.3. Find all real polynomials P satisfying P x 2 1) = P x) 2 1 for all x R. Exercise 1.5.4. 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 1.5.5. Prove that the polynomial P x) = x 2 x) 22015 1 is irreducible. Exercise 1.5.6. Prove that there is no polynomial P C[x] so that the set {P z) z = 1} in complex plane forms a polygon. Exercise 1.5.7. Find all co-prime polynomials P, Q, R with complex coefficients such that P 3 Q 4 = R 5 Exercise 1.5.8. 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 1.5.9. 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 1.5.10. If P is a complex polynomial such that P z) R for all z C with z = 1, then P is constant.
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 2.1.1. 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 2.1.2. 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 2.1.3. 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)
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 2.1.4. 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 2.2.1. 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 2.2.2. 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 2.2.3. 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 2.2.4. 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 2.2.5. 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 2 11 9 a 2 b 2 c 2 ) 0 Example 2.2.6. 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 1 3 2 Example 2.2.7. 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
2 INEQUALITY 8 2.3 Look before you leap Exercise 2.3.1. If x, y, z are non-zero reals so that x y z = xyz, then ) x 2 2 ) 1 y 2 2 ) 1 z 2 2 1 4 x y z Exercise 2.3.2. 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 2.3.3. 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 2.3.4. Let a, b, c be positives satisfying a b c = 3. Then a 5 b 5 c 5 3 3 a 3 b 3 c 3 3 ) Exercise 2.3.5. 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 2.3.6. 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 2.3.7. 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 2.3.8. 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