Polynomials In One Variable

c 2007 The Author(s) and The IMO Compendium Group

Polynomials in One Variable
Duˇan Djuki´ s c

Contents
1 2 3 4 5 6 7 8 9 General Properties . . . . . . . . . . Zeros of Polynomials . . . . . . . . . Polynomials with Integer Coefficients Irreducibility . . . . . . . . . . . . . Interpolating polynomials . . . . . . . Applications of Calculus . . . . . . . Symmetric polynomials . . . . . .. . Problems . . . . . . . . . . . . . . . Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 4 6 8 10 11 13 15 17

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General Properties

If only two or three of the above summands are nonzero, P is said to be a binomial and trinomial, respectively. The constants a0 , . . . , an in (∗) are the coefficients of polynomial P. The set of polynomials withthe coefficients in set A is denoted by A[x] - for instance, R[x] is the set of polynomials with real coefficients. We can assume in (∗) w.l.o.g. that an = 0 (if an = 0, the summand an xn can be erased without changing the polynomial). Then the exponent n is called the degree of polynomial P and denoted by deg P. In particular, polynomials of degree one, two and three are called linear, quadraticand cubic. A nonzero constant polynomial has degree 0, while the zero-polynomial P(x) ≡ 0 is assigned the degree −∞ for reasons soon to become clear. Example 1. P(x) = x3 (x + 1) + (1 − x2)2 = 2x4 + x3 − 2x2 + 1 is a polynomial with integer coefficients of degree 4.√ Q(x) = √ 2 − 2x + 3 is a linear polynomial with real coefficients. 0x √ R(x) = x2 = |x|, S(x) = 1 and T (x) = 2x + 1 are notpolynomials. x Polynomials can be added, subtracted or multiplied, and the result will be a polynomial too: A(x) = a0 + a1x + · · · + an xn , B(x) = b0 + b1 x + · · · + bm xm A(x) ± B(x) = (a0 − b0) + (a1 − b1 )x + · · · , A(x)B(x) = a0 b0 + (a0b1 + a1 b0 )x + · · · + anbm xm+n . The behavior of the degrees of the polynomials under these operations is clear:

A Monomial in variable x is an expression ofthe form cxk , where c is a constant and k a nonnegative integer. Constant c can be e.g. an integer, rational, real or complex number. A Polynomial in x is a sum of finitely many monomials in x. In other words, it is an expression of the form P(x) = an xn + an−1 + · · · + a1x + a0. (∗)

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Theorem 1. If A and B are two polynomials then: (i) deg(A ±B) ≤ max(deg A, deg B), with the equality if deg A = deg B. (ii) deg(A · B) = deg A + degB. The conventional equality deg 0 = −∞ actually arose from these properties of degrees, as else the equality (ii) would not be always true. Unlike a sum, difference and product, a quotient of two polynomials is not necessarily a polynomial. Instead, like integers, they can be divided with a residue. Theorem2. Given polynomials A and B = 0, there are unique polynomials Q (quotient) and R (residue) such that A = BQ + R and deg R < deg B. Proof. Let A(x) = an xn + · · · + a0 and B(x) = bk xk + · · · + b0, where an bk = 0. Assume k is fixed and use induction on n. For n < k the statement is trivial. Suppose that n = N ≥ k and that the statement is true for n < N. Then A1 (x) = A(x) − an xn−k B(x) is apolynomial of degree less than n (for its bk coefficient at xn iz zero); hence by the inductive assumption there are unique polynomials Q1 and R such that A1 = BQ1 + R and deg R. But this also implies A = BQ + R, where Q(x) = an n−k x + Q1(x) . bk

Example 2. The quotient upon division of A(x) = x3 + x2 − 1 by B(x) = x2 − x − 3 is x + 2 with the residue 5x + 5, as x3 + x2 − 1 5x + 5 = x+2+ 2 . x2...