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. I

    MATLAB Polynomials in Matlab - Plotting Results

    y = [0.053 1.477 1.970 3.279 4.153 4.934 5.178 5.828 6.082 6.484]; % data coef = polyfit(x,y,3) X=0:.1:9; Y=polyval(coef,X); plot(x,y,'o',X,Y) that's my code. did i do it right?
  2. NicolasPan

    Difference between Taylor Series and Taylor Polynomials?

    Hello,I've been reading my calculus book,and I can't tell the difference between a Taylor Series and a Taylor Polynomial.Is there really any difference? Thanks in advance
  3. mr.tea

    Taylor Polynomials Homework: Solving (a,b) w/ Error <1/100

    Homework Statement In the attached file. (a,b) Homework Equations \cos(x)=\sum_{k=0}^{n}\frac{(-1)^kx^{2k}}{(2k)!} Pn- Taylor expansion of order n The Attempt at a Solution I know that in this case, in order to get an error less than 1/100, I need 18 terms/order 18(according to Wolfram...
  4. bcrowell

    Curvature polynomials vanish for plane waves?

    Geroch 1968 touches on the Kundt type I and II curvature invariants. If I'm understanding correctly, then type I means curvature polynomials. Type II appears to be something else that I confess I don't understand very well. (I happen to own a copy of the book in which the Kundt paper appeared. I...
  5. S

    Adding and subtracting polynomials

    Homework Statement Write each polynomial in standard form. Then name each polynomial based on its degree and number of terms.[/B] 1.4y^3 -4y^2+3-y 2.x^2=x^4-6 3.x+2Homework EquationsThe Attempt at a Solution 4y+3-y Don't know where else to go from here. Would appreciate a nudge in the right...
  6. A

    High Order Polynomials: Questions & Answers

    Hello I am working on high order polynomials and I have two questions:1. Is there a method other than the one based on the "synthetic division with testing the signs" for identifying the lower and upper limits of the real roots? 2. Is there a way for identifying the smallest interval on the...
  7. DeldotB

    A few questions about a ring of polynomials over a field K

    Homework Statement Consider the ring of polynomails in two variables over a field K: R=K[x,y] a)Show the elements x and y are relatively prime b) Show that it is not possible to write 1=p(x,y)x+q(x,y)y with p,q \in R c) Show R is not a principle ideal domain Homework Equations None The...
  8. M

    Factoring Polynomials of Any Degree: Can Complex Numbers Help?

    Can any polynomial in any degree of x be factored into a product of the form (leading coefficient)(x-a)(x-b) ... (x-z) as long as we can use complex numbers for a,b, etc.? Thanks
  9. 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...
  10. ognik

    MHB Proving Self-Adjoint ODE for Legendre Polynomials

    (I haven't encountered these before, also not in the book prior to this problem or in the near future ...) Show that the 1st derivatives of the legendre polynomials satisfy a self-adjoint ODE with eigenvalue $\lambda = n(n+1)-2 $ Wiki shows a table of poly's , I don't think this is what the...
  11. M

    MHB Identities of Chebyshev polynomials

    Hey! :o We are given the polynomial functions $$T_0(x)=1, T_1(x)=x, x \in \mathbb{R} \\ T_{n+1}(x)=2xT_n(x)-T_{n-1}(x), n \in \mathbb{N}, x \in \mathbb{R}$$ (Chebyshev polynomials) Using induction I have to show that: the degree of $T_n$ is $n$ $\forall n \in \mathbb{N}$ : $T_n(1)=1$...
  12. M

    MHB What Polynomial Pairs Satisfy These Complex Functional Equations?

    Find all pairs of polynomials p(x) and q(x) with real coefficients for which both equations are satisfied: p(x^2+1)=q(x)^2+2x and q(x^2+1)=p(x)^2. These equations are set for all real x. I tried to substitute x for -x and others numbers like -1,1 etc. but nothing happened... I need your help
  13. E

    Linear Transformation and isomorphisms

    Homework Statement Suppose a linear transformation T: [P][/2]→[R][/3] is defined by T(1+x)= (1,3,1), T(1-x)= (-1,1,1) and T(1-[x][/2])=(-1,2,0) a) use the given values of T and linearity properties to find T(1), T(x) and T([x][/2]) b) Find the matrix representation of T (relative to standard...
  14. Math Amateur

    MHB Solutions to Irreducible Polynomials & Quotient Rings

    I have just finished a post entitled: http://mathhelpboards.com/linear-abstract-algebra-14/irreducible-polynomials-quotient-rings-rotman-proposition-3-116-a-16163.htmlon the Linear and Abstract Algebra Forum ... I want to have the following code recognised: k(z) \subseteq \text{ I am } \phi...
  15. Math Amateur

    MHB Irreducible Polynomials and Quotient Rings - Rotman Proposition 3.116

    I am reading Joseph J. Rotman's book: A First Course in Abstract Algebra with Applications (Third Edition) ... I am currently focused on Section 3.8 Quotient Rings and Finite Fields ... I need help with an aspect of the proof of Proposition 3.116 Proposition 3.116 and its proof reads as...
  16. Math Amateur

    MHB Software for Graphing Polynomials in Two Variables

    Does anyone know any very simple to operate (intuitive) software for graphing polynomials in two variables. If the software allows you to plot two polynomials simultaneously then all the better ... Be a bonus if it allowed the finding of zeros also ... Peter
  17. Math Amateur

    MHB Polynomials Over a Field - Rotman, Lemma 3.87

    I am reading Joseph J. Rotman's book: A First Course in Abstract Algebra with Applications (Third Edition) ... I am currently focused on Section 3.6 Unique Factorization ... I need help with an aspect of the proof of Lemma 3.87 ... The relevant text from Rotman's book is as follows: In the...
  18. Math Amateur

    MHB Rational Functions - Polynomials Over a Field - Rotman Proposition 3.70

    I am reading Joseph J. Rotman's book: A First Course in Abstract Algebra with Applications (Third Edition) ... I am currently focused on Section 3.5 From Numbers to Polynomials ... I need help with an aspect of the proof of Lemma 3.70 ... The relevant text from Rotman's book is as follows:In...
  19. Math Amateur

    MHB Polynomials Over a Field - Rotman, Lemma 3.67

    I am reading Joseph J. Rotman's book: A First Course in Abstract Algebra with Applications (Third Edition) ... I am currently focused on Section 3.5 From Numbers to Polynomials ... I need help with an aspect of the proof of Lemma 3.67 ... The relevant text from Rotman's book is as follows:In...
  20. Math Amateur

    MHB Lemma 2 - Factor Rings of Polynomials Over a Field - Nicholson, Lemma 2, Page 224

    I am reading W. Keith Nicholson's book: Introduction to Abstract Algebra (Third Edition) ... I am focused on Section 4.3:Factor Rings of Polynomials over a Field. I need some help with the proof of Lemma 2 on page 223-224. The relevant text from Nicholson's book is as...
  21. Math Amateur

    MHB Factor Rings of Polynomials Over a Field - Nicholson, Lemma 1, Page 224

    I am reading W. Keith Nicholson's book: Introduction to Abstract Algebra (Third Edition) ... I am focused on Section 4.3:Factor Rings of Polynomials over a Field. I need some help with the proof of Part 1 of Lemma 1 on page 223-224. The relevant text from Nicholson's book is as follows...
  22. Math Amateur

    MHB Unique Factorization of Polynomials Over a Field - Nicholson Theorem 12 - Page 218

    I am reading W. Keith Nicholson's book: Introduction to Abstract Algebra (Third Edition) ... I am focused on Section 4.2:Factorization of Polynomials over a Field. I need some help with the proof of Part 1 of Theorem 12 on page 218 The relevant text from Nicholson's book is as...
  23. Math Amateur

    MHB Irreducible Polynomials and Products on Nonzero Polynomials - Nicolson Theorem 11

    I am reading W. Keith Nicholson's book: Introduction to Abstract Algebra (Third Edition) ... I am focused on Section 4.2:Factorization of Polynomials over a Field. I need some help (with an apparently very simple issue) with the proof Theorem 11 on pages 217 - 218 ... just seem to have a...
  24. Math Amateur

    MHB Factorization of Polynomials Over a Field - Nicholson Example 10, Page 215

    I am reading W. Keith Nicholson's book: Introduction to Abstract Algebra (Third Edition) ... I am focused on Section 4.2:Factorization of Polynomials over a Filed. I need some help with Example 10 on page 215 ... The relevant text from Nicholson's book is as follows:In the above text, we read...
  25. B

    Finding Coefficients of Orthogonal Quadric Equations

    Given a quadric equation (F(x,y) = 0), exist other quadric equation (G(x,y) = 0) such that the poinst of intersection between the graphics are ortogonals. So, how to find the coefficients of the new quadic equation? EDIT: I think that F and G needs to satisfy∇F • ∇G = 0. So, if F is known, how...
  26. M

    Solving Polynomials: Factoring

    Homework Statement Solve for x. a) 2x^3-3x^2-5x+6=0 Homework Equations -Find possible values of x -Divide that factor by 2x^3-3x^2-5+6 using long division The Attempt at a Solution ƒ(2)=2(2)^3-3(2)^2-5(2)+6 ƒ(2)=16-12-10+6 ƒ(2)=0 Now using long division divide 2x^3-3x^2-5x+6 by (x-2)...
  27. Math Amateur

    MHB GCDs of Polynomials: Reading Rotman's Corollary 3.58

    I am reading Joseph J.Rotman's book, A First Course in Abstract Algebra. I am currently focused on Section 3.5 From Polynomials to Numbers I need help with the statement and meaning of Corollary 3.58 The relevant section of Rotman's text reads as...
  28. 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...
  29. C

    Hermite Polynomials: What Are the Initial Values?

    I'm currently reading a text which uses Hermite polynomials defined in the recursive manner. The form of the polynomials are such that C0 C1 are the 0th and 1st terms of a taylor series that generate the remaining coefficients. The author then says the standard value of C1 and C0 are used, but...
  30. evinda

    MHB How to Multiply Two Polynomials Using Divide-and-Conquer Algorithms?

    Hello! (Wave) Show how to multiply two linear polynomials $ax+b$ and $cx+d$ using only three multiplications. Give a divide-and-conquer algorithms for multiplying two polynomials of degree-bound $n$ that runs in time $\Theta(n^{\lg 3})$. The algorithm should divide the input polynomial...
  31. G

    Challenging exercise about polynomials

    Homework Statement If ##P,Q## are polynomials of ##\mathbb{Z}[X]##, and ##p## is a prime number that divides all the coefficients of ##PQ##, show that ##p## divides the coefficients of ##P## or the coefficients of ##Q##. Homework Equations ##c_n = \sum_{k=0}^{n} a_k b_{n-k} ## is the n-th...
  32. G

    What is the Greatest Common Divisor of Two Polynomials?

    Homework Statement What is the greatest common divisor of ##X^a - 1 ## and ## X^b - 1 ##, ##(a,b) \in \mathbb{N}^\star## ? Homework EquationsThe Attempt at a Solution Assuming that ## a\le b ##, I find by euclidian division of ##b## by ##a## that ## b = an + r \Rightarrow X^b - 1 = (X^a-1)...
  33. G

    Find the polynomials P in R[X]

    Homework Statement Find the polynomials P in R[X] such that ## X(X+1) P'' +(X+2) P' -P = 0 ## Homework Equations If P is a solution, I assume ## P = \sum_{n=0}^N a_nX^n## The Attempt at a Solution I get the answer ##\{\alpha (X+2),\alpha\in\mathbb{R}\} ##, but I've done many calculation...
  34. M

    MHB Field extensions and roots of polynomials

    Let F be a field extension of Q (the rationals) with [F:Q] = 24. Prove that the polynomial x^5+2x^4-16x^3+6x-10 has no roots in F. Proof: Let a be a root of x^5+2x^4-16x^3+6x-10. Since the polynomial has degree 5 by theorem we know that [Q(a):Q]=5. If a \in F and [F:Q]=24 then by theorem we...
  35. A

    MATLAB Roots of Polynomials by loop in matlab

    Dear Friends! I need to find roots of polynomials with variable coefficients, The command I used is w=0:50 A=w^2 B=w^3+2 C=w+2*w^2 D=w E=w./2 ss=[A B C D E] xi=roots(ss) by this I find all the roots of equation, I want to find velocities by setting v1=w/xi(1) v2=w/xi(2) v3=w/xi(3) v4=w/xi(4)...
  36. B

    Polynomials over a ring evaluated at a value?

    In ring theory, a polynomial over a rings, say ## R[x] ## is presented as an abstract object of the form: ## p(x) = a_{n}x^{n} + ...+ a_{1}x + a_{0} ## where the coefficients ## a_{n}...a_{0} ## are from a ring R with unity and ##x## is a formal symbol. So what is the significance of ## p(x+1)...
  37. C

    How can I factor these polynomials?

    I'm in desperate need of help with factoring. Basically, how do you do it? Below is an example of what I'm up against. 54c^2d^5e^3; 81d^3e^2 It wants me to find the greatest common factor. http://www.rempub.com/80-activities-to-make-basic-algebra-easier That is the book I'm working out of.
  38. M

    Factorising High order polynomials

    Find roots of: I need to set =0 then factorise. I know that this polynomial has coefficients of 1,2,4,8 and there is a rule to factorise this however, i don't know it. Also, i believe a high order polynomial will be included in my exams. Are there any other "special" polynomials such as this...
  39. K

    MHB Algebra help about polynomials

    I really need help. f(x) is a fourth degree polynomial function f(x) has zeros of plus or minus 2 and plus or minus 3i f(0)=-108 Find an equation for f(x) in general form
  40. ognik

    Legendre polynomials in the reverse direction

    I have just written a program to calculate Legendre Polynomials, finding for Pl+1 using the recursion (l+1)Pl+1 + lPl-1 - (2l+1).x.Pl=0 That is working fine. The next section of the problem is to investigate the recursive polynomial in the reverse direction. I would solve this for Pl-1 in...
  41. Rithikha

    Factor Theorem Question: Find a and b for P(x) as a factor of T(x)

    Homework Statement If the polynomial P(x) = x^2+ax+1 is a factor of T(x)=2x^3-16x+b, find a, b Homework EquationsThe Attempt at a Solution Let (px+q) be a factor of P(x), p can possibly be 1 and so can q, according to factor theorem, Hence, factors (x+1) or (x-1) P(1) = 0, substituting I got...
  42. L

    MHB Evaluating Integrals for 5th and 4th order polynomials

    Hi! I have a dataset that I fit to a 5th order and 4th order polynomial -- I was just trying to get the function that best fit the data. However, I realized that when I evaluate the integral for these 2 different functions (between 200 and 400), the answers are vastly different. I assumed...
  43. C

    Dividing Polynomials: How to Solve (4s^3+4s^2+72)/(s+3)

    Mod note: Moved from a technical math section, so missing the template. I have this question and the answer but my mathXL does not show me how it came to this conclusion. (4s3+4s2 + 72)/ s+3I got all the way to the answer 4s2 - 8s The correct answer is 4s2 - 8s + 24 I just don't know the...
  44. S

    MHB Completeness of Laguerre polynomials

    How to establish the completeness of Laguerre polynomials?
  45. Coffee_

    Visualizing legendre polynomials in the hydrogen atom.

    1. The way we solved this problem was proposing that the wave function has to form of ##\Psi=\Theta\Phi R## where the three latter variables represent the anlge and radius function which are independent. The legendre polynomials were the solution to the ##\Theta## part. I am having some trouble...
  46. PsychonautQQ

    Rings and Polynomials and other voodoo

    Homework Statement Prove the following theorem: Let R be any ring and let f =! 0 and g =! 0 (they don't equal zero) be polynomials in R[x]. If the leading coefficient of either f or g is a unit in R, then: 1) fg =! 0 in R[x] 2) deg(fg) = deg(f) + deg(g) Homework EquationsThe Attempt at a...
  47. 22990atinesh

    Highest degree of a given polynomial is

    Homework Statement A polynomial p(x) is such that p(0)=5, p(1)=4, p(2)=9 and p(3)=20. the minimum degree it can have a) 1 b) 2 c) 3 d) 4 Homework EquationsThe Attempt at a Solution a) Not Possible can't connect these points using straight line b) Not even possible to connect these points using...
  48. B

    MHB Advanced Calculus - Differentiable and Converging Polynomials

    I could really use some help on this problem as well! Thanks!
  49. T

    Associated vs. Non-associated Laguerre Polynomials

    Homework Statement Could someone pls clarify if the value of x changes from just Laguerre polynomial to associated one? I am confused about the role of variable x. Homework Equations From what I have learned in the class, I understand that L1n(x) = d/dx Ln(x), n = 1, 2, 3... The Attempt at a...
  50. T

    Associated Laguerre polynomials .

    Homework Statement Show that L11(x) and L12(x) are precisely the polynomials for 1s and 2s orbitals. What is the role of variable x in each case? Homework Equations L1n(x) = d/dx Ln(x), n = 1, 2, 3... The Attempt at a Solution Because L1(x) = 1 - x L2(x) = 2 - 4x + x2: I did: L11(x) = d/dx...
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