What is Polynomial: Definition and 1000 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. stungheld

    Finding Solutions for Polynomial Division: Where to Begin?

    Homework Statement How many pairs of solutions make x^4 + px^2 + q = 0 divisable by x^2 + px + q = 0 Homework Equations x1 + x2 = -p x1*x2= q[/B] The Attempt at a Solution I tried making z = x^2 and replacing but got nowhere. I figure 0,1,-1 are 3 numbers that fit but I am not sure what's...
  2. J

    Polynomial Division (continued from osnarf's problem)

    Hello, My problem is the same as osnarf's problem in thread "Polynomial division proof", https://www.physicsforums.com/threads/polynomial-division-proof.451991/ But, I would like some further help. The problem: Prove that for any polynomial function f, and any number a, there is a polynomial...
  3. B

    MHB How do I solve a polynomial with a missing term?

    Hi, I'm trying to help a high-school sophomore with a math problem, and unfortunately my algebra days are long behind me. Here's the equation: x^3-9x-440=0 I know x=8, but I don't know how to find it. I'd appreciate some guidance. Thanks.
  4. astrololo

    Solving a polynomial with complex coefficients

    Homework Statement z^6+(2i-1)z^3-1-i=0 Homework EquationsThe Attempt at a Solution I know that I must put k=z^3 and solve the quadratic. But I'm not able to simplify the quadratic. I get the square root of (-8i+1) What am I supposed to do ?
  5. Mark Brewer

    Polynomial solution to Legendre's equation

    Homework Statement Starting from the recurrence relation, show that, when l is an integer, the polynomial solution to Legendre's equation is yl(x) = Kl ∑ from k = 0 to (l/2) of (((-1)k) / k!) (((2l - 2k)!) / (l-k)! (l - 2k)!) (xl-2k) where Kl is an arbitrary constant (depending on l) and x...
  6. K

    MHB Prove the polynomial f(x)=x^2-q is irreducible in F_p[x]?

    If p and q are prime numbers such that p is not a quadratic residue mod q. Show that if pq=-1 mod 4 then the polynomial f(x)=x^2-q is irreducible in F_p[x].
  7. aikismos

    Exploring Coefficient Pairing in Polynomials: A Scientific Approach

    Just to double check, but if one wanted to, like in partial fraction decomposition, associate literal coefficients of polynomials with corresponding unknowns on the other side of the equation, the justification for this action is the definition of equality of polynomials? EDIT: I know this...
  8. T

    MHB Factoring Polynomial Equations

    I would like to know if it is possible to determine if a polynomial has rational zeroes, or, in other words, is unfactorable using whole numbers. For example 4x^3+2x^2-4x+25. I know you can use trial and error to sub in the factors of 25, and I understand the rational root theorem. However, I...
  9. ognik

    Find the zeros of a generalised Laguerre polynomial

    Hi - does anyone know of a program library/subroutine - failing that some other source, to find the zeros of a generalised Laguerre polynomial? ie. ## L^{\alpha}_N (x_i) = 0 ##
  10. ognik

    MHB Zeros of generalised Laguerre polynomial

    Hi - does anyone know of a program library/subroutine/some other source, to find the zeros of a generalised Laguerre polynomial? ie. $ L^{\alpha}_N (x_i) = 0 $
  11. F

    Limit with trigonometric and polynomial function.

    Homework Statement For $$\lim _{ x\rightarrow \infty }{ \frac { { x }^{ 2 }+{ e }^{ -{ x }^{ 2 }\sin ^{ 2 }{ x } } }{ \sqrt { { x }^{ 4 }+1 } } } $$, determine whether it exists. If it does, find its value. if it doesn't, explain. Homework Equations Sand witch theorem and arithmetic rule...
  12. W

    Associated Legendre polynomial (I think)

    Homework Statement I'm not 100% sure what this type of problem is called, we weren't really told, so I'm having trouble looking it up. I'd really appreciate any resources that show solved examples, or how to find some! Anyway. For the solution to the spherical wave equation φ(t, θ, Φ) i)...
  13. Z

    Polynomial Degree n Basis: 1,x,x^2...x^n

    I knew that a polynomial of degree n has n+1 basis, i.e 1,x,x^2...x^n; But what if a0=0,i.e the constant term is 0, like x^3+x, then what is the dimension and the basis? Is there only x(one dimension) as the basis?
  14. F

    Python How can I input a polynomial equation of infinite terms in P

    I have been given a task to create an interpolating/extrapolating programme. I have completed the programme for linear interpolation (2 points) but now must make it usable for 3 or more points, ie a polynomial of n points. I think I have the equation in general for a polynomial as it is an...
  15. DeldotB

    Principle Ideals of a Polynomial Quotient Ring

    Homework Statement Let A be the algebra \mathbb{Z}_5[x]/I where I is the principle ideal generated by x^2+4 and \mathbb{Z}_5[x] is the ring of polynomials modulo 5. Find all the ideals of A Let G be the group of invertible elements in A. Find the subgroups of the prime decomposition.Homework...
  16. D

    Complex numbers and polynomial

    Homework Statement Hi,I have a problem regarding to one of the questions in my homework.Actually I am not trying to ask for the solution.I am just not sure what the question is asking for.Please see the attachedHomework EquationsThe Attempt at a Solution In 5(c),the summation notation stated...
  17. S

    Can a cubic polynomial be solved without arccos?

    I was reviewing the Cardano's method formula for a real cubic polynomial having 3 real roots. It seems that to do so, the arccos (or another arc*) of a term involving the p & q parameters of the reduced cubes must be done, and then followed by cos & sin of 1/3 of the result from that arccos -...
  18. E

    Polynomial Inequality Homework: Solving without Technology | Remainder Theorem

    Homework Statement solve 3x4+2x2-4x+6≥6x4-5x3-9x+2 Do not use technology (i.e.-graphing calculators) Homework Equations Remainder Theorem The Attempt at a Solution I set the inequality equal to zero -3x4+5x3+3x2+5x+4≥0 Checking all the Possible rational roots for a possible factors... none...
  19. A

    Integrating a polynomial with a square root

    1. Integrate the following: (4x - x^2)^1/2 dx 2. Any assistance would be appreciated.3. Honestly don't know where to start.
  20. T

    Can't remember how to solve equation with two variables

    Umm from memory I used to use...that triangle: 1 1 1 1 2 1 1 3 3 1 Fibonachii was it? Pathetic I can't even remember the name. To factorise...or was it expand...polynomials...anyway, I don't think that's elevant here. My question is; I had an...
  21. T

    An equality about derivative of a polynomial?

    Why is $$ \left(x^2-1\right)\frac{d}{dx}\left(x^2-1\right)^n = 2nx\left(x^2-1\right)^{n-1} $$? This is in a textbook and says that its proof is left as an exercise. It seems to be a difficult equality. I believe this should just be $$ \left(x^2-1\right)\frac{d}{dx}\left(x^2-1\right)^n =...
  22. C

    Factorizing Polynomials with Irrational Exponents

    I should factorize following polynomial: P(x)=x^2n + 2cos(naπ)x^n + 1 in ℝ if i know that a is irrational number. Things that confuse me here are following: 1. When factorizing polynomials, i have known exponents (unlike here, where i have 2n and n) so i don't know what to do with them? 2...
  23. Math Amateur

    MHB Morphisms (or polynomial maps or regular maps) of algebraic sets - Dummit and Foote

    I am trying to gain an understanding of the basics of elementary algebraic geometry and am reading Dummit and Foote Chapter 15: Commutative Rings and Algebraic Geometry ... At present I am focused on Section 15.1 Noetherian Rings and Affine Algebraic Sets ... ... I need someone to help...
  24. duc

    Coefficient of a polynomial defined by Legendre polynomial

    Homework Statement The polynomial of order ##(l-1)## denoted ## W_{l-1}(x) ## is defined by ## W_{l-1}(x) = \sum_{m=1}^{l} \frac{1}{m} P_{m-1}(x) P_{l-m}(x) ## where ## P_m(x) ## is the Legendre polynomial of first kind. In addition, one can also write ## W_{l-1}(x) = \sum_{n=0}^{l-1} a_n \cdot...
  25. Math Amateur

    MHB Is that correct so far ... ?Yes, that is correct. Good job!

    I am trying to gain an understanding of the basics of elementary algebraic geometry and am reading Dummit and Foote Chapter 15: Commutative Rings and Algebraic Geometry ... At present I am focused on Section 15.1 Noetherian Rings and Affine Algebraic Sets ... ... I need help to get...
  26. Taryn1

    MHB Real Zeros of a Polynomial Function

    So I'm supposed to find all the real zeros of this polynomial function: $\int$ $\left(x\right)$ = $x^3$ + 3$x^2$ - 4$x$ - 12 Usually, to find the zeros, I would use the quadratic function $\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ But what do I do with the 3 at the beginning of the function? I...
  27. B

    Form factored of the polynomial discriminant

    I wrote x² - (a + b)x + (ab) in the wolfram and polynomial discriminant was: a² - 2ab + b². Factoring: (a-b)² --- So, I wrote x³ - (a+b+c) x² + (bc+ca+ab) x - (abc) and the polynomial discrimant given was: Factoring: (b-c)² (c-a)² (a-b)² --- Now, I wrote x² - 2Ax + B² and the polynomial...
  28. Math Amateur

    MHB Polynomial Rings _ Bland - Theorem 6.3.17

    I am reading Paul E. Bland's book, "The Basics of Abstract Algebra". I am currently focused on Chapter 6: Polynomial Rings. I need help with an aspect of Theorem 6.3.17. Theorem 6.3.17 requires awareness of the notation of Definition 6.3.15 which reads as follows...
  29. Math Amateur

    MHB Polynomial Rings - Irreducibility

    I am reading Joseph J. Rotman's book: A First Course in Abstract Algebra with Applications (Third Edition) ... I am currently focused on Section 3.7 Irreducibility... I need help with an aspect of the proof of Theorem 3.97. Theorem 3.97 and its proof read as follows: Now, the first part of...
  30. B

    How to Express Cos 3x as a Polynomial?

    How to express ##\cos 3x## as a polynomial in ##\cos x##?
  31. D

    Polynomial Problem f(x^2+2)=x^4+10x^2+4....

    so i transferred to a new school and I'm collaborating with another pre-cal teacher. she is kind of helpful but i can tell she doesn't really want to share her work (even though collaboration is about helping each other out). She already made a unit test, but i had to make my own answer key. I...
  32. D

    Create a polynomial with desired characteristics, factoring

    Homework Statement Hello! I understand that this is a very simple thing, but somehow I can't find the key :) Please, take a look a pictures attached with a problem and an answer. The task is to create a polynomial f with real number coefficients which has all of the desired characteristics...
  33. Bill_Nye_Fan

    A solvable polynomial with no factors?

    I seem to have encountered a situation in which I have a quartic which has solutions, but no factors. The polynomial is: x^4 - 8x^2 + 224x - 160 = 0 I attempted to find the factors for this quartic in the following manor f(x) = x^4 - 8x^2 + 224x - 160 f(1) = (1)^4 - 8(1)^2 + 224(1) - 160...
  34. B

    Sine/cosine function and polynomial function

    Some values of sine and cosine can ben expressed how the root of a polynomial of nth degree. Example:http://www.wolframalpha.com/input/?i=cos%28%28180%2F7%29%C2%B0%29 (Roll the scroll still you find: "alternate forns" and see the associated polynomial: " x³ - 4 x² - 4 x + 1") So, where I can...
  35. C

    Why my endomorphisme between Polynomial fonction is not continuous?

    Hello let be $$E = \mathbb{R}[X]$$ with the norme $$||P|| = sup_{t \in \mathbb{R}}e^{-|t|}|P(t)|$$. Let be $$A \in E$$. How to show that $$\Psi_{A} : P \rightarrow AP$$ is not continue please? Thank you in advance and have a nice afternoon:oldbiggrin:.
  36. avikarto

    What is the coefficient of the second order term in this polynomial expression?

    I am trying to use a numerical polynomial root finding method, but I am unsure of the order of an expression. For example, if I have something that looks like x2+5x √(x2+3)+x+1=0 what is the coefficient of the second order (and potentially even the first order) term? Is the entire 5x√... term...
  37. Math Amateur

    MHB Solving a Polynomial Equation - Discussion in Fraleigh - Page 204

    I am reading John Fraleigh's book, A First Course in Abstract Algebra. I am at present reading Section 22: Rings of Polynomials. I need some help with an aspect of Fraleigh's discussion of "solving a polynomial equation" or "finding a zero of a polynomial" ... The relevant text in Fraleigh...
  38. Math Amateur

    MHB Kunz - Vanishing Ideal and Minimum Polynomial

    I am reading Ernst Kunz book, "Introduction to Plane Algebraic Curves" I need help with some aspects of Kunz' definition of the vanishing ideal of an algebraic curve and Kunz' definition of a minimal polynomial ... The relevant text from Kunz is as...
  39. B

    Polynomial system of 6 variables

    U = A a² V = 2 A a b W = A b² u = 2 A a c + B a v = 2 A b c + B b w = A c² + B c + C I'd like to solve this system for A, B, C, a, b, c. Is it possible!?
  40. Math Amateur

    MHB Polynomials and Polynomial Functions in I_m = Z/mZ

    I am reading Joseph J.Rotman's book, A First Course in Abstract Algebra. I am currently focused on Section 3. Polynomials I need help with the a statement of Rotman's concerning the polynomial functions of a finite ring such as \mathbb{I}_m = \mathbb{Z}/ m \mathbb{Z} The relevant section...
  41. L

    Integral of Exponential with Polynomial Argument

    How can I find an Integral of an exponential with Polynomial argument with finite limits: \int_0^\pi \exp^{-a x^2 -b x^4} dx \\ \int_0^\pi \exp^{-a x^2 -b x^4} (x - x^3)dx
  42. S

    Irrational Roots Theorems for Polynomial Functions

    Is any Irrational Roots Theorem been developed for polynomial functions in the same way as Rational Roots Theorems for polynomial functions? We can choose several possible RATIONAL roots to test when we have polynomial functions; but if there are suspected IRRATIONAL roots, can they be found...
  43. K

    MHB Showing a polynomial is irreducible

    Given $m,n \in \mathbb{N},$ how can I show that the polynomial $x^m+y^n-1$ is irreducible in $\mathbb C[x,y]$? I'm given the following hint, but I don't follow. Note: I know Eisenstein's Criterion. *Adapt Eisenstein's Criterion to work in $\mathbb C[x,y]$ by using irreducibles in $\mathbb...
  44. K

    MHB Divisibility of a polynomial by another polynomial

    I have this question: Find all numbers $n\geq 1$ for which the polynomial $x^{n+1}+x^n+1$ is divisible by $x^2-x+1$. How do I even begin? **So far I have that $x^{n+1}+x^n+1 = x^{n-1}(x^2-x+1)+2x^n-x^{n-1}+1,$ and so the problem is equivalent to finding $n$ such that $2x^n-x^{n-1}+1$ is...
  45. evinda

    MHB Is the Degree of the Product of Two Polynomials 2n?

    Hello! (Wave) For polynomial multiplication, if $A(x)$ and $B(x)$ are polynomials of degree-bound $n$, we say that their product $C(x)$ is a polynomial of degree-bound $2n-1$ such that $C(x)=A(x)B(x)$ for all $x$ in the underlying field. A way to express the product $C(x)$ is $$C(x)=...
  46. evinda

    MHB How can the polynomial $L_n(x)$ be used to solve the equation Laguerre?

    Hello! (Wave) The differential equation $xy''+(1-x)y'+ay=0, a \in \mathbb{R}$, that is called equation Laguerre, is given. Let $L_n$ be the polynomial $L_n(x)=e^x \frac{d^n}{dx^n} (x^n \cdot e^{-x})$ (show that it is a polynomial), $n=1,2,3, \dots$. Show that $L_n$ satisfies the equation...
  47. M

    What is the Solution to the Chebyshev Polynomial Problem?

    This is something Chebyshev polynomial problems. I need to show that: ##\sum_{r=0}^{n}T_{2r}(x)=\frac{1}{2}\big ( 1+\frac{U_{2n+1}(x)}{\sqrt{1-x^2}}\big )## by using two type of solution : ##T_n(x)=\cos(n \cos^{-1}x)## and ##U_n(x)=\sin(n \cos^{-1}x)## with ##x=\cos\theta##, I have form the...
  48. A

    MHB What is the sum of polynomial zeros?

    From Vieta's Formulas, I got: $a=2r+k$ $b=2rk+r^2+s^2$ $65=k(r^2+s^2)$ Where $k$ is the other real zero. Then I split it into several cases: $r^2 + s^2 = 1, 5, 13, 65$ then: For case 1: $r = \{2, -2, 1, -1 \}$ $\sum a = 2(\sum r) + k \implies a = 13$ Then for case 2: $r^2 + s^2 = 13$, it...
  49. Avatrin

    Irreducibility of polynomial (need proof evaluation)

    One thing I have seen several times when trying to show that a polynomial p(x) is irreducible over a field F is that instead of showing that p(x) is irreducible, I am supposed to show that p(ax + b) is irreducible a,b\in F . This is supposedly equivalent. That does make sense, and I have a...
  50. G

    Langrange interpolation polynomial and Euclidian division

    Homework Statement Let ##x_1,...,x_n## be distinct real numbers, and ## P = \prod_{i=1}^n(X-x_i)##. If for ##i=1...n ##, ##L_i = \frac{\prod_{j \neq i}^n(X-x_j)}{\prod_{j\neq i}(x_i-x_j)}##, show that for any polynomial A (single variable and real coefs), the rest of the euclidian division of A...
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