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.

View More On Wikipedia.org
Recent contents Top users
  1. R

    How to Find Maclaurin Polynomials for ln(x) in Sigma Notation?

    Homework Statement Find the Maclaurin polynomials of orders n=0, 1, 2, 3, and 4, and then find the nth Maclaurin polynomials for the function in sigma notation: f(x)= ln(1+x) Homework Equations pn(x)= f(0) + f'(0)x + [f''(0)/2!](x)^2 +[f'''(0)/3!](x)^3 + ... + [f^(n)(0)/n!](x)^n...
  2. N

    Linear Polynomials: Applications in Integer Programming & Pattern Recognition

    Dear experts. What do you know about applications of linear polynomials with integer coefficients? (For example, I know that these polynomials are applied in the field of integer programming and pattern recognition (clustering). Thanks in Advance.
  3. A

    How to prove Chepyshev's polynomials generating function identity?

    (1-xt)/(1-2xt+t^2)=sum(Tn(x)t^n) How can i prove this equation? Could you give me a hint or suggestion?
  4. N

    Polynomials and the Inner Product

    The question requires me to check whether the following formulae satisfy the properties of an inner product given the linear space of all real polynomials. f(1)g(1) \left(\int_{0}^{1}f(t)dt\right)\left(\int_{0}^{1}g(t)dt\right) The properties are satisfied in both cases (at least, that's my...
  5. X

    Cyclotomic polynomials and primitive roots of unity

    w_{n} is primitive root of unity of order n, w_{m} is primitive root of unity of order m, all primitve roots of unity of order n are roots of Cyclotomic polynomials phi_{n}(x) which is a minimal polynomial of all primitive roots of unity of order n , similarly, phi_{m}(y) is a minimal...
  6. M

    Lucas-Lehmer Test With Polynomials

    Does anyone know why the lucas-lehmer test won't work with numbers represented by polynomials? The modulus polynomial does not divide the last term correctly for some reason. Example (the code boxes are polynomial long division): the numbers will be represented in base 4, so 4 = x p = x + 1...
  7. A

    Legendre polynomials proof question.Help

    Hello everyone i had some questions about legendre polynomials. I have solved most of them but i had just two not answered question. I tried to solve this problem by rodriguez rule but it was really hard for me. Could anyone help me or give me some hints for this question...
  8. K

    Are the following polynomials irreducible over Z2?

    Are the following polynomials irreducible over Z2? (a) x2 + x + 1 (b) x2 + 1 (c) x2 + x
  9. S

    Proving Orthogonality of Legendre Polynomials P3 and P1

    To show that two Legendre polynomials(Pn and Pm) are orthogonal wht is the test that i have to use? is it this? \int_{-1}^{1} P_{n}(x)P_{m}(x) dx = 0 in that case to prove that P3 and P1 are orthogonal i have to use the above formula??
  10. M

    Vector Space Analysis of Polynomials & Matrices

    I am supposed to determine whether or not the following two sets constitute a vector space. 1) The set of all polynomials degree two. 2) The set of all diagonal 2 x 2 matrices. For the first one, it will not be a vector space because it does not satisfy the closure property. Also the...
  11. L

    Problem with the Laguerre polynomials

    My task is to explicitly write down the first three Laguerre polynomials by using a power series ansatz. What should this ansatz look like? Should it be the Rodrigues representation L_n (x) = \frac{e^x}{n!} \frac{d^n}{dx^n} x^n e^{-x} ?
  12. L

    Solving Differential Equation with Laguerre Polynomials

    What differential equation does \phi_n (x) := e^{-x/2} L_n (x) solve? L_n is a Laguerre polynomial. Please give me a hint on this one. I haven't got a clue where to start.
  13. L

    Expand a function in terms of Legendre polynomials

    Problem: Suppose we wish to expand a function defined on the interval (a,b) in terms of Legendre polynomials. Show that the transformation u = (2x-a-b)/(b-a) maps the function onto the interval (-1,1). How do I even start working with this? I haven't got a clue...
  14. L

    Proving Orthogonality of Legendre Polynomials

    Problem: Show that \int_{-1}^{1} x P_n(x) P_m(x) dx = \frac{2(n+1)}{(2n+1)(2n+3)}\delta_{m,n+1} + \frac{2n}{(2n+1)(2n-1)}\delta_{m,n-1} I guess I should use orthogonality with the Legendre polynomials, but if I integrate by parts to get rid of the x my integral equals zero. Any tip on...
  15. P

    Multivariable Taylor polynomials?

    In textbooks these polynomials are not normally presented as an infinite series (the single variables are). What is the reason for this and are they equally allowed to be in infinite series form hence infinite order just like the single variable Taylor Polynomials? Or are there more issues about...
  16. T

    Finding Roots for a Challenging Polynomial Equation

    What are the ROots: x^4+4x^3+14x^2=-4x-13 ok the 13 really causes a problem because you can't factor that. So I move the right side to the left and then you can't find a number that fits so that it equals zero so I tried factoring it somehow but can't do it can someone help?
  17. G

    Solution of hydrogen atom : legendre polynomials

    I was messing around with the \theta equation of hydrogen atom. OK, the equation is a Legendre differential equation, which has solutions of Legendre polynomials. I haven't studied them before, so I decided to take closed look and began working on the most simple type of Legendre DE. And the...
  18. G

    Legendre Polynomials Orthogonality Relation

    ...and orthogonality relation. The book says \int_{-1}^{1} P_n(x) P_m(x) dx = \delta_{mn} \frac{2}{2n+1} So I sat and tried derieving it. First, I gather an inventory that might be useful: (1-x^2)P_n''(x) - 2xP_n'(x) + n(n+1) = 0 [(1-x^2)P_n'(x)]' = -n(n+1)P_n(x) P_n(-x) = (-1)^n P_n(x)...
  19. B

    Is there an easier way to factor these polynomials?

    I have two equations, I'm pretty sure I can solve them, but the method I know of is was too long, and I've never had to use very long methods to solve any problems so far. I'm wondering if there's some shortcut method that I don't know, or that I'm forgetting. Here are the equations(I have solve...
  20. O

    Struggling to Understand Polynomials?

    Need Some Help With Polynomials
  21. M

    Problem with polynomials and spheres

    Hi all, I've got this problem: Consider the points P such that the distance from P to A(-1,5,3) is twice the distance from P to B(6,2,-2). Show that the set of all such points is a sphere, and find its center and radius. I think the setup should be this: sqrt[(x+1)^2 + (y-5)^2 +...
  22. B

    Division with polynomials involving the remainder theorem

    Two questions, first, I solved something, but I was just playing around with the numbers, and I didn't really know what I was doing, nor did I really understand it after I was done. The question is as follows: When x + 2 is divided into f(x), the remainder is 3. Determine the remainder when x...
  23. B

    Long division(with and without polynomials)

    I had to relearn long division recently, because I just started division with polynomials in my math class. I realized that I didn't quite understand the whole process of long division. I can follow the steps that the teacher sets out, I just don't really understand the steps, why you use this...
  24. R

    Efficiently Factor Polynomials: Solving 36(2x-y)^2 - 25(u-2y)^2

    Factoring polynomials(2nd problem) Factor: 36(2x-y)^2 - 25(u-2y)^2 Having trouble where to start...should I expand out?
  25. -Job-

    Something i didn't know about polynomials

    Sometimes i like to waste my time observing the behavior of certain functions and identify any patterns. Most of the time i just look at the differences (absolute value) between two consecutive elements of a sequence of numbers and look for a pattern. If no pattern is evident i take the...
  26. A

    Modelling with polynomials

    5 A particle moves horizontally in a straight line according to the rule x(t) = t^3 − 5t^2 + 3t − 5, 0 ≤ t ≤ 7 where x metres is its displacement to the right of the origin at time t seconds. a What is the initial position of the particle? b After how many seconds does the particle pass...
  27. A

    Modelling with polynomials and rational functions

    4 The height, x metres, of a diver above a swimming pool at time t seconds after he has bounced from the diving board can be modeled by the function x(t)= 3- 3t (3t^2/2) a How long, in seconds, after he has bounced from the diving board does the diver reach his maximum height? b What is the...
  28. MathematicalPhysicist

    Solutions for roots polynomials.

    i wonder if there's a formula like for quadartic and cubic equations also for roots polynomials, like this equation: ax^(1/2)+bx+c=0 or like ax^(1/3)+bx^(1/2)+cx+d=0 ? and what are they?
  29. M

    Taylor Polynomials Questions

    Hi Given a function f(x) = sqrt(x) is the Taylor Polynomial of degree 2 for that function: \frac{x^2}{2} - 99x + 4901 where x = 100 ? Sincerely Fred
  30. M

    Is There an Easier Proof for Proving Irreducibility of Polynomials?

    im teeching algebra and had to prove that X^5 - 8 was irreducible over the rationals. so i did it using eisenstein. then more generally i seem to have proved that X^n - a is irreducible over the rationals whenever it has no rational root. but i used galois theory, and the course I am...
  31. E

    Do polynomials have asymptotes?

    First of all, do polynomials have asymptotes, including oblique ones? I know that rational functions have asymptotes, and it seems that most, if not all, of my book's exercises on this lesson contain only rational functions. So do polynomials have asymptotes? and if so, how do I determine...
  32. N

    Proving Hermite Equation with Hermite Polynomials

    Im stuck on this question :( The Hermite polynomials can be defined through \displaystyle{F(x,h) = \sum^{\infty}_{n = 0} \frac{h^n}{n!}H_n(x)} Prove that the H_n satisfy the hermite equation \displaystyle{H''_n(x) - 2xH'_n(x) + 2nH_n(x) = 0} Using...
  33. R

    Legendre polynomials application

    I need some help. I fitted a 7th order legendre polynomial and got the L0 to L7 coefficients for different ANOVA classes. How can I get a back transformation in order to plot each class using the estimated coefficients? Thanks to anybody. Roberto.
  34. G

    Numerical Analysis: Taylor Polynomials, Error, Bounds

    (a) I found the answer to be: 1/(1-x) = 1 + x + x^2 + x^3 + ... + [x^(n+1)]/(1-x) for x != 1 *Note: "^" precedes a superscript, "!=" means "does not equal" (b) Use part (a) to find a Taylor polynomial of a general (3n)th degree for: f(x) = (1/x)*Integral[(1/(1 + t^3), t, 0, x] *Note...
  35. B

    Solve Rook Polynomial: How Many Arrangements of k Nonattacking Rooks?

    If C is an m x n chessboard with m<=n. For a 0<=k<=m how many ways can we arrange k nonattacking rooks? and what is the rook polynomial r(C,k)?
  36. T

    Finding an Inner Product for Approximating Polynomials on a Set of Real Numbers

    Let S be a finite set of real numbers. What is a natural inner product to define on the space of all functions f:S->R? I want to approximate an arbitrary function with a polynomial of a fixed degree (both of which are defined only on S), and I want to use projections to do it, but I have no...
  37. D

    Solving cubic or higher degree polynomials by hand

    How does one find the roots of an equation, for instance: r^4 + r^3 - 7r^2 - r + 6 = 0 ...completely by hand. Is there some type of non-lengthy process or trick to find special circumstances for easy solving? Thanks.
  38. T

    Minimal polynomials and invertibility

    Let T\in L(V). Let g(x)\in F[x] and let m(x) be the minimal polynomial of T. Show that g(T) is invertible \Leftrightarrow \gcd (m(x),g(x))=1. Backwards is easy. For forwards, suppose I say that g(T) is invertible implies that g(T)(v)=0 \Rightarrow v=0 and therefore g(x) prime, therefore it is...
  39. W

    Back Transformation for Legendre Polynomials

    some body who can explain for me the Legndre polynomials:eek: :eek:
  40. H

    Step-by-Step Guide to Factoring Polynomials: Solving a Tricky Problem

    I am stuck on this one polynomail factoring problem and I was wondering if someone could help me with it: 16a^2 - 24ab + 9b^2 - 25c^2 I tried factoring the first 3 terms as a trinomial but I'm not sure if that's what I'm supposed to do. What I wrote down so far is: (4a -5c)(4a...
  41. H

    How to Factor Polynomials with Three Terms?

    I'm having trouble with this one factoring problem- x^2(a+b)-x(a+b)+(a+b) I thought it was 3 terms so I tried to do decomposition but it doesn't seem to work. If someone could just help me, I'd appreciate it. Thanks!
  42. T

    Mathematica Multiplying Polynomials in Mathematica

    How do I get mathematica to multiply out a bunch of polynomials? Like (1-x)(34+x)(32-x). When I hit return it simply gives me what I wrote.
  43. G

    Determine if (x3/11)+(x2/8) & 4x-22x+35x are Polynomials

    the directions are: determine whether each expression is a polynomial.:rolleyes: i got the easyones like: 12x3-2x2+0.5 is a polynomial but i can't figure these out: (x3/11)+(x2/8) I don't know if the variables can be fractions but i think this could be this: 1/11x3+1/8 x2 so i think...
  44. S

    Integrating Laguerre Polynomials - Fine structure hydrogen

    Hi I have the following problem: To calculate the fine structure energy corrections for the hydrogen atom, one has to calculate the expectation value for (R,R/r^m), where R is the solution of the radial part of the schroedinger equation (i.e. essentially associated laguerre polynomial) and...
  45. B

    Roots of Complex Polynomials

    Question that I came across and that has stumped me for about a week hehe. Let p(z)=z^n +i z^{n-1} - 10 if \omega_j are the roots for j=1,2,...,ncompute: \sum_{j=1}^n \omega_j} and \prod_{j=1}^n \omega_j}
  46. Astronuc

    Special Functions and Polynomials

    PF Member Careful pointed to the website of Gerardus 't Hooft, Dutch physicist and winner of 1999 Nobel Prize in Physics with Martinus J.G. Veltman. 't Hooft has a very interesting and useful website, which includes the following useful pdf file about 'Special Functions and Polynomials'...
  47. M

    Reducible polynomials over Zp.

    find a formula that depends on p that determines the number of reducible monic degree 2 polynomials over Zp. so the polynomials look like x^2+ax+b with a,b in Zp. I examined the case for Z3 and Z5 to try and see what was going on. in Z3 we had 9 monic degree 2 polynomials, 6 of them...
  48. R

    Linear Algebra and polynomials

    Let U and V denote, respectively, the spaces of even and odd polynomials in Pn. Show that dimU + dimV = n+1 [Hint: Consider T: Pn ---> Pn where T[p(x) - P(-x) ] So where to begin? I thought that i should let p(x) = a + a0x + a2x^2 + ... + anX^n So if U is the space of even polynomials...
  49. C

    V not vector space with degree 3 polynomials

    Okay, so i have this problem in my text, and I've almost figured it out (i think) but i need a little help "Let V be the set of all polynomials of degree 3. Define addition and scalar multiplication pointwise. Prove that V with respect to these operations of addiont and scalar multiplication...
  50. G

    Irreducible polynomials over finite fields

    Can someone explain to me why the following is true (ie, show me the proof, or at least give me a link to one): Over the field Zq the following polynomial: x^q^n-x is the product of all irreducible polynomials whose degree divides n Thanks.
Back
Top