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. D

    Geometry question form ax^2+bx+c

    I was tutoring a student and I came across the following question. I feel like I'm missing something obvious, but it seems like there are too many variables for an answer to be determined. The attached picture contains all of the question details.
  2. evinda

    MHB Proving $(n+a)^b=\Theta(n^b)$ Using Polynomials

    Hi! (Wave) I want to show that $\forall a,b$ with $b>0$: $$(n+a)^b=\Theta(n^b)$$ According to my notes, it is like that: $\forall n \geq n_0=\lceil |a| \rceil$ $$(1) \Rightarrow (n+a)^b \leq (2n)^b=2^b n^b=c n^b \Rightarrow (n+a)^b=O(n^b), \forall n \geq \lceil |a| \rceil \text{ and }...
  3. J

    MHB Polynomial Arithmetic in Lisp: How to Begin?

    Hi, I'm new to lisp and I've been set some coursework in it and I don't really know how to begin. I need to implement polynomial arithmetic so I can add, subtract and multiply polynomials. So like: $\left(x + y\right)\left(x + y\right) = \left({x}^{2} + 2xy + {y}^{2}\right)$ It also needs to...
  4. R

    Need Help With Factoring Polynomials

    (x2)3(2x3)3/(4x)2 I got 1/2x^13, is this correct?
  5. PsychonautQQ

    Division Algorithm for Polynomials in R[x] confusing me

    I will be using /= to mean 'does not equal'. From my textbook: Division Algorithm: Let R be any ring and let f(x) and g(x) be polynomials in R[x]. Assume that f(x) /= 0 and that the leading coefficient of f(x) is a unit in R. then unique determined polynomials q(x) and r(x) exist such that 1)...
  6. evinda

    MHB Showcasing $V(I \cap J)=V(I) \cup V(J)$ in Polynomials

    Hello! (Wave) I want to show that if $I,J$ ideals of $K[x_1, x_2, \dots , x_n]$, then $V(I \cap J)=V(I) \cup V(J)$. Do I have to show that $ V(IJ)=V(I\cap J)$ and then $V(IJ)=V(I)\cup V(J)$? (Thinking) If so, that's what I have tried: $x \in V(IJ) \leftrightarrow (f_i \cdot g_j)(x)=0$, where...
  7. B

    Associated Legendre polynomials

    m=1 and l=1 x = cos(θ) What would be the solution to this? Thanks.
  8. T

    Hermite and Legendre polynomials

    Hi, I am just curious, are Hermite and Legendre polynomials related to one another? From what I have learned so far, I understand that they are both set examples of orthogonal polynomials...so I am curious if Hermite and Legendre are related to one another, not simply as sets of orthogonal...
  9. Soumalya

    TextBooks for Some Topics in Mathematics

    Hi, I need suggestions for picking up some standard textbooks for the following set of topics as given below: Ordinary and singular points of linear differential equations Series solutions of linear homogenous differential equations about ordinary and regular singular points...
  10. T

    How to derive Legendre Polynomials?

    Homework Statement Could someone explain how Legendre polynomials are derived, particularly first three ones? I was only given the table in the class, not steps to solving them...so I am curious. Homework Equations P0(x) = 1 P1(x) = x P2(x) = 1/2 (3x2 - 1) The Attempt at a Solution ...
  11. M

    Details regarding Legendre Polynomials

    I just had a few questions not directly addressed in my textbook, and they're a little odd so I thought I would ask, if you don't mind. :) -Firstly, I was just wondering, why is it that Legendre polynomials are only evaluated on a domain of {-1. 1]? In realistic applications, is this a limiting...
  12. C

    Orthonormal Set spanning the subspace (polynomials)

    Homework Statement In the linear space of all real polynomials with inner product (x, y) = integral (0 to 1)(x(t)y(t))dt, let xn(t) = tn for n = 0, 1, 2,... Prove that the functions y0(t) = 1, y1(t) = sqrt(3)(2t-1), and y2 = sqrt(5)(6t2-6t+1) form an orthonormal set spanning the same subspace...
  13. M

    MHB Why must the ring $K[x]$ have infinitely many irreducible polynomials?

    Hey! :o Let $K$ be a field. I want to show that the ring $K[x]$ has infinitely many irreducible polynomials.I have done the following: We suppose that there are finite many irreducible polynomials, $f_1(x), f_2(x), \dots, f_n(x)$ with $deg f_i(x)>0$. Let $g(x)=f_1(x) \cdot f_2(x) \cdots...
  14. iamjon.smith

    What is the correct way to divide complex polynomials in two steps?

    Homework Statement (3y-2/y+3) - (3y+1)/(y2+6y-9) Homework EquationsThe Attempt at a Solution Ok, I have attempted to solve in 2 steps, step 1: solve 3y-2/y+3 step 2: solve 3y+1/y2+6y-9 and then subtract the answers. This doesn't seem to work, as I get: 3y-2/y+3 = 3 remainder 11 and...
  15. chwala

    Does Division of Polynomials Follow a Pattern of Degree Decrease?

    When you have a polynomial say ax^4+bx^3+cx^2+dx+e where a,b,c,d and e are constants and divide this by a polynomial say ax+b it follows that the quotient will be a cubic polynomial. Assuming that a remainder exists, then the remainder will be a constant because in my reasoning, the remainder...
  16. G

    Can you help me prove the integral for Hermite polynomials?

    Hi. I'm off to solve this integral and I'm not seeing how \int dx Hm(x)Hm(x)e^{-2x^2} Where Hm(x) is the hermite polynomial of m-th order. I know the hermite polynomials are a orthogonal set under the distribution exp(-x^2) but this is not the case here. Using Hm(x)=(-1)^m...
  17. caffeinemachine

    MHB Theorem: If Polynomials Converge, Roots Also Converge

    Proposition 5.2.1 in Artin states that: THEOREM. Let $p_k(t)\in \mathbf C[t]$ be a sequence of monic polynomials of degree $\leq n$, and let $p(t)\in \mathbf C[t]$ be another monic polynomial of degree $n$. Let $\alpha_{k,1},\ldots,\alpha_{k,n}$ and $\alpha_1,\ldots,\alpha_n$ be the roots...
  18. E

    A question about polynomials of degree 2

    Hey! In my calculus book they claim that a second degree polynomial always can be rewritten as x^2 - a^2 or as x^2 + a^2, if you use an appropriate change of variable. I was thinking about how this works. Let's say we have a second degree polynomial (on the general form?) ax^2 +bx + c = 0...
  19. T

    Calculating Taylor polynomials , Multiple Questions

    Whats up guys ! currently studying for calculas exam and could use someone going over my answers ! Homework Statement Q1. Calculate the taylor polynomial of degree 5 centred 0 for f(x) = e-x. Simply coeffcients and use the error formula to estimate the error when p5(0.1) Q.2 Q1...
  20. Math Amateur

    MHB Exercise 2.47 on Page 114: Showing a Polynomial Has Root in \mathbb{F}_4 - Peter

    I am reading Chapter 2: Commutative Rings in Joseph Rotman's book, Advanced Modern Algebra (Second Edition). I need help with Exercise 2.47 on page 114. Problem 2.47 reads as follows: I need help with showing that f(x) has a root \alpha \in \mathbb{F}_4 . My work on this part of the problem...
  21. STEMucator

    Legendre Equation & Polynomials

    I came across the Legendre differential equation today and I'm curious about how to solve it. The equation has the form: $$(1 - x^2)y'' - 2xy' + \nu(\nu +1)y = 0, (1)$$ Where ##\nu## is a constant. The equation has singularities at ##x_1 = \pm 1## where both ##p## and ##q## are not analytic...
  22. PsychonautQQ

    Polynomials in n variables subspaces and subrepresentations

    Homework Statement Trying to make sense of my notes... "A polynomial in n variables on an n-dimensional F-vector space V is a formal sum of the form: p(x)= ∑(C_i)x^β" so basically can somebody help me understand how polynomials represent vector spaces? Whatever degree the polynomial is...
  23. D

    Expansion of Cos(x) in Hermite polynomials

    [/itex]Homework Statement Find the first three coeficents c_n of the expansion of Cos(x) in Hermite Polynomials. The first three Hermite Polinomials are: H_0(x) = 1 H_1(x) = 2x H_0(x) = 4x^2-2The Attempt at a Solution I know how to solve a similar problem where the function is a polynomial of...
  24. J

    General form of the solutions of polynomials

    I was studying a article that solves the cube and quartic equation in the inverse sense: ##x = \sqrt[3]{A} + \sqrt[3]{B}## ##x = \sqrt[4]{A} + \sqrt[4]{B} + \sqrt[4]{C}## https://www.physicsforums.com/attachment.php?attachmentid=70239&stc=1&d=1401676309I found this relationship too...
  25. powerof

    Reason if polynomials A, B and C exist s.t. they satisfy the following

    Homework Statement Reason or prove whether there exist polynomials A, B and C such that the following is satisfied where y=e^{k\cdot arcsinx}: A\cdot y''+B\cdot y'+C\cdot y=0 Note that this is high school level calculus so it shouldn't be something too complicated. While I said...
  26. J

    Hydrogen Radial Equation: Recursion Relation & Laguerre Polynomials

    I'm in the first of 3 courses in quantum mechanics, and we just started chapter 4 of Griffiths. He goes into great detail in most of the solution of the radial equation, except for one part: translating the recursion relation into a form that matches the definition of the Laguerre polynomials...
  27. R

    Lagendre Polynomials - using the recursion relation

    (1) P_{l}(u) is normalised such that P_{l}(1) = 1. Find P_{0}(u) and P_{2}(u) We have the recursion relation: a_{n+2} = \frac{n(n+1) - l(l+1)}{(n+2)(n+1)}a_{n} I'm going to include a second similar question, which I'm hoping is solved in a similar way, so I can relate it to the above...
  28. S

    Write sin in terms of Hermite polynomials

    Homework Statement Write ##sin(ax)## for ##a \in \mathbb{R}##. (Use generating function for appropriate ##z##) Homework Equations ##e^{2xz-z^2}=\sum _{n=0}^{\infty }\frac{H_n(x)}{n!}z^n## The Attempt at a Solution No idea what to do. My idea was that since...
  29. M

    Finding Error on Taylor Polynomials using Taylor's Theorem

    (a) Use Taylor's Theorem to estimate the error in using the Taylor Polynomial of f(x)=sqrt{x} of degree 2 to approximate sqrt{8}. (The answer should be something like 1/2 * 8^{-7/2}. (b) Find a bound on the difference of sin(x) and x- x^{3}/6 + x^{5}/120 for x in [0,1]This is a problem on a...
  30. R

    Legendre Polynomials - how to find P0(u) and P2(u)?

    Pl(u) is normalized such that Pl(1) = 1. Find P0(u) and P2(u) note: l, 0 and 2 are subscript recursion relation an+2 = [n(n+1) - l (l+1) / (n+2)(n+1)] an n is subscript substituted λ = l(l+1) and put n=0 for P0(u) and n=2 for P2(u), didnt get very far please could someone...
  31. anemone

    MHB Polynomial Challenge: Find # of Int Roots of Degree 3 w/ Coeffs

    If $P(0)=3$ and $P(1)=11$ where $P$ is a polynomial of degree 3 with integer coefficients and $P$ has only 2 integer roots, find how many such polynomials $P$ exist?
  32. D

    Are fractional polynomials linearly independent?

    i.e., does the set of functions of the form, \{ x^{\frac{n}{m}}\}_{n=0}^{\infty} for some fixed m produce a linearly independent set? Either way, can you give a brief argument why or why not? Just curious :)
  33. D

    Roots of multivariate polynomials

    What is the maximum number of roots of a multivariate polynomial over a field? Is there a multivariate version of the fundamental theorem of algebra?
  34. evinda

    MHB Greatest common divisor of polynomials

    Hello! :cool: I want to find the greatest common divisor of $x^4+1$ and $x^2+x+1$. I applied the Euclidean division and found that $x^4+1=(x^2+x+1) \cdot (x^2-x)+(x+1)$. So,isn't it like that: $gcd(x^4+1,x^2+x+1)=x+1$ ? But.. in my textbook,the result is $1$! Which of the both results is...
  35. I

    Polynomials and complex numbers

    Homework Statement Suppose that u and v are real numbers for which u + iv has modulus 3. Express the imaginary part of (u + iv)^−3 in terms of a polynomial in v.Homework Equations The Attempt at a Solution |u+iv|=3 then sort(u^2+i^2) = 3 then u = 3 and v=0 or u=0 and v=3(0+3i)^-3 i swear i am...
  36. mente oscura

    MHB Polynomials and Roots: Properties and Analysis

    Hello. I open this 'thread', in number theory, but he also wears "calculation". I've done a little research, I share with you. Let \ r_1, r_2, \cdots, r_n, roots of the polynomial. P(x)=p_0x^n+p_1x^{n-1}+ \cdots+p_{n-1}x+p_n Let \ Q(x)=q_0 x^n+q_1x^{n-1}+ \cdots +q_n, such that its roots...
  37. F

    Simple Substitution Taylor Polynomials

    I've been taught that with the basic form of a function's maclaurin series, complex forms of the same series can be found. For example, the first three terms for arctan(x) are x-x^3/3 + x^5/5, meaning the first three terms for arctan(x^2+1) at a=0 should be (x^2+1) - ((x^2+1)^3)/3 +...
  38. J

    Can a Polynomial be Transformed to Eliminate its Quadratic and Linear Terms?

    Homework Statement I want to transform a polynomial of kind p(x)=ax³+bx²+cx+d in another like p(t)=At³+B. Is possible? Homework Equations Is possible to transform a polynomial of kind ax³+bx²+cx+d in another like t³+pt+q...
  39. V

    Volume Integral Orthogonal Polynomials

    Hello. Homework Statement Basically I want to evaluate the integral as shown in this document: Homework Equations The Attempt at a Solution The integral with the complex exponentials yields a Kronecker Delta. My question is whether this Delta can be taken inside the integral...
  40. schrodingerscat11

    Derivation: Normalization condition of Legendre polynomials

    Greetings! :biggrin: Homework Statement Starting from the Rodrigues formula, derive the orthonormality condition for the Legendre polynomials: \int^{+1}_{-1} P_l(x)P_{l'}(x)dx=(\frac{2}{2l + 1}) δ_{ll'} Hint: Use integration by parts Homework Equations P_l=...
  41. M

    MHB Maximum error for the Lagrange interpolating polynomials

    Hey :o ! Could you help me at the following exercise? $k, n \in \mathbb{N}$ $f(x)=cos(k \pi x), x \in [0,1]$ $x_i=ih, i=0,1,2,...,n, h=\frac{1}{n}$ Let $p \in \mathbb{P}_n$ the Lagrange interpolating polynomials of $f$ at the points $x_i$. Calculate an upper bound of the maximum error...
  42. alyafey22

    MHB Group of polynomials with coefficients from Z_10.

    Contemporary Abstract Algebra by Gallian This is Exercise 14 Chapter 3 Page 69 Question Let $G$ be the group of polynomials under the addition with coefficients from $Z_{10}$. Find the order of $f=7x^2+5x+4$ . Note: this is not the full question, I removed the remaining parts. Attempt...
  43. M

    Taylor Polynomials and Numerical Analysis

    Homework Statement Use a Taylor Polynomial about pi/4 to approximate cos(42){degrees} to an accuracy of 10^-6. *To get an accuracy of 10^-6, use the error term to determine an nth Taylor Polynomial to use. Homework Equations x = 45 or pi/4, x0 = 42 or 7pi/30 cos(x) = Pn(x) + Rn(x)...
  44. O

    Electric field and Legendre Polynomials

    Homework Statement I want to varify that the components of a homogenous electric field in spherical coordinates \vec{E} = E_r \vec{e}_r + E_{\theta} \vec{e}_{\theta} + E_{\varphi} \vec{e}_{\varphi} are given via: E_r = - \sum\limits_{l=0}^\infty (l+1) [a_{l+1}r^l P_{l+1}(cos \theta) - b_l...
  45. K

    First derivative of the legendre polynomials

    show that the first derivative of the legendre polynomials satisfy a self-adjoint differential equation with eigenvalue λ=n(n+1)-2 The attempt at a solution: (1-x^2 ) P_n^''-2xP_n^'=λP_n λ = n(n + 1) - 2 and (1-x^2 ) P_n^''-2xP_n^'=nP_(n-1)^'-nP_n-nxP_n^' ∴nP_(n-1)^'-nP_n-nxP_n^'=(...
  46. Sudharaka

    MHB Euclidean Space of Polynomials

    Hi everyone, :) Here's a question I encountered and I need your help to solve it. Question: Let \(V\) be the space of real polynomials of degree \(\leq n\). a) Check that setting \(\left(f(x),\,g(x)\right)=\int_{0}^{1}f(x)g(x)\,dx\) turns \(V\) to a Euclidean space. b) If \(n=1\), find...
  47. Math Amateur

    MHB Residue Class Rings (Factor Rings) of Polynomials _ R Y Sharp

    I am reading R Y Sharp: Steps in Commutative Algebra. In Chapter 3 (Prime Ideals and Maximal Ideals) on page 44 we find Exercise 3.24 which reads as follows: ----------------------------------------------------------------------------- Show that the residue class ring S of the ring of...
  48. Y

    MHB Understanding Legendre Polynomials: A Guide for Students

    Does anyone understand this project? I desperately need your help! Please let me know. Appreciate a lot!
  49. T

    Find T-cyclic subspace, minimal polynomials, eigenvalues, eigenvectors

    Homework Statement Let T: R^6 -> R^6 be the linear operator defined by the following matrix(with respect to the standard basis of R^6): (0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 ) a) Find the T-cyclic subspace generated by each standard basis vector...
  50. B

    Expressing gcd of two polynomials as a linear combination

    Homework Statement Find the ##gcd(x^3+x^2-x, x^5+x^4+2x^2-x-1) ##and write it as a linear combination. Homework Equations The Attempt at a Solution I know the ##gcd(x^3+x^2-x, x^5+x^4+2x^2-x-1)=1## What I have so far is ##1. x^5+x^4+2x^2-x-1=(x^3+x^2-x)(x^2+1)+(x^2-1)## ##2...
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