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. caffeinemachine

    MHB Gcd of polynomials is 1. There is an nxn matrix with determinant....

    Let $F$ be any field. Let $p_1,\ldots, p_n\in F[x]$. Assume that $\gcd(p_1,\ldots,p_n)=1$. Show that there is an $n\times n$ matrix over $F[x]$ of determinant $1$ whose first row is $p_1,\ldots,p_n$. When $n=2$ this is easy since then there exist $a_1,a_2\in F[x]$ such that $p_1a_1+p_2a_2=1$...
  2. T

    Is there a way to transform a polynomial into a vector?

    Ok for the longest while I've been at war with polynomials and isomorphisms in linear algebra, for the death of me I always have a brain freeze when dealing with them. With that said here is my question: Is this pair of vector spaces isomorphic? If so, find an isomorphism T: V-->W. V= R4 ...
  3. M

    Linear Algebra: Linear indepency of a set of Polynomials

    Homework Statement Let {p, q} be linearly independent polynomials. Show that {p, q, pq} is linearly independent if and only if deg p ≥ 1 and deg q ≥ 1. Homework Equations λ1p + λ2q = 0 ⇔ λ1 = λ2 = 0 The Attempt at a Solution λ1p + λ2q + λ3pq = 0 I know if λ3 = 0, then the coefficients of...
  4. C

    Solve Infinite Primes with Quadratic Polynomials

    My teacher said that, No one knows of any quadratic polynomial that produces an infinite amount of primes. I was thinking could we use a polynomial like x^2+1 and then do a trick similar to Euclids proof of the infinite amount of primes and assume their are only finitely many of them...
  5. Fernando Revilla

    MHB Solving an Unsolved Math Problem: Ring A & Polynomials

    I quote an unsolved problem posted in another forum on December 5th, 2012.
  6. D

    Multilinear Functions and Polynomials

    A function f : \mathbf{R}^n\rightarrow\mathbf{R} is multilinear if it's linear in every variable. Is there a multilinear function that's not a multilinear polynomial? Given a function defined on the n dimensional hypercube, values of which are 0 or 1, there is a unique multilinear extension...
  7. H

    Integral involving product of derivatives of Legendre polynomials

    Anyone how to evaluate this integral? \int_{-1}^{1} (1-x^2) P_{n}^{'} P_m^{'} dx , where the primes represent derivative with respect to x ? I tried using different recurrence relations for derivatives of the Legendre polynomial, but it didn't get me anywhere...
  8. I

    Irreducible polynomials over different fields

    Problem: Let f be monic irreducible polynomial over a field F, k be monic irreducible polynomial over a field K, deg f = deg k. Let u be common root of f and k. Prove (or disprove by counterexample), that f=k over field (F intersection K), i.e. polynomials f and k are identical. Proof would be...
  9. A

    Proving ƩP(n)/n! x^n = P(x)e^x for Polynomials

    You have: Ʃ(n+1)/n! x^n = (1+x)e^x Is it in general true with a polynomium that: ƩP(n)/n! x^n = P(x)e^x ?
  10. L

    Legendre polynomials and binomial series

    Homework Statement Where P_n(x) is the nth legendre polynomial, find f(n) such that \int_{0}^{1} P_n(x)dx = f(n) {1/2 \choose k} + g(n)Homework Equations Legendre generating function: (1 - 2xh - h^2)^{-1/2} = \sum_{n = 0}^{\infty} P_n(x)h^n The Attempt at a Solution I'm not sure if that...
  11. E

    Roots of linear sum of Fibonacci polynomials

    For what complex numbers, x, is Gn = fn-1(x) - 2fn(x) + fn+1(x) = 0 where the terms are consecutive Fibonacci polynomials? Here's what I know: 1) Each individual polynomial, fm, has roots x=2icos(kπ/m), k=1,...,m-1. 2) The problem can be rewritten recursively as Gn+2 = xGn+1 +...
  12. L

    Calculators TI-89 won't factor or expand polynomials

    When I use the factor or expand functions on my TI-89 it outputs a matrix with values that are seemingly coming from no where. For example, if I ask my calculator to expand (3-x)^2 it gives the matrix [45, 12; 12,13]. Why is it doing this? How do I fix it?
  13. B

    Polynomials of odd degree can have no critical point.

    Are these assertions true? I am referring to polynomials with real coefficients. 1. There exists of polynomial of any even degree such that it has no real roots. 2. Polynomials of odd degree have atleast one real root which implies that polynomial of even degree has atleast one critical...
  14. U

    Evaluating high-degree polynomials

    Hi all. I am trying to evaluate high-degree Chebyshev polynomials of the first kind. It is well known that for each Chebyshev polynomial T_n, if -1\le x\le1 then -1\le T_n(x)\le 1 However, when I try to evaluate a Chebyshev polynomial of a high degree, such as T_{60}, MATLAB gives...
  15. M

    Approximating accuracy of Taylor polynomials

    Homework Statement Determine the values of x for which the function can be replaced by the Taylor polynomial if the error cannot exceed 0.001. e^x ≈ 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} For x < 0 Homework Equations Taylor's Theorem to approximate a remainder: |R(x)| =...
  16. B

    Factoring polynomials in general

    Homework Statement I was wondering how people intuitively see how to decompose functions? For example: x^2 + 5x - 14, how do you solve that to be (x+7)(x-2) without a calculator? Do you use a specific method or do you just sit for a while trying and failing? The question is a...
  17. Math Amateur

    MHB Factorization of Polynomials - Irreducibles - Anderson and Feil

    I am reading Anderson and Feil - A First Course in Abstract Algebra. On page 56 (see attached) ANderson and Feil show that the polynomial f = x^2 + 2 is irreducible in \mathbb{Q} [x] After this they challenge the reader with the following exercise: Show that x^4 + 2 is irreducible in...
  18. H

    Find Degree 3 Taylor Polynomial Approximation of 5ln(sec(x))

    Homework Statement Find the degree 3 Taylor polynomial approximation to the function f(x)=5ln(sec(x)) about x=0. Homework Equations the taylor polynomial equation The Attempt at a Solution Here are my derivatives f(x)=5ln(secx) f'(x)=5tanx f''(x)5sec^2(x)...
  19. H

    Solve T4(x) for Taylor Polynomials of f(x)=arctan(11x)

    Homework Statement Find T4(x), the Taylor polynomial of degree 4 of the function f(x)=arctan(11x) about x=0. (You need to enter a function.) Homework Equations The taylor polynomial equation Tn(x)= f(x)+(fn(x)(x-a)^n)/n!... The Attempt at a Solution When I take every...
  20. D

    Eigenvalues and characteristic polynomials

    Hello guise. I am familiar to a method of diagonalizing an nxn-matrix which fulfills the following condition: the sum of the dimensions of the eigenspaces is equal to n. As to the algorithm itself, it says: 1. Find the characteristic polynomial. 2. Find the roots of the characteristic...
  21. A

    Intuition for Quotient Ring in Polynomials

    I just had a discussion with someone who said he thought about quotient rings of polynomials as simply adjoining an element that is a root of the polynomial defining the ideal. For example, consider a field, F, and a polynomial, x-a, in F[x]. If we let (x-a) denote the ideal generated by x-a...
  22. C

    How Do You Calculate Error Bounds for Maclaurin Polynomial Approximations?

    Now I'm trying to get my head around this question. I just know they're going to give us a large degree question like this in the exam... Let's say: I = ∫[e^(x^2)]dx with nodes being x=0 to x=0.5 The 5th degree polynomial is 1 + x^2 + (1/2)(x^4) So my queries are: How would I go about...
  23. A

    A question about minimal polynomials

    Let A \in M_n(F) and v \in F^n. Also...[g \in F[x] : g(A)(v)=0] = Ann_A (v) is an ideal in F[x], called the annihilator of v with respect to A. We know that g \in Ann_A(v) if and only if f|g in F[x]. Let V = Span(v, Av, A^2v, ... , A^{k-1}v).. V is the smallest A-invariant subspace containing...
  24. F

    Factoring Polynomial z^4-4z^3+6z^2-4z-15 =0

    z^4-4z^3+6z^2-4z-15 =0 How can i factor this polynomial in order to find the solutions?? I tried with the ruffini' rule. and i reached the following equation [(z+1)(-z^3-5z^2+11z-15)] =0 now how can i factor (-z^3-5z^2+11z-15) ? i tried it, but i can not solve it... :/
  25. C

    Recurrence relations for Associated Legendre Polynomials

    Homework Statement I'm working on problem 6.11 in Bransden and Joachain's QM. I have to prove 4 different recurrence relations for the associate legendre polynomials. I have managed to do the first two, but can't get anywhere for the last 2 Homework Equations Generating Function: T(\omega...
  26. Fantini

    MHB Solving Congruences with Polynomials: A Prime Challenge

    I'm having trouble with the following question: Construct a polynomial $q(x) \neq 0$ with integer coefficients which has no rational roots but is such that for any prime $p$ we can solve the congruence $q(x) \equiv 0 \mod p$ in the integers. Any hints on how to even start the problem will be...
  27. R

    Proof Involving Matrix Polynomials and Matrix Multiplication

    Homework Statement Let A be an nxn matrix, and C be an mxm matrix, and suppose AB = BC. (a) Prove the following by induction: For every n∈ℕ, (A^n)B = B(C^n). What property of matrix multiplication do you need to prove this? Homework Equations The four basic properties of matrix...
  28. Z

    Degree-Raising Formulas for Bernstein Polynomials

    Two part question... Homework Statement Question 1: Let v = (a, b, c)T be a column vector which represents a coordinate vector of a polynomial in P2 with respect to the Bernstein basis. Find the 4 × 3 matrix which transforms v to the standard basis of P3. (Hint: First transform v to...
  29. T

    Finding a basis for a set of polynomials and functions

    Find a basis for and the dimension of the subspaces defined for each of the following sets of conditions: {p \in P3(R) | p(2) = p(-1) = 0 } { f\inSpan{ex, e2x, e3x} | f(0) = f'(0) = 0} Attempt: Having trouble getting started... So I think my issue is interpreting what those sets...
  30. R

    Proving Irreducibility of Polynomials over the Integers

    If i have to show a polynomial x^2+1 is irreduceable over the integers, is it enough to show that X^2 + 1 can only be factored into (x-i)(x+i), therefore has no roots in the integers, and is subsequently irreduceable?
  31. D

    Showing orthogonal polynomials are unique

    Homework Statement We are given that a set of polynomials on [-1,1] have the following properties and have to show they are unique by induction. I have a way to show they are unique, but is not what he is looking for. I honestly have never seen it presented this way. P_n(x) = Ʃa_in*x^i All...
  32. D

    Why Does the Integral of Legendre Polynomials Yield a Kronecker Delta?

    I am doing a Laplace's equation in spherical coordinates and have come to a part of the problem that has the integral... ∫ P(sub L)*(x) * P(sub L')*(x) dx (-1<x<1) The answer to this integral is given by a Kronecker delta function (δ)... = 0 if L...
  33. matqkks

    Inner product of polynomials

    In inner product spaces of polynomials, what is the point of finding the angle and distance between two polynomials? How does the distance and angle relate back to the polynomial?
  34. matqkks

    MHB Finding Angle & Distance Between Polynomials: Exploring Inner Product Spaces

    In inner product spaces of polynomials, what is the point of finding the angle and distance between two polynomials? How does the distance and angle relate back to the polynomial?
  35. S

    Re-Expressing a Quotient of Polynomials

    This is a basic question but I don't think I've ever seen anything like it before. If Q(x) \ = \ b_0(x \ - \ a_1)(x \ - \ a_2)\cdot \ . \ . \ . \ \cdot (x \ - \ a_s) then \frac{P(x)}{Q(x)} \ = \ R(x) \ + \ \sum_{i=1}^s \frac{P(a_i)}{(x \ - \ a_i)Q'(a_i)}I just don't understand where the P(aᵢ)...
  36. J

    MHB No. of Polynomials of Degree 5 Divisible By x2-x+1

    Find the number of polynomials of degree $5$ with distinct coefficients from the set $\left\{1, 2, 3, 4, 5, 6, 7, 8\right\}$ that are divisible by $x^2 - x + 1$
  37. J

    Homomorphisms of Polynomials Over Integral Domains

    Let A be an integral domain. If c ε A, let h: A[x] → A[x] be defined by h(a(x))=a(cx). Prove that h is an automorphism iff c is invertible. This one really had me stumped. I have a general idea of what the function is doing. Now, assuming that h is an automorphism, we want to show that...
  38. M

    How Do You Find the Maclaurin Polynomials for cos(πx)?

    1. Find the Maclaurin polynomials of order n = 0, 1, 2, 3, and 4, and then find the nth MacLaurin polynomials for the function in sigma notation. cos(∏x) 2. Here is what I did: p0x = cos (0∏) = 1 p1x = cos(0∏) - ∏sin(0∏)x = 1 p2x = cos(0∏) - ∏sin(0∏)x -\frac{∏2(cos∏x)(x2)}{2!}(...
  39. F

    Complete Factoring of x^2 - 4x + 4 - 4y^2: Homework Solution & Explanation

    Homework Statement Factor the polynomial x^2 - 4x + 4 -4y^2 completely. Homework Equations The Attempt at a Solution Rearranging, I get x^2 - 4y^2 - 4x + 4 Then, I know that it is equal to (x -2y)(x+2y) - 4(x-1) and that is my final answer. but, my teacher only considered the...
  40. J

    Identify power series with coeffs. that are palindromic polynomials of a param.

    Can someone help me to identify the type of power series for which the coefficients are palindromic polynomials of a parameter? More specifically, for a particular function f(x;a) with x, a, and f() in ℝ1, a > 0, and an exponential power series representation F0 + F1x/1! + F2x2/2! +...
  41. A

    Is it always possible to find the G.C.D of two polynomials?

    Suppose that we've been given two polynomials and we want to find their Greatest Common Divisor. For integers, we have the Euclidean algorithm which gives us the G.C.D of the two given integers. Could we generalize the Euclidean algorithm to be used to find the G.C.D of any two given polynomials...
  42. R

    What is the exact form of the zeros of Hermite polynomials?

    So I was working on eigenvalues of tridiagonal matrices, interestingly I get hermite polynomials as the solution. Is it possible to get an exact form for the zeros of hermite polynomials?
  43. S

    Farrel Polynomials: Is It the Same as "Feral"?

    Is there such a thing as a "feral polynomial" ? Saw it mentioned on an Internet forum where someone claimed to be studying "feral polynomials". Closes I could find were "Farrel polynomials" .
  44. X

    Solving Polynomials of Increasing Degree

    x^2+2\\\\ \frac{2}{3} x^3 + \frac{13}{3} x\\\\ \frac{1}{3} x^4 + \frac{14}{3} x^2 + 2\\ \\ \frac{2}{15} x^5 + \frac{10}{3} x^3 + \frac{83}{15} x\\ \\ \frac{2}{45} x^6 + \frac{16}{9} x^4 + \frac{323}{45} x^2 + 2\\\\ \dots ?
  45. C

    Is there such a thing as an uncountable polynomial?

    Is it possible to have a polynomial with an uncountable number of turns? Like we could think of cos(x) as having a countable number of turns. could I have y=x^{\aleph_1} My question may not even make sense.
  46. G

    Roots of polynomials as nonlinear systems of equations

    Ok, to start off I have been examining the structure of polynomials. For instance, consider the general polynomial P(x)=\sum^{n}_{k=0}a_{k}x^{k} (1) Given some polynomial, the coefficients are known. Without the loss of generality...
  47. J

    Connection between roots of polynomials of degree n

    Homework Statement The two polynomial eqns have the same coefficients, if switched order: a_0 x_n+ a_1 x_n-1 + a_2 x_n-2 + … + a_n-2 x_2 + a_n-1 x + a_n = 0 …….(1) a_n x_n+ a_n-1 x_n-1 + a_n-2 x_n-2 + … + a_2 x_2 + a_1 x + a_0 = 0 …….(2) what is the connection between the roots of...
  48. AnTiFreeze3

    Multiplying Polynomials to Find Values of 'k'

    Homework Statement Find the value or values of k that make the equation true. Homework Equations Starting Equation: (2x + k)(x - 2k) = 2x2 + 9x - 18 The Attempt at a Solution (2x + k)(x - 2k) = 2x2 + 9x - 18 2x2 - 3xk - 2k2 = 2x2 + 9x - 18 <- I just foiled the left side...
  49. B

    The set P of polynomials

    Homework Statement Consider the set P4 of all real polynomials if degree <= 4. 1)Prove that P4 is a subspace of the vector space of all real polynomials 2)What is the dimension of the vector space P4. Prove answer by demonstrating a basis and verifying the proposed set is really a basis...
  50. C

    Four proof that exercises about taylor polynomials

    Four "proof that" exercises about taylor polynomials Homework Statement Definition: A function f is called C^n if f n times derivable and if the n-time derivable f^(n) is continuos. If is from class C^n then its called f ε C^n Exercise 1) Be f ε C^n in the interval [a,x]. Be P a polynomial...
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