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. F

    I Adapting Hermite polynomials

    Hello everyone. I am working with generalized polynomial chaos. To represent a Normal random variable, the Hermite polynomials are used. However, as far as I understand, these represent N(0,1); if what I have read is correct, if I want to work with any other mean and variance, I shoud simply...
  2. J

    I Legendre polynomials in boosted temperature approximation

    Hi all, In S. Weinberg's book "Cosmology", there is a derivation of the slightly modified temperature of the cosmic microwave background as seen from the Earth moving w.r.t. a frame at rest in the CMB. On Page 131 (1st printing), an approximation (Formula 2.4.7) is given in terms of Legendre...
  3. L

    Legendre polynomials, Hypergeometric function

    Homework Statement _2F_1(a,b;c;x)=\sum^{\infty}_{n=0}\frac{(a)_n(b)_n}{(c)_nn!}x^n Show that Legendre polynomial of degree ##n## is defined by P_n(x)=\,_2F_1(-n,n+1;1;\frac{1-x}{2}) Homework Equations Definition of Pochamer symbol[/B] (a)_n=\frac{\Gamma(a+n)}{\Gamma(a)} The Attempt at a...
  4. chwala

    What Are the Roots of a Given Quartic Polynomial?

    Homework Statement Given ## x^4+x^3+Ax^2+4x-2=0## and giben that the roots are ## 1/Φ, 1/Ψ, 1/ξ ,1/φ## find AHomework EquationsThe Attempt at a Solution ## (x-a)(x-b)(x-c)(x-d)=0## where a,b,c and d are the roots
  5. V

    MHB Laguerre polynomials Roots

    Hi - does anyone know of a program library/subroutine/some other source, to find the zeros of a generalised Laguerre polynomial? ie. LαN(xi)=0
  6. opus

    B Upper and Lower Bounds of Polynomials

    Say we have a polynomial ##f(x)=2x^3+3x^2-14x-21## and we want to find the upper and lower bounds of the real zeros of this polynomial. If no real zero of ##f## is greater than b, then b is considered to be the upper bound of ##f##. And if no real zero of ##f## is less than a, then a is...
  7. A

    MHB Problem about Rodrigues' formula and Legendre polynomials

    using Rodrigues' formula show that \int_{-1}^{1} \,{P}_{n}(x){P}_{n}(x)dx = \frac{2}{2n+1} {P}_{n}(x) = \frac{1}{2^nn!}\frac{d^n}{dx^n}(x^2-1)^n my thoughts \int_{-1}^{1} \,{P}_{n}(x){P}_{n}(x)dx = \frac{1}{2^{2n}(n!)^2}\int_{-1}^{1} \,\frac{d^n}{dx^n}(x^2-1)^n\frac{d^n}{dx^n}(x^2-1)^ndx...
  8. opus

    B Solving for Zeros in Polynomials of Higher Degree

    Please see the attached image which is presented in my text. This is the end step after using the Rational Zeros Theorem to find possible rational zeros, testing by synthetic division, and then factoring. What I don't understand here is that we have the term...
  9. A

    MHB What is Associated Legendre polynomials

    hey i have doubt about Legendre polynomials and Associated Legendre polynomials what is Associated Legendre polynomials ? It different with Legendre polynomials ?
  10. Math Amateur

    MHB Submodules of R-Module of Polynomials ....

    I am reading Paul E. Bland's book: Rings and Their Modules and am currently focused on Section 2.1 Direct Products and Direct Sums ... ... I need help with Problem 8 of Problem Set 2.1 ... Problem 8 of Problem Set 2.1 reads as follows: " 8. Verify Examples 4 and 5 " I am working on...
  11. ertagon2

    MHB Induction and Polynomials

    Can someone check if my answers are right and help me with the missing ones.
  12. J

    MHB Solve Part (b) of Nonzero Polynomials | Help Needed

    Hi all, I have solved part (a) which require verification if it is correct. However, for part b, I am not sure how to do. Appreciate your help. Thank you.
  13. C

    I Defining Legendre polynomials in (1,2)

    Hello everyone. The Legendre polynomials are defined between (-1 and 1) as 1, x, ½*(3x2-1), ½*(5x3-3x)... My question is how can I switch the domain to (1, 2) and how can I calculate the new polynomials. I need them to construct an estimation of a random uniform variable by chaos polynomials...
  14. C

    I Splitting ring of polynomials - why is this result unfindable?

    Assume that ##P## is a polynomial over a commutative ring ##R##. Then there exists a ring ##\tilde R## extending ##R## where ##P## splits into linear factor (not necessarily uniquely). This theorem, whose proof is given below, is difficult to find in the literature (if someone know a source, it...
  15. opus

    Domain and Range of a Function and Its Inverse- Polynomials

    Homework Statement Consider the function ##f\left(x\right)=\sqrt {x+2}##. Determine if the function is a one-to-one function, If so, find ##f^{-1}\left(x\right)## and state the domain and range of ##f\left(x\right)## and ##f^{-1}\left(x\right)## Homework Equations N/A The Attempt at a...
  16. another_dude

    Problem with a product of 2 remainders (polynomials)

    Homework Statement [/B] Polynomial P(x) when divided by (x-2) gives a remainder of 10. Same polynomial when divided by (x+3) gives a remainder of 5. Find the remainder the polynomial gives when divided by (x-2)(x+3). 2. Homework Equations Polynomial division, remainder theorem The Attempt...
  17. C

    I Is there a geometric interpretation of orthogonal functions?

    Hi all. So to start I'll say I'm just dealing with functions of a real variable. In my linear algebra courses one thing was drilled into my head: "Algebraic invariants are geometric objects" So with that in mind, is there any geometric connection between two orthoganal functions on some...
  18. J

    I Primes and Polynomials

    Does there exist a polynomial P(x) with rational coefficients such that for every composite number x, P(x) takes an integer value and for every prime number x, P(x) does not take on an integer value? Can someone please guide me in the right direction? I've tried to consider the roots of the...
  19. J

    How to Solve for x: Factoring Polynomials Homework Statement

    Homework Statement Solve for x: 3x^3 + 2x^2 + 75x - 50 = 0 The Attempt at a Solution I have tried substituting multiple values for "x" so that we get a factor, f(x)=0 I cannot seem to find an "x" value that will make this function=0. Is there a way to factor this function or did the book...
  20. O

    Find inflection points of polynomials.

    Homework Statement Find critical points and inflection points of: 1/[x(x-1)] Homework Equations 1/[x(x-1)] The Attempt at a Solution using quotient rule, we obtain (0-(2x-1)/(x^2-x)^2 set -2x+1=0 we get 1/2 for critical point. for second derivative, i get...
  21. O

    Find maxima/minima of polynomials

    Homework Statement find maxima/minima of following equation. Homework Equations -(x+1)(x-1)^2 The Attempt at a Solution (-x-1)(x-1)^2 Using product rule, we obtain, -1(x-1)^2+(-x-1)*2(x-1) I don't know where to go from here. The software's factoring I had never seen before.
  22. O

    Derivative of two polynomials, one of them being squared

    Homework Statement find derivative of (x-2)(x-3)^2 Homework Equations using product rule. The Attempt at a Solution 1(x-3)^2+2(x-3) x^2-6x-9 +2x-6 x^2-4x-15 doesn't factor.
  23. V

    I Predicting new polynomials from known ones

    Not too sure which forum this would be best suited to. Say I have lots of polynomials that have been obtained through conducting experiments, with the different coefficients in the polynomial representing different physical properties that have been changed in each case. How could I use this...
  24. Mr Davis 97

    Prove that roots of trig polynomials are denumerable

    Homework Statement Prove that the roots of trigonometric polynomials with integer coefficients are denumerable. Homework EquationsThe Attempt at a Solution The book does not define what a trig polynomial is, but I am assuming it is something of the form ##\displaystyle a_0 + \sum^N_{n=1}a_n...
  25. H

    A Problems with identities involving Legendre polynomials

    I am studying the linear oscillation of the spherical droplet of water with azimuthal symmetry. I have written the surface of the droplet as F=r-R-f(t,\theta)\equiv 0. I have boiled the problem down to a Laplace equation for the perturbed pressure, p_{1}(t,r,\theta). I have also reasoned that...
  26. B

    Water Bottle Design Using Polynomials

    Homework Statement [/B] I am to design a 600mL water bottle by drawing one side (bottle lying horizontally). Three types of functions must be included (different orders). The cross-sectional view would be centred about the x-axis, and the y-axis would represent the radius of that particular...
  27. awholenumber

    B What are the different methods of factoring polynomials?

    Methods of factoring . Method of common factors Factorization by regrouping terms Factorization using identities Factors of the form ( x + a) ( x + b) Factor by Splitting Is this all the factoring methods out there ? Or are there more ? I am also looking for a book with lots of...
  28. Alan Sammarone

    I Integration of Legendre Polynomials with different arguments

    Hi everybody, I'm trying to calculate this: $$\sum_{l=0}^{\infty} \int_{\Omega} d\theta' d\phi' \cos{\theta'} \sin{\theta'} P_l (\cos{\gamma})$$ where ##P_{l}## are the Legendre polynomials, ##\Omega## is the surface of a sphere of radius ##R##, and $$ \cos{\gamma} = \cos{\theta'}...
  29. Math Amateur

    MHB Separable Polynomials - Dummit and Foote - Proposition 37 .... ....

    I am reading David S. Dummit and Richard M. Foote : Abstract Algebra ... I am trying to understand the proof of Proposition 37 in Section 13.5 Separable and Inseparable Extensions ...The Proposition 37 and its proof (note that the proof comes before the statement of the Proposition) read as...
  30. Math Amateur

    I Separable Polynomials - Dummit and Foote - Proposition 37

    I am reading David S. Dummit and Richard M. Foote : Abstract Algebra ... I am trying to understand the proof of Proposition 37 in Section 13.5 Separable and Inseparable Extensions ...The Proposition 37 and its proof (note that the proof comes before the statement of the Proposition) read as...
  31. Math Amateur

    MHB Separable Polynomials - Remarks by Dummit and Foote .... ....

    Dummit and Foote in Section 13.5 on separable extensions make some remarks about separable polynomials that I do not quite follow. The remarks follow Corollary 34 and its proof ... Corollary 34, its proof and the remarks read as follows: https://www.physicsforums.com/attachments/6639 In the...
  32. Math Amateur

    I Separable Polynomials - Remarks by Dummit and Foote .... ....

    Dummit and Foote in Section 13.5 on separable extensions make some remarks about separable polynomials that I do not quite follow. The remarks follow Corollary 34 and its proof ... Corollary 34, its proof and the remarks read as follows: In the above text by D&F, in the remarks after the...
  33. Math Amateur

    MHB Separable Polynomials - Paul E Bland's definition and example ....

    I am reading Paul E Bland's book: The Basics of Abstract Algebra and I am trying to understand his definition of "separable polynomial" and his second example ... Bland defines a separable polynomial as follows:https://www.physicsforums.com/attachments/6636... and Bland's second example is as...
  34. Math Amateur

    I Separable Polynomials - Paul E Bland's definition and exampl

    I am reading Paul E Bland's book: The Basics of Abstract Algebra and I am trying to understand his definition of "separable polynomial" and his second example ... Bland defines a separable polynomial as follows: ... and Bland's second example is as follows: I am uncomfortable with, and do...
  35. Math Amateur

    MHB Splitting Fields and Separable Polynomials ....

    I am reading both David S. Dummit and Richard M. Foote : Abstract Algebra and Paul E. Bland's book: The Basics of Abstract Algebra ... ... I am trying to understand separable polynomials ... ... but D&F and Bland seem to define them slightly differently and interpret the application of the...
  36. Math Amateur

    I Splitting Fields and Separable Polynomials ....

    I am reading both David S. Dummit and Richard M. Foote : Abstract Algebra and Paul E. Bland's book: The Basics of Abstract Algebra ... ... I am trying to understand separable polynomials ... ... but D&F and Bland seem to define them slightly differently and interpret the application of the...
  37. D

    Matlab expressions for two types of odd polynomials

    Homework Statement i am trying to find the expressions for the "equiripple" and "maximal flat" polynomials for the equiripple :[/B] for maximal flat: Homework EquationsThe Attempt at a Solution i know that for maximal flat its butterworth polinomials, but for P_3 i don't get the same...
  38. M

    I Linearization with Polynomials

    Hi everyone. I started to look at different linearization techniques like: -linear interpolation - spline interpolation - curve fitting... Now Iam wondering (and I guess its very stupid) : As polynomials with a degree > 1 are not linear, why can I use them for linearization? With the...
  39. Mr Davis 97

    I Why are polynomials defined the way they are in algebra?

    I've always been curious about why we define polynomials the way we do. On the surface, it seems that they are expressions that naturally arise from combining the standard arithmetic operations on indeterminates. However, there are some points that I am generally confused about. Why are...
  40. R

    Normalizing Hermite Polynomials

    Homework Statement Evaluate the normalization integral in (22.15). Hint: Use (22.12) for one of the $H_n(x)$ factors, integrate by parts, and use (22.17a); then use your result repeatedly.Homework Equations (22.15) ##\int_{-\infty}^{\infty}e^{-x^2}H_n(x)H_m(x)dx = \sqrt{\pi}2^nn!## when ##n=m##...
  41. Mr Davis 97

    When do quadratic polynomials generate the same ideal?

    Homework Statement When do two quadratic polynomials in ##\mathbb{Z}_3 [x]## generate the same ideal? Homework EquationsThe Attempt at a Solution I feel like they generate the same ideal only when they have the same coefficients, but am not sure how to show this.
  42. Math Amateur

    MHB Quadratic Polynomials and Irreducibles and Primes

    I am reading "Introductory Algebraic Number Theory"by Saban Alaca and Kenneth S. Williams ... and am currently focused on Chapter 1: Integral Domains ... I need some help with the proof of Theorem 1.2.2 ... Theorem 1.2.2 reads as follows: https://www.physicsforums.com/attachments/6515 In the...
  43. Math Amateur

    I Quadratic Polynomials and Irreducibles and Primes ....

    I am reading "Introductory Algebraic Number Theory"by Saban Alaca and Kenneth S. Williams ... and am currently focused on Chapter 1: Integral Domains ... I need some help with the proof of Theorem 1.2.2 ... Theorem 1.2.2 reads as follows: In the above text from Alaca and Williams, we read the...
  44. lfdahl

    MHB Expressing a polynomial P(x)=(x−a)^2(x−b)^2+1 by two other polynomials

    Let $a$ and $b$ be two integer numbers, $a \ne b$. Prove, that the polynomial: $$P(x) = (x-a)^2(x-b)^2 + 1$$ cannot be expressed as a product of two nonconstant polynomials with integer coefficients.
  45. HOLALO

    B Taylor Polynomials and decreasing terms

    Hi, I have a question about taylor polynomials. https://wikimedia.org/api/rest_v1/media/math/render/svg/09523585d1633ee9c48750c11b60d82c82b315bfI was looking for proof that why every lagrange remainder is decreasing as the order of lagnrange remainder increases. so on wikipedia, it says, for a...
  46. R

    A Legendre Polynomials -- Jackson Derivation

    Hello all, I'm reading through Jackson's Classical Electrodynamics book and am working through the derivation of the Legendre polynomials. He uses this ##\alpha## term that seems to complicate the derivation more and is throwing me for a bit of a loop. Jackson assumes the solution is of the...
  47. Yiming Xu

    I Express power sums in terms of elementary symmetric function

    The sum of the $k$ th power of n variables $\sum_{i=1}^{i=n} x_i^k$ is a symmetric polynomial, so it can be written as a sum of the elementary symmetric polynomials. I do know about the Newton's identities, but just with the algorithm of proving the symmetric function theorem, what should we do...
  48. K

    I Can Taylor series be used to get the roots of a polynomial?

    I'm using this method: First, write the polynomial in this form: $$a_nx^n+a_{n-1}x^{n-1}+...a_2x^2+a_1x=c$$ Let the LHS of this expression be the function ##f(x)##. I'm going to write the Taylor series of ##f^{-1}(x)## around ##x=0## and then put ##x=c## in it to get ##f^{-1}(c)## which will be...
  49. D

    A Integral involving Hermite polynomials

    Hello. I've an integral: \int_{-\infty}^{0}x\exp(-x^2)H_n(x-a)H_n(x+a)dx Of course for any given n it can be calculated, but I'm interested if there is some general formula for arbitrary n. Could someone with access type that into Mathematica? In case that there exists general formula, idea how...
  50. Quadrat

    Factoring a four term polynomial

    Homework Statement I just want to know how get from ##4x^3+3x^2-6x-5=0 ## to ##(x+1)^2(4x-5)=0##. What's the trick when dealing with these nasty polynomials? I got the answer by taking another approach (solving a root equation) but I noted this is also a way to go, but I can't figure out the...
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