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.

View More On Wikipedia.org
Recent contents Top users
  1. F

    Finding a Basis for Polynomials of Degree n-1

    Homework Statement Let W equal the set of all polynomials in F[x] with degree less than or equal to n-1 such that the sum of the coefficients of the terms is 0. Find a basis of W over F. The Attempt at a Solution I don't know where to begin to find the basis. Do I use the fact that the sum...
  2. N

    Viewing Polynomials from k[x,y] as Elements of k[y][x]

    "Let f is a polynomial from k[x,y], where k is a field. Suppose that x appears in f with positive degree. We view f as an element of k(y)[x], that is polynomial in x whose coefficients are rational functions of y." I think I am missing something...why do we need rational functions here? can't...
  3. L

    Factorization of Polynomials over a field

    I don't understand how to factor a polynomial over Z3 [x], Z7 [x], and Z11 [x] I need to factor the polynomail x3 - 23x2 - 97x + 291 PLEASE HELP!
  4. H

    Limits of Polynomials: x3-6x2+11x - 6 & x3-4x2+19x + 14

    Homework Statement lim x-> -1 x3-6x2+11x - 6 x3-4x2+19x + 14 Homework Equations The Attempt at a Solution
  5. E

    Lagrange Polynomials Questions

    Lagrange Polynomals are defined by: lj(t)= (t-a0) ...(t-aj-1)(t-aj+1)...(t-an) / (aj-a0)...(aj-aj-1)(aj-aj+1)...(aj-an) A) compute the lagrange polynomials associated with a0=1, a1=2, a2=3. Evaluate lj(ai). B) prove that (l0, l1, ... ln) form a basis for R[t] less than or equal to n...
  6. H

    Integral involving Hermite polynomials

    Hello! Is there any way of calculating the integral of H_n(x) * H_m(x) * exp(-c^2 x^2) with x going from -infinity to +infinity and c differs from unity. I'm aware that c=1 is trivial case of orthogonality but I'm really having a problem with the general case. (I should say that this isn't a...
  7. M

    Equation about polynomials that implies polynomials are zero

    Hi everyone, I have to demonsrate that for every real polynomial, P Q and R, I have : P²=X(Q²+R²) ==imply==> P=Q=R=0 Using degrees, we can easily demonsrate the above. However, I'm looking for another way, without using that.
  8. Mentallic

    Can the Roots of Polynomials Be Proven?

    It is basic knowledge that if a polynomial P(x) of nth degree has a root or zero at P(a), then (x-a) is a factor of the polynomial. However, can this be proved? or is this more of a definition of roots of polynomials?
  9. V

    How is the multiplication map used to compute the resultant of n polynomials?

    Hello, I'm having trouble understanding the computation involved in the resultants from algebraic geometry. I've checked out the textbook Using Algebraic Geometry by Cox et al, so I'll be using terminologies from that text. For systems of 2 polynomials I can compute the resultant the "long...
  10. T

    GCD of Polynomials: Is 1 Always the Solution?

    Would the GCD of x^2+x+c and (x-a)^2+(x-a)+c always be 1?
  11. D

    Spherical Harmonics, Hermite and Other Special Polynomials etc.

    Whats a good book to learn about these topics?
  12. B

    Some confusion of the concepts about polynomials

    I'm not reading a text in English, so I should clarify some notation first: P[X], is the set of all polynomials. P[[X]], is the set of formal power series, which may infinitely many nonzero coefficients. I'm asked to prove that X is the only prime(?correct term?) element in P[[X]]. By saying...
  13. B

    Normalisation of associated Laguerre polynomials

    I'm looking right now at what purports to be the normalisation condition for the associated Laguerre polynomials: \int_0^\infty e^{-x}x^k L_n^k(x)L_m^k(x)dx=\frac{(n+k)!}{n!}\delta_{mn} However, in the context of Schroedinger's equation in spherical coordinates, I find that my...
  14. X

    Find all the monic irreducible polynomials

    Homework Statement Find all the monic irreducible polynomials of degree \leq 3 in \mathbb{F}_2[x], and the same in \mathbb{F}_3[x].Homework Equations n/aThe Attempt at a Solution n/a
  15. N

    Understanding Polynomials: Explaining Degree and Real Roots

    Hello again my favourite helpers (or should I say, saviours!) :tongue: Long time no speak, but I am in more need of an explanation than an answer. In this mathematics textbook I have, it gives an explanation under the heading Polynomials in general. It goes as follows: "If f(x)=...
  16. M

    Find lowest common denominator of these polynomials

    Homework Statement 3/x^2+2x - 2/x^2+x-2 + 4/x^2(x-1) Find the lowest common denominator and solve. Homework Equations The Attempt at a Solution I factored x(x+2) - (x-1)(x+2) + x^2(x-1) It looks like (x+2), (x-1) are common but what to do with the x & x^2 left over? Thank...
  17. P

    Find a basis for polynomials with complex coefficients

    Homework Statement Let V=Pn(C) (polynomials of nth degree with complex coefficients), where n >=1. Find a basis for W=V s.t. every f(x) belonging to the basis satisfies f(0)f(1)= -1. (Demonstrate that the set you find is linearly independent and spans W.) Homework Equations For...
  18. L

    Symmetry of Odd-Degree Polynomials: Conditions for Symmetry in the Plane

    Hi! Brief question: I wonder which conditions should a polynomial function of odd degree fulfill in order to be symmetric to some point in the plane. Are there such conditions?
  19. S

    Stability of Polynomials: Hurwitz-Routh & Nyquist Locus Curve

    1) As far as i think i understand, stability of a polynomial means; the polynomial correspond to a (its inverse laplace transform) differential equation, and the differential equations solution is dependant on the coefficiants of this polynomial? if the polynomial is unstable the solution of...
  20. L

    What is the depressed polynomial for f(x) with an obvious root?

    how would i write the depressed polynomial for the obvious root of f(x) = x5 - .5x4 - 5.5x3 - 3.5x2 - 6.5x - 3
  21. A

    Decompositoin of f(x) in Legendre polynomials

    Hi, In Wikipedia it's stated that "... Legendre polynomials are useful in expanding functions like \frac{1}{\sqrt{1 + \eta^{2} - 2\eta x}} = \sum_{k=0}^{\infty} \eta^{k} P_{k}(x) ..." Unfortunately, I am failing to see how this can be true. Is there a way of showing this...
  22. A

    Modelling Logistics, Polynomials using data.

    Okay, I'm doing my yr 11 Mathematics assignemnt and I need help. How do you find a Logistic or Polynomial function/equation using data? (in steps please!) I could just give some example data as... an example, but I think I'll just throw 1 small section of my assignment, use the data or...
  23. P

    Proof Normalization Hermite Polynomials

    Can anyone PROOVE how to find out the normalisation of hermite polynomial?
  24. H

    Eigenfunction expansion in Legendre polynomials

    Homework Statement How to use eigenfunction expansion in Legendre polynomials to find the bounded solution of (1-x^2)f'' - 2xf' + f = 6 - x - 15x^2 on -1<= x <= 1 Homework Equations eigenfunction expansion The Attempt at a Solution [r(x)y']' + [ q(x) + λ p(x) ]...
  25. T

    Finding the Number of Primitive Polynomials in Finite Fields

    Can anyone tell me how to find the exact number of primitive polynomials of degree n over a finite field F_q? I believe the answer is φ(q^n-1)/n, but I cannot find a proof of this. Thanx in advance.
  26. D

    Hermite Polynomials: Spans All Polynomials f from R to R?

    Since the Hermite Polynomials are orthogonal, could one state that they span all polynomials f where f : R \rightarrow R? This would be EXTREMELY useful for the harmonic oscillator potential in quantum mechanics...
  27. J

    Can all polynomials in Z modulus 9 be written as the product of two polynomials?

    I'm trying to figure out how to prove that every polynomial in \mathbb{Z}_9 can be written as the product of two polynomials of positive degree (except for the constant polynomials [3] and [6]). This basically is just showing that the only possible irreducible polynomials in \mathbb{Z}_9 are the...
  28. J

    Could the polynomials p(x) exist?

    Are there any ploynomials p(x) such that p(x)^2 -1 = p(x^2+1) for all x? To cut it short: With CAS software I have verified that there are no solutions except p(x) = 1.618... and p(x) = -0.618... (constants) up to order 53 or so, but I have to prove this (or find the other solutions)...
  29. E

    Relatively Prime Polynomials in Extension Fields

    Homework Statement Recall that two polynomials f(x) and g(x) from F[x] are said to be relatively prime if there is no polynomial of positive degree in F[x] that divides both f(x) and g(x). Show that if f(x) and g(x) are relatively prime in F[x], they are relatively prime in K[x], where K is an...
  30. C

    Connection between polynomials and groups

    Hey Everyone, I'm reading a paper by Claude LeBrun about exotic smoothness on manifolds and he is talking about a connection between polynomials and groups that I am not familiar with (or at least I think that's what he's talking about). He's creating a line bundle (which happens to be...
  31. Q

    MATLAB Evaluating polynomials in Matlab

    I defined x as a syms, then work with polynomials involving x. But then I can't find an evaluation command for these polynomials at some x value. Anyone knows how I can get it to work? Thank you in advance. There's another system where you can define polynomials just like matrices, e.g. x^2+1...
  32. MathematicalPhysicist

    How can I show the expansion of Hermite Polynomials using exponential functions?

    I need to show that: \sum_{n=0}^{\infty}\frac{H_n(x)}{n!}y^n=e^{-y^2+2xy} where H_n(x) is hermite polynomial. Now I tried the next expansion: e^{-y^2}e^{2xy}=\sum_{n=0}^{\infty}\frac{(-y)^{2n}}{n!}\cdot \sum_{k=0}^{\infty}\frac{(2xy)^k}{k!} after some simple algebraic rearrangemnets i...
  33. M

    Is it 2nd order polynomials, or 2nd order quadratics?

    I thought this was rather odd, and wanted to just show it to see what you all thought of it. Well, also, if anyone knows what I should read to exactly understand what I did. 1: Let's define each answer of a polynomial such as (ax + b)(cx + d) as x1 = (-b/a), x2 = (-d/c). 2: The...
  34. Z

    Abstract algebra: irreducible polynomials

    Homework Statement Prove that f(x)=x^3-7x+11 is irreducible over Q Homework Equations The Attempt at a Solution I've tried using the eisenstein criterion for the polynomial. It doesn't work as it is written so I created a new polynomial...
  35. P

    Rational Polynomials over a Field

    Homework Statement Suppose there are two polynomials over a field, f and g, and that gcd(f,g)=1. Consider the rational functions a(x)/f(x) and b(x)/g(x), where deg(a)<deg(f) and deg(b)<deg(g). Show that if a(x)/f(x)=b(x)/g(x) is only true if a(x)=b(x)=0. Homework Equations None The Attempt at...
  36. J

    Prove: Non-constant Polynomials as Products of Irreducibles

    Homework Statement For polynomials over a field F, prove that every non constant polynomial can be expressed as a product of irreducible polynomial. Homework Equations No relevant equations. The Attempt at a Solution Well a hint the teacher gave me was that the degree of the...
  37. I

    Are Hermite Polynomials Always Cubic When Used for Interpolation?

    are hermite interpolationg polynomials necessarily cubic even when used to interpolate between two points? this page would have me believe so in calling it a "clamped cubic" : http://math.fullerton.edu/mathews/n2003/HermitePolyMod.html
  38. C

    How Do You Calculate a Degree 3 Taylor Polynomial for e^x?

    Hello, I'm having trouble with this question and was wondering if someone could give me hints or suggestions on how to solve it. Any help would be greatly appreciated thankyou! :) Find the Taylor polynomial of degree 3 of f (x) = e^x about x = 0 and hence find an approximate value for...
  39. MathematicalPhysicist

    Legendre Polynomials: Expansion and Series Generation

    I need to expand the next function in lengendre polynomial series: f(x)=1 x in (0,1] f(x)=0 x=0 f(x)=-1 x in [-1,0). Now here's what I did: the legendre series is given by the next generating function: g(x,t)=(1-2tx+t^2)^(-1/2)=\sum_{0}^{\infty}P_n(x)t^n where P_n are legendre...
  40. C

    Applications of Taylor polynomials to Planck's Law

    Due to too much wrong information being posted on my behalf, I am resubmitting a cleaned up version of my last post. I have 2 hours to get this problem done :(. Essentially, I don't know at all how to find the Taylor Polynomial for g(x) = \frac{1}{x^5 ( e^{b/x} -1)} [/URL]
  41. C

    Applications of Taylor polynomials

    Homework Statement f(\lambda) = \frac{8\pi hc\lambda ^{-5}}{e^{hc/\lambda kT}-1} Is Planck's Law where h\ =\ Planck's\ constant\ =\ 6.62606876(52)\ \times\ 10^{-34}\ J\ s; c\ =\ speed\ of\ light\ =\ 2.99792458\ \times\ 10^{8}\ m\ s^{-1}; and\ Boltzmann's\ constant\ =\ k\ =\ 1.3806503(24)\...
  42. O

    Algebta (factoring polynomials)

    [SOLVED] Algebta (factoring polynomials) Homework Statement x^2 * [ (1/2) * (1-x^2)^(-1/2) * (-2x)] + (1 - x^2)^(1/2) * (2x) factor this down. Homework Equations n/a The Attempt at a Solution -x^3*(1- x^2)^(-1/2) + 2x(1 - x^2)^(1/2) I understand how you get to this part...
  43. H

    Ignoring Fundamental Theorem of Polynomials (a^n + b^n)

    Homework Statement Okay, basically why does (a_{}n + b_{}n) ignore the Fundamental Theory of Polynomials? Homework Equations ... I could post them here, but basically when n is odd (a_{n} + b_{n}) = a series that looks like this: (a+b) (a_{n-1} b_{0} - a_{n-2}b +a_{n-3}b_{2} + ... +...
  44. M

    Irreducible polynomials

    (1): Find all irreducible polynomials of the form x^2 + ax +b , where a,b belong to the field \mathbb{F}_3 with 3 elements. Show explicitly that \mathbb{F}_3(x)/(x^2 + x + 2) is a field by computing its multiplicative monoid. Identify [\mathbb{F}_3(x)/(x^2 + x + 2)]* as an abstract group...
  45. D

    Decidability of Polynomials with Integer Coefficients and at Least 1 Real Root?

    Homework Statement Show that the set of polynomials with integer coefficients with at least 1 real root is decidable. The Attempt at a Solution The question did not ask for specific language, just an intuitive finite algorithm will do.
  46. W

    Can matrices be used to solve systems with different degree variables?

    Hello! First post here. My question is, is it possible to use a matrix to solve a system where you have the same variable, but a different degree. i.e. 2x^2 + 2x + y = 2 -3x^2 - 6x + 2y = 4 4x^2 + 6x - 3y = 6 Now, I know this is possible other ways, but seeing as each of those would...
  47. E

    Product of monic polynomials

    Homework Statement Show that x^{p^n}-x is the product of all the monic irreducible polynomials in \mathbb{Z}_p[x] of a degree d dividing n.Homework Equations The Attempt at a Solution So, I want to prove that the zeros of all such monic polynomials are also zeros of x^{p^n}-x and vice versa. I...
  48. D

    Limits of polynomials at infinity

    Help, I am infinitly confused :) When solving the limit for this type and factoring the largest power of a variable in the polynomial in order to make its coefficient become a limit multiplied by another limit of Infinity I get lost. I just do not understand how (Infinity)(5 + Infinity +...
  49. Repetit

    Product of Legendre Polynomials

    Hey! Could someone please help me find out how to express the product of two Legendre polynomials in terms of a sum of Legendre polynomials. I believe I have to use the recursion formula (l+1)P_{l+1}(x)-(2l+1) x P_l(x) + l P_{l-1}(x)=0 but I am not sure how to do this. What is basically...
  50. M

    Root finding methods for Polynomials.

    Hi , is there a method to obtain the roots of Polynomials: P(x)=a_{0}+a_{1}x+a{2}x^{2}+...+a_{n}x^{n} i know there are , but my problem is this if we knew that are complex roots of the form z=a+ib , would be a method to obtain the complex root with BIGGER and SMALLER real part ?? , i mean...
Back
Top