What is Polynomials: Definition and 783 Discussions

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example in three variables is x3 + 2xyz2 − yz + 1.
Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry.

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  1. Demon117

    Orthogonality of Legendre Polynomials from Jackson

    Hello all! I am trying to work through and understand the derivation of the Legendre Polynomials from Jackson's Classical electrodynamics. I have reached a part that I cannot get through however. Jackson starts with the following orthogonality statement and jumps (as it seems) in his proof...
  2. N

    How do I determine whether a set of polynomials form a basis?

    Homework Statement Are the following statements true or false? Explain your answers carefully, giving all necessary working. (1) p_{1}(t) = 3 + t^{2} and p_{2}(t) = -1 +5t +7t^{2} form a basis for P_{2} (2) p_{1}(t) = 1 + 2t + t^{2}, p_{2}(t) = -1 + t^{2} and p_{3}(t) = 7 + 5t -6t^{2}...
  3. B

    Proof by contradiction - polynomials and infinite primes

    Homework Statement Two Questions: 1. Prove, by contradiction, that if a and b are integers and b is odd,, then -1 is not a root of f(x)= ax^2+bx+a. 2. Prove, by contradiction, that there are infinitely many primes as follows. Assume that there only finite primes. Let P be the largest...
  4. L

    Matrix polynomials and inverses- Linear Algebra

    Homework Statement For p(x)=x4-2x3+3x2-3x+1 and A= 1 1 1 -1 -1 0 -2 1 0 0 1 0 1 0 0 0 you can check that P(A)=0 using this find a polynomial q(x) so that q(A)=A-1. The point is A4-2A3+3A2-3A=A(-A3+2A2-3A+3I)=I a) What is q(x)? I don't really understand how to approach...
  5. T

    Norms and orthogonal Polynomials

    Homework Statement Thanks very much for reading. I actually have two problems, I hope it's ok to state both of the in the same thread. 1. Let Vn be the space of all functions having the n'th derivitve in the point x0. I've been given the semi-norm (holds all the norm axioms other than ||v|| =...
  6. T

    Visual basic algorithm for computing hermite polynomials

    Please I need Visual Basic algorithm for computing Hermite polynomials. Any one with useful info? Thanks.
  7. H

    How to Find the Inverse of a Polynomial: Step-by-Step Guide

    Homework Statement Let f(x) = 2x^3 + 5x + 3 Find the inverse at f^-1(x) = 1 Homework Equations N/AThe Attempt at a Solution The only way that I know how to solve inverses is by solving for X, then replacing it by Y. Then I supposed I would sub 1 into the inverted polynomial. However I'm...
  8. G

    Are These Polynomials Irreducible Over Q?

    Homework Statement determine whether the following polynomials are irreducible over Q, i)f(x) = x^5+25x^4+15x^2+20 ii)f(x) = x^3+2x^2+3x+5 iii)f(x) = x^3+4x^2+3x+2 iv)f(x) = x^4+x^3+x^2+x+1 Homework Equations The Attempt at a Solution By eisensteins criterion let...
  9. V

    MATLAB Integration of a product of legendre polynomials in matlab

    I am trying to find a way to integrate the following expression Integral {Ylm(theta, phi) Conjugate (Yl'm'(theta, phi) LegendrePolynomial(n, Cos[theta])} dtheta dphi for definite values of l,m,n,l',m' . You normally do this in Mathematica very easily. But it happens that I need to use this...
  10. F

    Solutions to the Harmonic Oscillator Equation and Hermite Polynomials

    How are Hermite Polynomials related to the solutions to the Schrodinger equation for a harmonic oscillator? Are they the solutions themselves, or are the solutions to the equation the product of a Hermite polynomial and an exponential function? Thanks!
  11. G

    Calculus II - Approximating Functions With Polynomials

    Hi, If I'm given something like this for a problem, Approximate the given quantities using Taylor polynomials with n=3 sqrt(101) how do I know what I should set f(x) equal to? I could set it to many different things, sqrt(x), sqrt(x+100), sqrt(x+50). My answer would be very different...
  12. D

    Fortran Implementing Generalized Laguerre Polynomials in Fortran

    Hi! Im trying to do some rather easy QM-calculations in Fortran. To do that i need a routine that calculates the generalized Laguerre polynomials. I just did the simplest implementation of the equation: L^l_n(x)=\sum_{k=0}^n\frac{(n+l)!(-x^2)^k}{(n-k)!k!} I implemented this in the...
  13. M

    Existence of Roots for Quadratic Forms Modulo Prime Numbers

    I've been doing some work and I keep running into polynomials of the following form: P(x,y,z) = ax^2 + by^2 + cz^2 + 2(exy + fxz + gyz) \mod p where a,b,c \in \mathbb{Z}_p/ \{0\} and d , e, f \in \mathbb{Z}_p . It would be great if I knew anything about the existence of roots of P ...
  14. 6

    Student t orthogonal polynomials

    I've just read a paper that references the use of student-t orthogonal polynomials. I understand how the Gauss-Hermite polynomials are derived, however applying the same process to the weight function (1 + t^2/v)^-(v+1)/2 I can't quite get an answer that looks anything like a polynomial...
  15. J

    Working with rational polynomials in Maple

    Hi Could someone please explain how to best handle rational polynomials in Maple? I have matrix of rational polynomials and for some reason Maple keeps grumbling i.e. "error, (in, linearalgebra:- HermiteForm) expecting a matrix of rational polynomials" The matrix I am working with is...
  16. V

    What is the condition for a circular orbit?

    while attempting to solve for radius of a circular orbit, I ended up getting P(x)=0 and dP(x)/dx=0 where P is a fourth order polynomial. I am not sure how can I solve it. Can someone shed some light on it. Thanks
  17. B

    Irreducible Polynomials over Finite Fields

    Hi, yet another question regarding polynomials :). Just curious about this. Let f(x), g(x) be irreducible polynomials over the finite field GF(q) with coprime degrees n, m resp. Let \alpha , \beta be roots of f(x), g(x) resp. Then the roots of f(x), g(x), are \alpha^{q^i}, 0\leq i \leq n-1...
  18. B

    Classes of polynomials whose roots form a cyclic group

    Hi, I'm currently doing a project and this topic has come up. Are there any known famous classes of polynomials (besides cyclotomic polynomials) that fit that description? In particular, I'm more interested in the case where the polynomials have odd degree. I know for example that the roots of...
  19. M

    Does this set of polynomials span P3?

    hey i want to find out if the set s = {t2-2t , t3+8 , t3-t2 , t2-4} spans P3 For vectors, i would setup a matrix (v1 v2 v3 v4 .. vn | x) where x is a column vector (x , y ,z .. etc) and reduce the system. If a solution exists then the vectors span the space, if there are no solutions then...
  20. Z

    Inverse problem for Orthogonal POlynomials

    given a set of orthogonal polynomials \int_{-\infty}^{\infty}dx P_{m} (x) P_{n} (x) w(x) = \delta _{m,n} the measure is EVEN and positive, so all the polynomials will be even or odd my question is if we suppose that for n-->oo \frac{ P_{2n} (x)}{P_{2n}(0)}= f(x) for a known...
  21. P

    Legendre Polynomials and Complex Analysis

    Hi all, I am currently a 2nd year mathematics and physics student. I am working, for the first time, on my own research and just sort of getting my feet wet. I got in touch with a professor that studies Special Functions and he led me to the Legendre functions and associated Legendre...
  22. D

    Rational Root Theorem for Factoring Polynomials

    Hi I was wondering since i have problems factoring any polynomial past 2nd degree i was wondering if anyone can show a way i can remember for finals ^_^. IE. let's say we have a 3rd degree polynomial. X^3 - 3X^2 +4 i tried looking it up but most don't show how they did the work so i can...
  23. B

    What is an inner product and how can it be verified for polynomials?

    [-1]int[1]P(x)Q(x)dx P,Q\inS verify that this is an inner product.
  24. W

    Orthogonality in Legendre polynomials

    Homework Statement There is a recursion relation between the Legendre polynomial. To see this, show that the polynomial x p_k is orthogonal to all the polynomials of degree less than or equal k-2. Homework Equations <p,q>=0 if and only if p and q are orthogonal. The Attempt at a...
  25. G

    Splitting Polynomials into Even and Odd Parts: A Unique Direct Sum Decomposition

    1. Let \mathbb{R}[x]_n^+ and } \mathbb{R}[x]_n^- denote the vector subspaces of even and odd polynomials in \mathbb{R}[x]_n Show \mathbb{R}[x]_n=\mathbb{R}[x]_n^+ \oplus\mathbb{R}[x]_n^- 3. For every p^+(x) \in \mathbb{R}[x]_n^+ \displaystyle p^+(x)=\sum_{m=0}^n a_m x^m=p^+(-x)...
  26. C

    Minimal vs Characteristic Polynomials

    Let T:V to V be a linear operator on an n-dimensional vector space V. Let T have n distinct eigenvalues. Prove that the minimal polynomial and the characteristic polynomial are identical up to a factor of +/- 1 I'm probably over thinking this, but it seems that if you have n distinct...
  27. H

    Obtaining Polynomials: Probability & Ways

    In how many ways can obtain polynomial from [PLAIN]http://im3.gulfup.com/2011-05-05/1304543619801.gif notes that c any coffieceints is in{0.1} also in how many ways can obtain even ploynomials?whats the probability that we can obtain P(1,1,1)=0
  28. V

    Mathematical induction proof concerning polynomials in Z2.

    Homework Statement For each n\,\in\,\mathbb{N}, let p_n(x)\,\in\,\mathbb{Z}_2[x] be the polynomial 1\,+\,x\,+\,\cdots\,x^{n\,-\,1}\,+\,x^n Use mathematical induction to prove that p_n(x)\,\cdot\,p_n(x)\,=\,1\,+\,x^2\,+\,\cdots\,+x^{2n\,-\,2}\,+\,x^{2n}Homework Equations Induction steps...
  29. S

    Polynomials in Z6[x]: Find & Explain Deg 0 Product

    Homework Statement Find two polynomials, each of degree 2, in Z6[x] whose product has degree 0. Can you repeat the same in Z7[x]? Explain. Homework Equations In Z6[x] and Z7[x] can the only variable be x? The Attempt at a Solution I know Z6 consists of {0,1,2,3,4,5} and Z7...
  30. P

    Solving polynomials for variable (x)

    Homework Statement I'm doing work finding the centroids of 2d graphs. I'm working these problems using double integrals of regions that are horizontally or vertically simple. To do this I have to be able to convert line equations from one variable to the other. Some are simple but others...
  31. Z

    Solving Hermite Polynomials: Find Form from Definition

    In a past exam paper at my uni I am asked to show that the hermite polynomials are solutions of the hermite diff. equation but first there is the following \Phi(x,t)=\exp (2xh-h^2)=\sum_{n=0}^{\infty} \frac{h^n}{n!}H_n(x) So I need to find the form of H_n first, and I'm stuck. I tried...
  32. R

    Factoring x^{16}-x in F_8[x] and Proving Equivalency in F_2[x]

    Homework Statement Factor x^{16}-x in F_8[x] Homework Equations The Attempt at a Solution I know how to do it in F_2[x] . I also feel the factorizations are the same in the two fields..but not sure how to prove it.
  33. M

    Finding an Orthogonal Polynomial to x^2-1/2 on L2[0,1]

    Find a polynomial that is orthogonal to f(x)=x2-1/2 using L2[0,1]. I have looked all in the textbook and all over the internet and have found some hints if the interval is [-1,1], but still do not even know where to start here. I think I was gone the day our professor taught this because I do...
  34. K

    Exploring Degree Odd Polynomials and Extension Fields of K

    1.Let F be an extension field of K and let u be in F. Show that K(a^2)contained in K(a) and [K(u):K(a^2)]=1 or 2. 2.Let F be an extension field of K and let a be in F be algebraic over K with minimal polynomial m(x). Show that if degm(x) is odd then K(u)=K(a^2). 1. I was thinking of...
  35. V

    Abstract Algebra - Polynomials: Irreducibles and Unique Factorization

    Homework Statement Show that x^2\,+\,x can be factored in two ways in \mathbb{Z}_6[x] as the product of nonconstant polynomials that are not units.Homework Equations Theorem 4.8 Let R be an integral domain. then f(x) is a unit in R[x] if and only if f(x) is a constant polynomial that is a...
  36. D

    What are the primitive elements in GF(9)?

    Homework Statement Hi, I need to show that \alpha+1=[x] is a primitive element of GF(9)= \mathbb{Z}_3[x]/<x^{2}+x+2> I have already worked out that the function in the < > is irreducible but I do not know where to go from this. Homework Equations there are 8 elements in the...
  37. B

    Fast Construction of Irreducible Polynomials of degree n over any Finite Field

    Hello, I'm currently doing an undergrad project on this topic and I was wondering if any of you guys know what is the fastest algorithm (asymptotically) that has been discovered so far, for such purpose. Here is paper by Shoup (1993) which gave the fastest algorithm up to then...
  38. B

    Fast Construction of Irreducible Polynomials of degree n over any Finite Field

    Hello, I'm currently doing an undergrad project on this topic and I was wondering if any of you guys know what is the fastest algorithm (asymptotically) that has been discovered so far, for such purpose. Here is paper by Shoup (1993) which gave the fastest algorithm up to then...
  39. O

    Completeness of Legendre Polynomials

    I've recently been working with Legendre polynomials, particularly in the context of Spherical Harmonics. For the moment, it's enough to consider the regular L. polynomials which solve the differential equation [(1-x^2) P_n']'+\lambda P=0 However, I've run into a problem. Why in the...
  40. G

    Linear Algebra - Characteristic Polynomials and Nilpotent Operators

    Homework Statement If the characteristic polynomial of an operator T is (-1)^n*t^n, is T nilpotent? Homework Equations The Attempt at a Solution My first instinct for this question is that the answer is yes, because the matrix form of T must have 0's on the diagonal and must...
  41. G

    Characteristic Polynomials and Nilpotent Operators

    If the characteristic polynomial of an operator T is (-1)^n*t^n, is T nilpotent? My first instinct for this question is that the answer is yes, because the matrix form of T must have 0's on the diagonal and must be either upper triangular or lower triangular. This is what I found when I tried...
  42. L

    Invariant Polynomials on complexified bundles with connection

    I would like to know if the following correct. Suppose I start with a connection on a real vector bundle and extend it to the complexification of the bundle. The curvature forms of the complexification seem to be the same as curvature 2 forms of the real bundle. From this it seems that the...
  43. R

    A polynomials with coefficients in a field

    Homework Statement Prove that a polynomial f of degree n with coefficients in a field F has at most n roots in F. Homework Equations The Attempt at a Solution So we could prove this by induction by using a is a root of f if and only if x-a divides f. My question is: why do...
  44. alyafey22

    Why should the exponents of polynomials just be whole numbers

    Why should the exponents of polynomials just be whole numbers ?
  45. L

    What are the possible values of m and n for which Q divides P?

    Hi everybody! I have this problem: Either P = (X+2)m+(X+3)n and Q = x2+5x+7; Determine m, n such that Q | P;( m, n = ? (Q divide P)); May you help me please? Thank You!
  46. T

    The notation of the norm of polynomials

    Homework Statement attached Homework Equations The Attempt at a Solution what is x_i? is it the coefficient of x or simply add up 1-5? i found the notation different from http://mathworld.wolfram.com/PolynomialNorm.html so i am confused. Thx!
  47. I

    Finding the Big O of polynomials

    Homework Statement 1. Use the definition of "f(x) is O(g(x))" to show that 2^x + 17 is O(3^x)'). 2. Determine whether the function x^4/2 is O(x^2) 2. The attempt at a solution 1. From my understanding, I would say of course 2^x + 17 is O(3^x) because the constant is of such low...
  48. R

    Irreducible polynomials over the reals

    Homework Statement How to prove that the only irreducible polys over the reals are the linear ones and the quadratic ones no real roots? What about the ones with higher degree? I feel that I'm missing something that's really obvious. Homework Equations The Attempt at a Solution
  49. P

    Orthogonality limits of Bessel Polynomials

    Anyone who knows the limits of orthogonality for Bessel polynomials? Been searching the Internet for a while now and I can't find a single source which explicitly states these limits (wiki, wolfram, articles, etc). One thought: since the Bessel polynomials can be expressed as a generalized...
  50. M

    Question on Chebyshev polynomials?

    Question A Chebyshev polynomial is Tn(x) = cos(arccos^(-1)(x)) My questions are: 1. what are the domain(s) and range(s) of this function? 2. Give equivalent polynomial definitions for Tn(x) when n = 0; 1; 2; 3. That is: show that the definition for Tn above really is a polynomial...
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