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. P

    Polynomials, Kernels and Derivatives

    Is there a simple way to show that when we differentiate the following expression (call this equation 1): Y(x) = \frac{1}{n!} \int_0^x (x-t)^n f(t)dt that we will get the following expression (call this equation 2): Y'(x) = \frac{1}{(n-1)!}\int_0^x (x-t)^{n-1}f(t)dt It's simple...
  2. B

    Can P(x) be factored into irreducible polynomials over Z_5?

    Homework Statement Write P(x) = x^3+2x+3 as the product of Irreducible Polynomials over Z_5 Homework Equations Polynomial division The Attempt at a Solution I start out by taking out a factor of x+3 That is x+3 \div x^3+2x+3 I get P(x) = x^2-3x+1 which has zero...
  3. I

    Normal forms of polynomials over a semiring

    Let R be a commutative semiring. That is a triple (R,+,.) such that (R,+) is a commutative monoid and (R,.) is a commutative semigroup. Let {\mathbf \alpha}_i = \alpha_1,\alpha_2,\ldots,\alpha_n . The n-variate indeterminate is just free monoid on n letters. However, it is common to...
  4. N

    X,Y are polynomials of n, and n is an element of N, then

    Homework Statement X={4^n-3n-1/ n belongs to N} Y={9(n-1)/ n belongs to N } Homework Equations then, XUY is equals to X or Y or N or None of these
  5. Telemachus

    Solve 3rd Degree Taylor Polynomial for \sqrt[ ]{9.03}

    Homework Statement Find an approximated value for \sqrt[ ]{9.03} using a Taylors polynomial of third degree and estimate the error. Homework Equations The Attempt at a Solution I thought of solving it by using f(x)=\sqrt[]{x} centered at x_0=9 So...
  6. W

    Set of degree 2 polynomials a subspace

    Homework Statement Which of the subsets of P2 given in exercises 1 through 5 are subspaces of P2? Find a basis for those that are subspaces. (P(t)|p(0) = 2) Homework Equations The Attempt at a Solution The solution manual says that this subset is not a subspace because it...
  7. M

    Roots of Polynomials: Finding g(y) with y1, y2, y3

    Homework Statement Let x1, x2, x3 are the roots of the polynomial f(x)=x3+px+q, where f(x)\inQ[x], p\neq0. Find a polynomial g(y) of third degree with roots: y1=x1/(x2+x3-q) y2=x2/(x1+x3-q) y3=x3/(x1+x2-q) Homework Equations The Attempt at a Solution Any ideas? Thank you.
  8. O

    Abstract Algebra: Polynomials problem

    Homework Statement Let f(x)=x5-x2-1 \in C and x1,...,x5 are the roots of f over C. Find the value of the symmetric function: (2x1-x14).(2x2-x24)...(2x5-x54) Homework Equations I think, that I have to use the Viete's formulas and Newton's Binomial Theorem. The Attempt at a...
  9. J

    Polynomials: Number of Roots

    I'm trying to prove that a polynomial function of degree n has at most n roots. I was thinking that I could accomplish this by induction on the degree of the polynomial but I wanted to make sure that this would work first. If someone could let me know if this approach will work, I would...
  10. S

    Unique Factorization for polynomials

    Homework Statement Prove unique factorization for hte set of polynomials in x with integer coefficients Homework Equations The Euclidean algorithm may be of some use The Attempt at a Solution Let's say that the polynomial is of the form anx^n + a(n-1)x^(n-1) ... a1x + a0 There...
  11. M

    How Do I Factorize Large Polynomials by Hand?

    Basically, i am doing some cryptography, i need to show that a polynomial i have, which is not irreducibale, implies it is not primitive. I am having trouble factorising these rather large polynomials. I have checked to see whether the following polynomials are irreducible and found there...
  12. clope023

    Linear operator on the set of polynomials

    Homework Statement Let L be the operator on P_3(x) defined by L(p(x)) = xp'(x)+p"(x) if p(x) = a_0(x)+a_1(x)+a_2(1+x^2) calculate L^n(p(x)) Homework Equations stuck between 2 possible solutions i) as powers of x decrease the derivatives of p(x) increase ii) as derivatives...
  13. T

    Integral problems with roots of polynomials

    Homework Statement How do i solve this integral ? \int \big( \sqrt{x^{3}+1} + \sqrt[3] {x^{2}+2x} \big) \ dx Homework Equations The Attempt at a Solution what is the appropriate substitution to make here
  14. estro

    Understanding Taylor Polynomials for Calc-2

    Hi, I'm doing calc-2, and I have hard time understanding and visualizing the idea of Taylor approximation in my head. By the same time I have no problems solving homework on this topic. Can someone please explain how I should visualize and think about approximations using Taylor Polynomials...
  15. estro

    Understanding Maclaurin Polynomials: Exploring Substitution Techniques

    I'm trying to understand the reminder of Maclaurin polynomials http://estro.uuuq.com/0.png http://estro.uuuq.com/1.png [PLAIN]http://estro.uuuq.com/2.png [PLAIN][PLAIN]http://estro.uuuq.com/3.png [PLAIN][PLAIN]http://estro.uuuq.com/4.png Here I show few attempts to use substitution...
  16. T

    Electrostatic potential in Legendre polynomials

    Homework Statement Two spherical shells of radius ‘a’ and ‘b’ (b>a) are centered about the origin of the axes, and are grounded. A point charge ‘q’ is placed between them at distance R from the origin (a<R<b). Expand the electrostatic potential in Legendre polynomials and find the Green...
  17. N

    Generating function for Legendre polynomials

    Homework Statement Using binomial expansion, prove that \frac{1}{\sqrt{1 - 2 x u + u^2}} = \sum_{k} P_k(x) u^k. Homework Equations \frac{1}{\sqrt{1 + v}} = \sum_{k} (-1)^k \frac{(2k)!}{2^{2k} (k!)^2} v^k The Attempt at a Solution I simply inserted v = u^2 - 2 x u, then...
  18. Z

    Bernoulli polynomials evaluated at 1/4

    Homework Statement The problem is to prove the identity B_k(1/4) = 2^{-k} B_k(1/2) for even k. Homework Equations The Bernoulli polynomials B_k(y) are defined by the generating function relation \frac{xe^{xy}}{e^x-1} = \sum_{k=0}^{\infty} \frac{B_k(y) x^k}{k!}. The Attempt at a Solution...
  19. I

    Approximating ln(1.75) with Taylor/Maclaurin Polynomials Using 6 Terms

    Homework Statement Find a Taylor or Maclaurin polynomial to apporximate ln(1.75) using 6 terms. Homework Equations The Attempt at a Solution I now that a MacLaurin polynomial is as follows.. c=0 and a Talyor polynomial is as follows.. so do I assume I'm working...
  20. P

    Polynomials in Zn[x]: Degree & n^2

    I am wondering how you determine how many polynomials of degree, let's say b, are in Zn[x]. From what I gather, it looks like it does not depend on what b is, but rather what n is. Namely, n^2. Is this correct?
  21. M

    Maple Symmetric polynomials in Maple?

    Does anyone know if it possible to generate elementary symmetric polynomials in Maple (I am using version 12), and if so, how? I have scoured all the help files, and indeed the whole internet, but the only thing I have found is a reference to a command "symmpoly", which was apparently...
  22. P

    Writing a polynomial in terms of other polynomials (Hermite, Legendre, Laguerre)

    Homework Statement The first 3 parts of this 4 part problem were to derive the first 5 Hermite polynomials (thanks vela), The first 5 Legendre polynomials, and the first 5 Laguerre polynomials. Here is the last part: Write the polynomial 2x^4-x^3+3x^2+5x+2 in terms of each of the sets of...
  23. T

    MATLAB Plotting multiple polynomials in matlab

    Ok, I can plot a single polynomial easy enough such as 3*(x^2)-1 using fplot, but I want to graph multiple polynomials. When I try to use the plot it doesn't work even for one though. The graph is completely wrong. ie I make a new m-file. x = [-1:1]; y = 3*x.^2 - 1; Then call the...
  24. P

    Indefinite integral (Hermite polynomials)

    Homework Statement I need to evaluate the following integral: \int_{-\infty}^{\infty}x^mx^ne^{-x^2}dx I need the result to construct the first 5 Hermite polynomials. Homework Equations The Attempt at a Solution First I tried arbitrary values for "m" and "n". I was not able to...
  25. Z

    Limit of orthogonal polynomials for big n

    given the 'normalized' Chebyshev and Legendre Polynomials \frac{L_{2n}(x)}{L_{2n}(0)} and \frac{T_{2n}(x)}{T_{2n}(0)} for n even and BIG 2n--->oo then it would be true that (in this limit) \frac{L_{2n}(x)}{L_{2n}(0)}=\frac{sin(x)}{2x} and \frac{T_{2n}(x)}{T_{2n}(0)}=J_{0}(2x) here...
  26. R

    Finding the basis for a set of polynomials (linear algebra)

    Hi. Thanks for the help. Homework Statement Find a basis for the set of polynomials in P3 with P'(1)=0 and P''(2)=0. Homework Equations P' is the first derivative, P'' is the second derivative. The Attempt at a Solution The general form of a polynomial in P3 is ax^3+bx^2+cx+d...
  27. J

    Linear Algebra: Polynomials subspaces

    U and W are subspaces of V = P3(R) Given the subspace U{a(t+1)^2 + b | a,b in R} and W={a+bt+(a+b)t^2+(a-b)t^3 |a,b in R} 1) show that V = U direct sum with W 2) Find a basis for U perp for some inner product Attempt at the solution: 1) For the direct sum I need to show that it...
  28. P

    Residue of a ratio of polynomials

    Homework Statement The problem is to find the inverse laplace of \frac{s^2-a^2}{(s^2+a^2)^2} I am supposed to use the residue definition of inverse laplace (given below) The poles of F(s) are at ai and at -ai and they are both double poles. Homework Equations f(t) =...
  29. H

    Finding Derivatives Using Taylor/Maclaurin Polynomials

    Homework Statement Compute the 6th derivative of f(x) = arctan((x^2)/4) at x = 0. Hint: Use the Maclaurin series for f(x). Homework Equations The maclaurin series of arctanx which is ((-1)^n)*x^(2n+1)/2n+1 The Attempt at a Solution I subbed in x^2/4 for x into the maclaurin...
  30. K

    Division algorithm for polynomials

    Homework Statement M and N are positive integers with M>N. The division algorithm for integers tells us there exists integers Q and R such that M=QN+R with 0\leqR<N. The division algorithm for real polynomials tells us that there exist real polynomials q and r such that xM - 1 = q(xN - 1) +...
  31. Z

    Linear Algebra - Characteristic polynomials and similar matrices question

    Homework Statement For each matrix A below, let T be the linear operator on R3 thathas matrix A relative to the basis A = {(1,0,0), (1,1,0), (1,1,1)}. Find the algebraic and geometric multiplicities of each eigenvalues, and a basis for each eigenspace. a) A =...
  32. F

    Real roots of complex polynomials

    Homework Statement Let f be a polynomial of degree n >= 1 with all roots of multiplicity 1 and real on R. Prove that f has at most one more real root than f' f' has no more nonreal roots than f Homework Equations We are given the Gauss Lucas theorem: Every root of f' is contained in...
  33. G

    Is it possible to interpolate between two polynomials?

    Hi everyone! Having spent many fruitless hours Googling this I stumbled upon this forum, and am hoping you may be able to help... I'm looking for a way to interpolate between two polynomials. These two lines are related and run along in a near-parallel fashion, and I want to divide the gap...
  34. C

    Mastering Factoring Polynomials: Tips, Tricks, and Examples to Help You Succeed!

    Alright, I'll be honest. I was extremely tired and slept all through the lesson in Algebra today lol. And now I need help with factoring polynomials. Example problems that I need help on: 7h3+448 Perfect square factoring - y4-81 Grouping - 3n3-10n2-48n+160 You don't have to answer...
  35. M

    What Are the Rank and Nullity of a Linear Transformation?

    Homework Statement find the rank and nullity of the linear transformation T:U -> V and find the basis of the kernel and the image of T Homework Equations U=R[x]<=5 V=R[x]<=5 (polynomials of degree at most 5 over R), T(f)=f'''' (4th derivative) The Attempt at a Solution Rank = 2...
  36. S

    Finding g(t) for Characteristic Polynomial f(t) = t2 - 5t + 4 and Matrix A

    Homework Statement Suppose A is a 2x2 real matrix with characteristic polynomial f(t) = t2 - 5t +4. Find a real polynomial g(t) of degree 1 such that (g(A))2 = A. Suppose A is a 2x2 complex matrix with A2 ≠ O. Show that there is a complex polynomial g(t) of degree 1 such that (g(A))2 = A...
  37. Y

    Question on Rodrigues' equation in Legendre polynomials.

    I have problem understand in one step of deriving the Legendre polymonial formula. We start with: P_n (x)=\frac{1}{2^n } \sum ^M_{m=0} (-1)^m \frac{2n-2m)}{m!(n-m)(n-2m)}x^n-2m Where M=n/2 for n=even and M=(n-1)/2 for n=odd. For 0<=m<=M \Rightarrow \frac{d^n}{dx^n}x^2n-2m =...
  38. L

    Dividing Polynomials With Exponents

    Nevermind Solved it.
  39. K

    MATLAB Integrating and Differentiating Polynomials in MATLAB

    Hello, First of all, I am not trying to "spam" subforums. I found out that my thread shouldn't be posted under homework. Anyways, here it is. Integration Let say there's a polynomial, 5x+6 and you want to integrate from 0 to 3 respect to x, how do you input in MATLAB? (I guess you can't...
  40. K

    Linear algebra - matrices and polynomials

    Homework Statement Prove for each square matrix B there is a real polynomial p(x) (not the zero polynomial) so p(B)=0 Homework Equations Rank-nullity? dimv = r(T) + n(T) The Attempt at a Solution I've found the dimension for nxn square matrices (n²) and a basis (1 in one place and...
  41. J

    Cubic Polynomials: Solving w/o Rational Roots

    Homework Statement I think I saw another thread answer this question, but I was a little lost whilst reading it. I have just recently learned of the rational root theorem and was using it quite happily; figuring out what possibly answers went with cubic and quartic polynomials gave new...
  42. N

    Calculating Residues of Reciprocal Polynomials

    I have need to calculate the residues of some functions of the form \frac{f(x)}{p(x)} where p(x) is a polynomial. To be more specific I have already calculated the 2 residues of \frac{1}{x^2+a^2}. That one was quite easy. Now I'm asked to calculate the residues of...
  43. Z

    A question about zeros of polynomials

    POlynomials (or Taylor series ) of the form P(x)= \sum_{n}a_{2n}X^{2n} with a_{2n}\ge 0 strictly have ALWAYS pure imaginary roots ?? it happens with sinh(x)/x cos(x) could someone provide a counterexample ? is there an hypothesis with this name ??
  44. F

    Integrating legendre polynomials with weighting function

    Homework Statement I recently came across this integral while doing a problem in electromagnetism (I'm not sure if there exists a nice analytic answer): \int_{0}^{\pi}P_m(\cos(t))P_n(\cos(t)) \sin^2(t) = \int_{-1}^{1}P_m(x) P_n(x) \sqrt{1-x^2}, Homework Equations P_m(x) is the m^th...
  45. G

    Prove Lagrange Polynomials Basis of $\mathbb{R}_{n}[X]

    hello everyone:smile: for i=1,2,...,(n+1) let P_{i}(X)=\frac{\prod_{1\leq j\leq n+1,j\neq i}(X-a_j)}{\prod_{1\leq j\leq n+1,j\neq i}(a_i-a_j)} prove that (P_1,P_2,...P_{n+1}) is basis of \mathbb{R}_{n}[X] . i already have an answer but i don't understand some of it. ... we have...
  46. B

    A subspace spanned by polynomials 1 and x

    1. Let W be the linear subspace spanned by the polynomials 1 and x. Find an orthogonal projection of the polynomial p(x) = 1+x^2 to W. Find a basis in the space W(perp) My problem is that I don't know how to represent W as a matrix so that I could apply the orthogonal projection formula...
  47. O

    Why is simpsons rule exact for 3rd degree polynomials?

    this is really perplexing. how can it be exact? simpsons rule uses quadratics to approximate the curve. how can it be exact if I am approximating a cubic with a quadratic?
  48. E

    Polynomials f(x) & g(x) in Z[x] Relatively Prime in Q[x]

    trying to show that polynomials f(x), g(x) in Z[x] are relatively prime in Q[x] iff the ideal they generate in Z[x] contains an integer.Thanks .Not homework
  49. Z

    A courious thing about orthogonal polynomials

    given a set of orthogonal polynomials p_{n} (x) with respect to a certain positive measure \mu (x) > 0 on a certain interval (a,b) then i have notices for several cases that f(z) defined by the integral transform \int_{a}^{b}dx\mu(x)cos(xz)=f(z) has ALWAYS only real roots ¡¡ *...
  50. B

    Solving complex exponential polynomials

    Are there any general methods to solve the following complex exponential polynomial without relying on numerical methods? I want to find all possible solutions, not just a single solution. e^(j*m*\theta1) + e^(j*m*\theta2)+e^(j*m*\theta3) + e^(j*m*\theta4) + e^(j*m*\theta5) = 0 where...
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