What is Polynomial: Definition and 1000 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. T

    Polynomial that satisfies a differential equation

    Homework Statement I must show that t H_n satisfies a diferential equation. By diferentiating H_n(x) = -xH_(n-1)(x) - (n - 1)H_(n-2)(x) (1) and using induction on n, show that, for n >= 1, H'_n(x) = -nH_(n-1)(x) (2) I have to use (2) to express H_(n-1) and H_(n-2) in terms of derivatives...
  2. C

    Finding the eigenvalues of maps of polynomial vector spaces

    Homework Statement Let V be the vector space of all real-coefficient polynomials with degree strictly less than five. Find the eigenvalues and their geometric multiplicities for the following maps from V to V: a) G(f) = xD(f), where f is an element of V and D is the differentiation map...
  3. I

    Characteristic Polynomial of Matrix

    Homework Statement Let J be the nxn matrix all of whose entries are equal to 1. Find the minimal polynomial and characteristic polynomial of J and the eigenvalues. Well, I figure the way I'm trying to do it is more involved then other methods but this is the easiest method for me to...
  4. C

    Taylor polynomial of 1/(2+x-2y)

    Homework Statement Find the Taylor polynomial of degree 3 of \frac{1}{2+x-2y} near (2,1). Homework Equations The Attempt at a Solution I have already solved this problem by evaluating the R^2 Taylor series; I'm mostly curious about another aspect of the problem. By substituting u = x-2y, it...
  5. T

    Understanding the Recurrence Relation of H_n(x) and Its Properties

    Homework Statement The polynomials are defined as follows H_n(x) for n = 0; 1; : : : as follows: fi rst, set H_0(x) =1 and H_1(x) = -x; then, for n >= 2, H_n is defined by the recurrence H_n(x) = -xH_(n-1)(x) - (n - 1)H_(n-2)(x): (1) I have to use (1) to verify that H_2(x) = x^2 -1 and...
  6. J

    Complex Analysis: polynomial coefficients

    Homework Statement Show that if the polynomial p(z)=anzn+an-1zn-1+...+a0 is written in factored form as p(z)=an(z-z1)d1(z-z2)d2...(z-zr)dr, then (a) n=d1+d2+...+dr (b) an-1=-an(d1z1+d2z2+...+drzr) (c) a0=an(-1)rz1d1z2d2...zrdr Homework Equations Taylor form of a polynomial? p(z) = \sum...
  7. H

    What is the Derivative of the Inverse of a Polynomial?

    1. I am having trouble finding a inverse of a polynomial. http://img218.imageshack.us/i/problemqk.png/ http://img218.imageshack.us/i/problemqk.png/
  8. G

    Solve that without 4th order polynomial?

    I was trying to calculate when two ellipses of the form ((x-x0)/a)^2+((y-y0)/b)^2=1 have intersections. Most of the time I get 4th order equations. Actually I only want to know a condition IF they just touch. Any ideas on this? After playing around I had various equations. For example the...
  9. C

    Slope of a polynomial function

    Homework Statement Slope of: y=.00002715x^2-.04934171x+44.18240907 Homework Equations d/dx The Attempt at a Solution d/dx[.00002715x^2-.04934171x+44.18240907] = .0000543x-.04934171 This is the derivative (slope) of the function though it's looking for a numerical value. It is...
  10. Z

    Polynomial function - different degrees don't understand

    Homework Statement Hello. I don't understand this: Let f(x) be a polynomial function of degree k+1, then f(x) has the form ak+1xk+1 + ... + a1x + a0 Now the polynomial function has degree h(x) = f(x) - ak+1(x-a) has degree <= k How? Homework Equations The Attempt at a...
  11. S

    Minimum degree of polynomial time NP complete problem algorithm

    So no one is quite sure that P != NP, although they tend to favor that relation. But I was curious, has anyone proved a minimum degree order to any algorithm that solves NP complete problems in polynomial time? In other words, they don't know if it can be done in polynomial time, but do they...
  12. K

    What are the roots of x^(p-1) in Z_p?

    Let p be a prime number. Find all roots of x^(p-1) in Z_p I have this definition. Let f(x) be in F[x]. An element c in F is said to be a root of multiplicity m>=1 of f(x) if (x-c)^m|f(x), but (x-c)^(m+1) does not divide f(x). I'm not sure if I use this idea somehow or not.
  13. J

    Finding Roots of Complex Polynomials

    Started my first year of Electronic Engineering a few months back and I'm already struggling with the mathematics. I have been to this forum a number of times over the last year and finally decided to join just 10 minutes ago!Homework Statement Find the roots of P(x) given that two roots are...
  14. R

    Small oscillations around equilibrium point in polynomial potential

    Hi guys i am a bit confused about this problem, a particle of mass, m, moves in potential a potential u(x)=k(x4 - 7 x2 -4x) I need to find the frequency of small oscillations about the equilibrium point. I have worked out that x=2 corresponds to the equilibrium point as - dU/dx = F =...
  15. R

    Splitting field of a polynomial over a finite field

    Homework Statement Assume F is a field of size p^r, with p prime, and assume f \in F[x] is an irreducible polynomial with degree n (with both r and n positive). Show that a splitting field for f over F is F[x]/(f). Homework Equations Not sure. The Attempt at a Solution I know from...
  16. I

    Solving Polynomial Question with Gauss Method

    Hey guys, I am new to this forum and I have just found you after some long and unsuccessful research on the following question: Homework Statement The question is a combined matrix and polynomial question. First I am given the following matrix A: 2 1 1 5 4 -3 2 1 3...
  17. J

    Autonomous polynomial differential equation

    Homework Statement Is it possible to find general solution for the following 3rd degree polynomial differential equation: dx/dt=-a1*x+a2*x^2+a3*x^3 Homework Equations The Attempt at a Solution I understand that its is possible to integrate 1/(-a1*x+a2*x^2+a3*x^3), however, end equation...
  18. estro

    Calculating Maclaurin Polynomial of 3rd Order for ln(cosx)

    I have hard time to come with Maclaurin Polynomial of a given order [lets say 3] for a composite function like ln(cosx). Will appreciate help of how to approach such a problem.
  19. J

    Complex Variables Polynomial

    Find a (complex) polynomial function f of x and y that is differentiable at the origin, with df/dz = 1 at the point z=0, and differentiable at all points on the unit circle x^2 + y^2=1, but is not differentiable at any other point in the complex plane. (Bruce Palka, Page 101) I think we use...
  20. T

    Find the cubic polynomial

    Question: Find the cubic polynomial, which take the following values: y(0) = 1, y(1)=0, y(2)=1 and y(3)=10 Hence obtain y(4).
  21. Z

    Are trig functions polynomial fuctions?

    Homework Statement Is 3cos22x + cosx2 - 1 a polynomial function? Homework Equations The Attempt at a Solution
  22. N

    Network function. 6th degree polynomial to complex numbers

    Homework Statement the problem that i attached bellow is related to how you can obtain a transfer function from its squared magnituded. my question is not on the problem it self as its just a solved example from my book. what i find difficult to understand as you can see from my...
  23. Demon117

    Optimizing Polynomial Approximations for C2 Functions on Closed Intervals

    1. Suppose that f:R-->R is of class C2. Given b>0 and a positive number \epsilon, show that there is a polynomial p such that |p(x)-f(x)|<\epsilon, |p'(x)-f'(x)|<\epsilon, |p"(x)-f"(x)|<\epsilon for all x in [0,b]. The Attempt at a Solution First I choose a polynomial q...
  24. B

    Finding Characteristic Polynomial of Matrix B

    Homework Statement B = |a 1 -5 | |-2 b -8 | |2 3 c | Find the characteristic polynomial of the following matrix. Homework Equations None The Attempt at a Solution So basically I have to find the det(B-λI). No matter what I do to the matrix I can't make the...
  25. Q

    Invariants of a characteristic polynomial

    Hi: There are 3 invariants. The first one is a trace. The third one is a determinant. So they are invariants. The strange thing is the 2nd one. It is a hybrid term. Why is it also an invariant?
  26. O

    Polynomial division proof

    Spivak's "Calculus," chapter 3 - problem 7 - a Homework Statement Prove that for any polynomial function f, and any number a, there is a polynomial function g, anad a number b, such that f(x) = (x - a)*g(x) + b for all x. (The idea is simply to divide (x - a) into f(x) by long division, until...
  27. O

    Polynomial question from spivak's calculus

    Question from spivak's calculus - 3rd edition - chapter 3, question 6(a). Homework Statement If x1, ..., xn are distinct numbers, find a polynomial function fi of degree n - 1 which is 1 at xi and 0 at xj for j =/= i. Hint, the product of all (x - xi) for j =/= i, is 0 at xj if j =/= i...
  28. N

    Polynomial Rings/Fields/Division Rings

    Homework Statement Let F be a field, F[x] the ring of polynomials in one variable over F. For a \in F[x], let (a) be all the multiples of a in F[x] (note (a) is an ideal). If b \in F[x], let c(b) be the coset of b mod (a) (that is, the set of all b + qa, where q \in F[x]). F[x]/(a), then is the...
  29. X

    A is a root of order of polynomial p iff p(a)=p'(a)= =[p^(k-1)](a)=0

    a is a root of order k of the polynomial p provided that k is a natural number such that p(x)=[(x-a)^k]r(x), r is a polynomial and r(a) not equal to 0. Prove a is a root of order k of the polynomial p iff p(a)=P'(a)=...=[p^(k-1)](a)=0 and [p^(k)](a) not equal to 0. Note: [p^(k-1)](a) :=...
  30. S

    Minimal and characteristic polynomial

    Let V =Mn(k),n>1 and T:V→V defined by T(M)=Mt (transpose of M). i) Find the minimal polynomial of T. Is T diagonalisable when k = R,C,F2? ii) Suppose k = R. Find the characteristic polynomial chT . I know that T2=T(Mt))=M and that has got to help me find the minimal polynomial
  31. A

    Power Series Representation of a Function when a is a polynomial

    Power Series Representation of a Function when "r" is a polynomial Homework Statement Find a power series representation for the function and determine the radius of convergence. f(x)=\stackrel{(1+x)}{(1-x)^{2}} Homework Equations a series converges when |x|<1...
  32. K

    The Maclaurin Series of an inverse polynomial function

    Let f(x)=\frac{1}{x^2+x+1} Let f(x)=\sum_{n=0}^{\infty}c_nx^n be the Maclaurin series representation for f(x). Find the value of c_{36}-c_{37}+c_{38}. After working out the fraction, I arrived at the following, f(x)=\sum_{n=0}^{\infty}x^{3n}-\sum_{n=0}^{\infty}x^{3n+1} But I dun get how to...
  33. W

    What's the difference between polynomials and polynomial functions?

    What's the difference between polynomials (as elements of a ring of polynomials) and polynomial functions??
  34. N

    Taylor Polynomial for f(x)=ln3x

    Homework Statement Find the Taylor Polynomial of degree 3 for the function f(x) = ln3x about a = 1/3Homework Equations NoneThe Attempt at a Solution I have found up to the fourth derivative of f(x) along with the values of the derivatives at x = 1/3. At this point i get Σ{(-1)kk!fk(1/3)}, but...
  35. J

    Find polynomial when given a complex root

    Hi there, I have over the last couple of month worked my way through Algebra Demystified and College Algebra Demystified (well almost), and I just completed the chapter on polynomial functions and I’m stuck at one of the questions in the chapter review where I’m given a complex root and the...
  36. M

    Factoring a higher order polynomial

    Homework Statement x^4 + 4x^3 - x^2 + 16x - 12 I know that with some higher order polynomials you can substitute say x^4 as a = x^2 thereby making it easier to break the thing apart and find its factors. I know I am looking for 4 roots, but my little substitution method doesn't really work...
  37. M

    Is this a polynomial function?

    f(x) = (x^2)/100 + (x^3)/1000 + (x^4)/10000 + ... till the power infinity
  38. B

    Complex zeros of polynomial with no real zeros

    I have to find the complex zeros of the following polynomial: x^6+x^4+x^3+x^2+1 This evidently doesn't have any real solution so I tried to facto it with long division and by guessing I came up with: (x^2-x+1)(x^4+x^3+x^2+x+1) How can I factor the 4th degree polynomial now? Or how can I...
  39. E

    Terrifying question about polynomial in analysis

    the textbook says that: "a non-constant analytic polynomial cannot be real-valued, for then both the partial derivative with respect to x and y would be real and the cauchyriemann equation cannot be satisfied." why??there's no explanation in the book and this sentence is written as an example...
  40. M

    Derivatives and Polynomial Functions

    Homework Statement Show that there is a polynomial function f of degree n such that: 1. f('x) = 0 for precisely n-1 numbers x 2. f'(x) = 0 for no x, if n is odd 3. f'(x) = 0 for exactly one x, if n is even 4. f'(x) = 0 for exactly k numbers, if n-k is odd Homework Equations The...
  41. Telemachus

    Taylor polynomial of third degree and error estimation

    Homework Statement It seems that I'm a little bit lost about this exercise. It says: Find the taylors polynomial of third degree centered at the origin for z=\cos y \sin x. Estimate the error for: \Delta x=-0.15,\Delta y=0.2. So, I did the first part (the easy one), the taylors polynomial for...
  42. T

    Solving polynomial equation using induction

    Homework Statement If f(x)=(x+1)p(x) where f(x)=x^{2n}+2nx+2n-1, what is p(x)? Answer given: x^{2n-1}-x^{2n-2}+...-x^2+x+2n-1 Homework Equations The Attempt at a Solution I tried the long division and managed to some terms correct. Is there any other methods of finding this...
  43. QuarkCharmer

    Finding the Inverse of a Cubic Polynomial

    Homework Statement Find the inverse of the function. f(x)=2x^{3}+5Homework Equations Possibly the quadratic equation.The Attempt at a Solutionf(x)=2x^{3}+5 y=2x^{3}+5 -2x^{3}=-y+5 x^{3}= \frac{-y+5}{-2} x= \pm\sqrt[3]{\frac{-y+5}{-2}} y= \pm\sqrt[3]{\frac{-x+5}{-2}}So the solution is two...
  44. B

    Can a 4th Order Polynomial be Factored Without a Computer?

    Factor a 4th order polynomial (Solved) Homework Statement Find the roots of: x^5-1=0 Homework Equations Polynomial long division. The Attempt at a Solution x^5-1 = (x-1)(x^4+x^3+x^2+x+1) = 0 x^4+x^3+x^2+x+1 = (x^2+1)^2+x^3+x-x^2 (x^2+1)^2+x^3+x-x^2 = (x^2+1)^2+x(x^2+1)-x^2...
  45. I

    Existence of polynomial in R^2

    Here is a potentially neat problem. Let x(t),y(t) (for all t\in \mathbb{R}) be polynomials in t. Prove that for any x(t),y(t) there exists a non-zero polynomial f(x,y) in 2 variables such that f(x(t),y(t))=0 for all t. The strategy is to show that for n sufficiently large, the polynomials...
  46. L

    Can absolute value functions be considered polynomial functions?

    Are lx3l or lxl3 polynomials? If not, what would be a good example of a cubic polynomial function (R \rightarrow R) that doesn't cover all real numbers in its codomain?
  47. S

    Big Oh: showing that a polynomial grows faster than any log function.

    Big Oh: showing that a polynomial "grows faster" than any log function. Homework Statement is n^0.01 big oh of (log(n))^10 ? ***Bear in mind, the log is a binary log. Homework Equations the formal statement would be... does n^0.01 < c(log(n))^10 hold for all n greater than a...
  48. G

    Proving that data follows a polynomial function

    I can prove that data follows a curve of the form y=Ax^n and y=Ae^x by using log log and natural log transformations. I have some data that I believe is more complex, something of the form y=anx^n+an-1x^n-1+...+a1x+a0, in other words a polynomial function. Is there any way I can prove that it...
  49. M

    Very strange about the zero degree polynomial

    Hi everybody f(x) = aX0 is the form of any constant polynomial... right?? eg: f(x) = 3 is actually f(x) =3X0 where X belongs to R... ok?? since 00 is an unspecified quantity.. therefore on graphing a constant.. it should exists a hole on y-axis... and the y-intercept should not satisfy...
  50. F

    What is a Basis for a Polynomial Subspace with Specific Roots?

    Homework Statement Let P_4(\mathbb{R}) be the vector space of real polynomials of degree less than or equal to 4. Show that {{f \in P_4(\mathbb{R}):f(0)=f(1)=0}} defines a subspace of V, and find a basis for this subspace. The Attempt at a Solution Since P_4(\mathbb{R}) is...
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