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. Mashiro

    B Finding polynomials with given roots

    Say we have the following conditions: For an any degree polynomial with integer coefficients, the root of the polynomial is n. There should be infinite polynomials that satisfy this condition. What is the general way to generate one of the polynomial?
  2. S

    I Analogy question for algebraists

    An "analogy question": Polynomials with one variable and coefficients in the field K are to finite dimensional K vector spaces as polynomials in several variables over the field K are to ....? As a teenager, I recall taking tests that had "analogy questions" on them. The format was: Thing A...
  3. G

    A Coefficients of Chebyshev polynomials

    Not long ago, I derived the formula for Chebyshev polynomials $$T_{n}\left( x\right)= \sum_{k=0}^{\lfloor \frac{n}{2} \rfloor}{n \choose 2k}x^{n-2k}\left( x^2-1\right)^{k}$$ How to extract the coefficients of this polynomial of degree n ? I tried using Newton's binomial but got a double sum...
  4. T

    I Antisymmetrizing a Factorized Polynomial Vanishes?

    Hi all, I am having trouble understanding the argument below equation (3.5) in https://arxiv.org/pdf/cond-mat/9605145.pdf where they claim that "Upon antisymmetrization, however, a term with ##k## factors of ##(z_{i}-z_{j})## would have to antisymmetrize ##2k## variables with a polynomial that...
  5. nomadreid

    I Want to understand how to express the derivative as a matrix

    In https://www.math.drexel.edu/~tolya/derivative, the author selects a domain P_2 = the set of all coefficients (a,b,c) (I'm writing horizontally instead off vertically) of second degree polynomials ax^2+bx+c, then defines the operator as matrix to correspond to the d/dx linear transformation...
  6. pawlo392

    I Resolve the Recursion of Dickson polynomials

    I am trying to prove the expression for Dickson polynomials: $$D_n(x, a)=\sum_{i=0}^{\lfloor \frac{n}{2}\rfloor}d_{n,i}x^{n-2i}, \quad \text{where} \quad d_{n,i}=\frac{n}{n-i}{n-i\choose i}(-a)^i$$ I am supposed to use the recurrence relation: $$D_n(x,a)=xD_{n-1}(x,a)-aD_{n-2}(x,a)$$ I have...
  7. pawlo392

    A Differential equation and Appell polynomials

    Hello! Let $n$ be a natural number, $P_n(x)$ be a polynomial with rational coefficients, and $\deg P_n(x) = n$. Let $P_0(x)$ be a constant polynomial that is not equal to zero. We define the sequence ${P_n(x)}_{n \geq 0}$ as an Appell sequence if the following relation holds: \begin{equation}...
  8. V9999

    I A doubt about the multiplicity of polynomials in two variables

    Let ##P(x,y)## be a multivariable polynomial equation given by $$P(x,y)=52+50x^{2}-20x(1+12y)+8y(31+61y)+(1+2y)(-120+124+488y)=0,$$ which is zero at ##q=\left(-1, -\frac{1}{2}\right)##. That is to say, $$ P(q)=P\left(-1, -\frac{1}{2}\right)=0.$$ My doubts relie on the multiplicity of this point...
  9. B

    I Questions about algebraic curves and homogeneous polynomial equations

    It is generally well-known that a plane algebraic curve is a curve in ##\mathcal{CP}^{2}## given by a homogeneous polynomial equation ##f(x,y)= \sum^{N}_{i+j=0}a_{i\,j}x^{i}y^{j}=0##, where ##i## and ##j## are nonnegative integers and not all coefficients ##a_{ij}## are zero~[1]. In addition, if...
  10. V9999

    I May I use set theory to define the number of solutions of polynomials?

    Let ##Q_{n}(x)## be the inverse of an nth-degree polynomial. Precisely, $$Q_{n}(x)=\displaystyle\frac{1}{P_{n}(x)}$$, It is of my interest to use the set notation to formally define a number, ##J_{n}## that provides the maximum number of solutions of ##Q_{n}(x)^{-1}=0##. Despite not knowing...
  11. M

    I Polynomials can be used to generate a finite string of primes....

    F(n)=##n^2 −n+41## generates primes for all n<41. Questions: (1) Are there polynomials that have longer lists? (2) Is such a list of polynomials finite (yes, no, unknown)? (3) Same questions for quadratic polynomials?
  12. M

    Proof: Integer Divisibility by 3 via Polynomials

    Proof: Let ## P(x)= \Sigma^{m}_{k=0} a_{k} x^{k} ## be a polynomial function. Then ## N=a_{m}10^{m}+a_{m-1}10^{m-1}+\dotsb +a_{1}10+a_{0} ## for ## 0\leq a_{k}\leq 9 ##. Since ## 10\equiv 1\pmod {3} ##, it follows that ## P(10)\equiv P(1)\pmod {3} ##. Note that ## N\equiv (a_{m}+a_{m-1}+\dotsb...
  13. S

    How Do You Simplify and Analyze Taylor Polynomials for Higher Degree Functions?

    f(x) = 4 + 5x - 6x^2 + 11x^3 - 19x^4 + x^5 question a almost seems too easy as I end up 'removing' the x^4 and x^5 terms a. T_{2} (x) = 4 + 5x - 6x^2 b. = R_{2} (x) = 11x^3 - 19x^4 + x^5 c. i don't understand what i need to do here. To find the maximum value of a function, we...
  14. D

    MHB Can you factor the following two polynomials?

    Can you factor the following polynomials over integers? x^4 + 4 x^4 + 3 ~x^2~y^2 + 2 ~y^4 + 4 ~x^2 + 5 ~y^2 + 3 If not, you can get help from the following free math tutoring YouTube channel "Math Tutoring by Dr. Liang" https://www.youtube.com/channel/UCWvb3TYCbleZjfzz8HEDcQQ
  15. patric44

    Potential of a charged ring in terms of Legendre polynomials

    hi guys I am trying to calculate the the potential at any point P due to a charged ring with a radius = a, but my answer didn't match the one on the textbook, I tried by using $$ V = \int\frac{\lambda ad\phi}{|\vec{r}-\vec{r'}|} $$ by evaluating the integral and expanding denominator in terms of...
  16. M

    MHB Program to calculate the sum of polynomials

    Hey! 😊 A polynomial can be represented by a dictionary by setting the powers as keys and the coefficients as values. For example $x^12+4x^5-7x^2-1$ can be represented by the dictionary as $\{0 : -1, 2 : -7, 5 : 4, 12 : 1\}$. Write a function in Python that has as arguments two polynomials in...
  17. A

    Check that the polynomials form a basis of R3[x]

    I put it in echelon form but don't know where to go from there.
  18. D

    I Irreducible polynomials and prime elements

    let p∈Z a prime how can I show that p is a prime element of Z[√3] if and only if the polynomial x^2−3 is irreducible in Fp[x]? ideas or everything is well accepted :)
  19. R

    Expanding potential in Legendre polynomials (or spherical harmonics)

    Using the generating function for the legendre polynomial: $$ \sum_{n=0}^{\infty} P_{n}(x) t^{n}=\frac{1}{\sqrt{1-2 x t+t^{2}}} $$ It's possible to expand the coulomb potential in a basis of legendre polynomials (or even spherical harmonic ) like this: $$ \begin{aligned} &\frac{1}{\left.\mid...
  20. L

    Prove eigenvalues of the derivatives of Legendre polynomials >= 0

    The problem has a hint about finding a relationship between ##\int_{-1}^1 (P^{(k+1)}(x))^2 f(x) dx## and ##\int_{-1}^1 (P^{(k)}(x))^2 g(x) dx## for suitable ##f, g##. It looks they're the weighting functions in the Sturm-Liouville theory and we may be able to make use of Parseval's identity...
  21. Delta2

    I Proving the Existence of Roots in Complex Polynomials

    How do we prove that every polynomial (with coefficients from C) of degree n has exactly n roots in C? This is not a homework (I wish I was young enough to have homework) I guess this is covered in every typical undergraduate introductory algebra course but for the time being I can't find my...
  22. fresh_42

    Challenge Math Challenge - August 2021

    Summary: countability, topological vector spaces, continuity of linear maps, polynomials, finite fields, function theory, calculus1. Let ##(X,\rho)## be a metric space, and suppose that there exists a sequence ##(f_i)_i## of real-valued continuous functions on ##X## with the property that a...
  23. anemone

    MHB Find Polynomials: Real Coeffs Resulting in 1

    Find all pairs of polynomials $p(x)$ and $q(x)$ with real coefficients such that $p(x)q(x+1)-p(x+1)q(x)=1$.
  24. anemone

    MHB Prove Product of Polynomials: No Odd Degree Terms

    Prove that in the following product $P=(1-x+x^2-x^3+\cdots-x^{99}+x^{100})(1+x+x^2+x^3+\cdots+x^{99}+x^{100})$ after multiplying and collecting like terms, there does not appear a term in $x$ of odd degree.
  25. D

    I What can be deduced about the roots of this polynomial?

    Hello everyone, I'm currently doing some research about feedback systems in engineering and right now I'm playing around with special types of feedback matrices. In the process, I stumbled upon a potentially interesting polynomial, which is actually the characteristic polynomial of the system...
  26. issue

    Important help on the subject of polynomials of binomial arrangement

    [Mentor Note -- Multiple threads merged. @issue -- please do not cross-post your threads] Hi, everyone It is known that binomial distribution can also be solved by polynomials. i add document with a question I can not solve. Glad to get for help Thanks to all the respondents
  27. rcgldr

    Factoring polynomials with finite field coeffcients

    I'm not sure if I should post this here or in the mathematics section. I'm trying to find a way to implement a mapping of a larger finite field such as GF(2^64) to a composite field GF((2^32)^2). Let f(x) be a primitive polynomial for GF(2^64), with 1 bit coefficients. If the coefficients of...
  28. F

    A Problem calculating arbitrary Polynomial Chaos polynomials using SAMBA

    Hello everyone. I have recently read the following article (which title is SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos) since I have some data in the form of a histogram without knowing the probability distribution function of said data. I have been able to calculate...
  29. Barracuda

    How do I obtain a set of orthogonal polynomials up to the 7th term?

    Hello everyone, I need some help with this solution. I'm trying to obtain a set of orthogonal polynomials up to the 7th term. I think i got it up to the 6th term, but the integration is getting more complex. I'm not sure if I'm on the right track. Please help
  30. username123456789

    I Invertible Polynomials: P2 (R) → P2 (R)

    0 Let T: P2 (R) → P2 (R) be the linear map defined by T(p(x)) = p''(x) - 5p'(x). Is T invertible ? P2 (R) is the vector space of polynomials of degree 2 or less
  31. U

    I Finding a limit involving Chebyshev polynomials

    How could I show that this limit: ##\lim_{N\to\infty}\frac{\sum_{p=1}^N T_{4N} \left(u_0(N)\cdot \cos\frac{p\pi}{2N+1}\right)}{N}## is equal to 0? In the expression above ##T_{4N}## is the Chebyshev polynomials of order ##4N##, ##u_0(N)\geq 1## is a number such that ##T_{4N}(u_0)=b##, with...
  32. Mayhem

    I Finding global minima of nth degree polynomials

    Is it possible (read: reasonably easy) to find global minima of an nth degree polynomial of the general form $$a_nx^n + a_{n-1}x^{n-1} ... a_2x^2 +a_1x + a_0 = 0$$ It seems to have applications in computational chemistry as I have a "hunch" that polynomial regression could be used to somewhat...
  33. P

    Legendre Polynomials as an Orthogonal Basis

    If you had legendre polynomials defined in ##L^2([-1,1])##, with ##||Pn_2||^2=\frac{2}{2n+1}##, show that for any polynomial with p a set of ##L^2([-1,1])##, with degree less than n, we have the inner product of ##P_n## and p = 0. Find the polynomials ##P_0,... P_4## Tried to use the integral...
  34. anemone

    MHB Proving $x-1$ is a Factor of $P(x)$ with Polynomials

    If $P(x),\,Q(x),\,R(x),\,S(x)$ are polynomials such that $P(x^5)+xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x)$, prove that $x-1$ is a factor of $P(x)$.
  35. anemone

    MHB Find Polynomials Fulfilling Real Coefficient Equation

    Let $k\ne 0$ be an integer. Find all polynomials $P(x)$ with real coefficients such that $(x^3-kx^2+1)P(x+1)+(x^3+kx^2+1)P(x-1)=2(x^3-kx+1)P(x)$ for all real number $x$.
  36. fresh_42

    Constructive Proofs (open) Boundaries on the roots of splitting real polynomials

    Suppose all roots of the polynomial ##x^n+a_{n−1}x^{n−1}+\cdots+a_0## are real. Then the roots are contained in the interval with the endpoints $$ -\dfrac{a_{n-1}}{n} \pm \dfrac{n-1}{n}\sqrt{a_{n-1}^2-\dfrac{2n}{n-1}a_{n-2}}\,. $$ Hint: Use the inequality of Cauchy-Schwarz.
  37. LCSphysicist

    Orthogonality Relationship for Legendre Polynomials

    Suppose p = a + bx + cx². I am trying to orthogonalize the basis {1,x,x²} I finished finding {1,x,x²-(1/3)}, but this seems different from the second legendre polynomial. What is the problem here? I thought could be the a problem about orthonormalization, but check and is not.
  38. M

    MHB How Do Polynomial Functions Transform When Applied to Matrices?

    Hey! 😊 Let $\mathbb{K}$ be a field and $1\leq n\in \mathbb{N}$. For a polynom $\displaystyle{\sum_{i=0}^mc_it^i\in \mathbb{K}[t]}$ and a matrix $a\in M_n(\mathbb{K})$ the $f(a)\in M_n(\mathbb{K})$ is defined by \begin{equation*}f(a):=\sum_{i=0}^mc_ia^i=c_ma^m+c_{m-1}a^{m-1}+\ldots...
  39. G

    How to expand this ratio of polynomials?

    I could simplify the expressions in the numerator and denominator to (1+x^n)/(1+x) as they are in geometric series and I used the geometric sum formula to reduce it. Now for what value of n will it be a polynomial? I do get the idea for some value of n the simplified numerator will contain the...
  40. G

    MHB Understanding Cubic Equation Formula for Polynomials of Degree Three

    Hi, Can someone please help me in understanding few parameters of cubic equation formula for solving polynomial of degree three. I attached the formula in the screenshot. My questions are: (1) what is ". " dot in the end of the formula and what does it mean? (2) I want to use it only for real...
  41. Vick

    C/C++ What is happening to num and den in this C++ code for Stieltjes polynomials?

    I'm not at all C++ literate, but I need to understand what a C++ code is doing for a math problem regarding the Stieltjes polynomials. Especially, I want to know what is happening to the "num" and "den" in the code; the code is in this link: Stieltjes
  42. Vick

    A How Are Stieltjes Polynomials Related to Legendre Polynomials?

    I am looking for a recurrence relation and/or defining expression for the Stieltjes polynomials with regard to the Legendre polynomials. I found an article about it here: Legendre-Stieltjes but they do not offer a formula. For example a recurrence relation for the Gegenbauer polynomials is...
  43. C

    I Question about weights using Chebyshev polynomials as quadrature

    Hello everyone. I am studying this article since I am interested in optimization. The article makes use of Clenshaw–Curtis quadrature scheme to discretize the integral part of the cost function to a finite sum using Chebyshev polynomials. The article differentiates between the case of odd...
  44. C

    I Question about the roots of Chebyshev polynomials

    Hello everyone. I am trying to construct an optimization problem using Chebyshev pseudospectral method as described in this article. For that, I need to calculate the zeros of the Chebyshev polynomial of any order. In the article is sugested to do it as tk=cos(πk/N) k=0, ..., N...
  45. L

    Minimizing as a function of variables

    As promised, here is the original question, with the integral written in a more legible form.
  46. CrosisBH

    I Solving an ODE with Legendre Polynomials

    From Griffiths E&M 4th edition. He went over solving a PDE using separation of variables. It got to this ODE \frac{d}{d\theta}\left(\sin\theta\frac{d\Theta}{d\theta}\right)= -l(l+1)\sin \theta \Theta Griffths states that this ODE has the solution \Theta = P_l(\cos\theta) Where $$P_l =...
  47. M

    Vector space - polynomials vs. functions

    As per source # 1 ( link below), when treating polynomials as vectors, we use their coefficients as vector elements, similar to what we do when we create matrices to represent simultaneous equations. However, what I noticed in Source #2 was that, when functions are represented as vectors, the...
  48. R

    B Properties of roots of polynomials

    i have some doubts from chapter 1 of the book Mathematical methods for physics and engineering. i have attached 2 screenshots to exactly point my doubts. in the first screenshot...could you tell me why exactly the 3 values of f(x) are equal. the first is the very definition of polynomials...but...
  49. R

    B Roots of Polynomials: Understanding Mathematical Methods

    I was reading this book - " mathematical methods for physics and engineering" in it in chapter 1 its says "F(x) = A(x - α1)(x - α2) · · · (x - αr)," this makes sense to me but then it also said We next note that the condition f(αk) = 0 for k = 1, 2, . . . , r, could also be met if (1.8) were...
  50. C

    MHB Open neighbourhoods and equating coefficients of polynomials

    Hi all, I am trying to understand some examples given to me by my supervisor but am struggling with some bits. The part I don't understand is: if the equation $$ax+b\lambda=\bar{a}x-\bar{d}y$$ holds for any $x,y\in V$, an open neighbourhood of the origin, and $\lambda$ is a mapping from $V$ to...