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. F

    Linear Independence: Polynomial Example

    Hello everyone. I was going through my Linear Algebra Done Right textbook that threw me off. I hope this forum is appropriate for my inquiry; while there is no problem I'm trying to solve here, I don't know whether just asking for clarification would belong to the homework forum instead. If...
  2. S

    Help Needed: Calculating a Large Power of a Polynomial Modulo Another

    Hi all, I've been set some holiday work by my study director which is meant to be teaching us all about algorithms and a few other mathematical bits and bobs - unfortunately I've come unstuck on one of the bobs, and was hoping for some help! I've asked for help elsewhere but was given very...
  3. S

    Holomorphic function reduces to a polynomial

    Homework Statement Let f: C -> C be a holomorphic function such that there is a constant R such that |z| > R implies |f(z)| > R. Show that f is a polynomial. Homework Equations Not sure, I pulled this randomly from a complex analysis qualifying exam. The Attempt at a Solution So...
  4. J

    Factoring cubic polynomial help

    Homework Statement This is probably an easy question, but using the rational zero theorem I have not found any roots for this cubic polynomial. Factor the Following 6x^3-37x^2-8x+12 Homework Equations The Attempt at a Solution I have used all my knowledge of factoring and...
  5. C

    Finding a Polynomial with Nonnegative Coefficients

    Homework Statement find a polynomial P(x) which has nonnegative coefficients. If P(1)=1 and P(5)= 426, then wast is p(3) Homework Equations P(1)= 6 P(5)= 426 P(3)= xThe Attempt at a Solution I have tried to use guess and check. I can't find a way to solve algebraically.
  6. Z

    Polynomial Basis and Linear Transformation

    Homework Statement Let X be the vector space of polynomial of order less than or equal to M a) Show that the set B={1,x,...,x^M} is a basis vector b) Consider the mapping T from X to X defined as: f(x)= Tg(x) = d/dx g(x) i) Show T is linear ii) derive a matrix...
  7. W

    Integration of a natural log and polynomial

    Homework Statement Evaluate the integral when x > 0: indefinite integral of ln(x2+19x+84)dxHomework Equations I know I need to use some form of integration by parts: integral of u*dv=uv-(integral of(du*v)) The Attempt at a Solution I began by making u=ln(x2+19x+84) and dv=dx. Thus, (after...
  8. R

    Using interpolants to solve a polynomial.

    Homework Statement Show that a root of the equation x3 - 3x - 5 =0 lies in the interval [2,3], and then find the root using linear interpolation correct to one decimal place. Homework Equations n/a The Attempt at a Solution This is my first ever time using interpolants ( well at...
  9. Z

    Polynomial Mapping: Linear Algebra Basics for Beginners

    Hi I'm a new student and don't have any basics for linear algebra. thanks Homework Statement Q1. Let X be the vector space of polynomials of order less than or equal to M. a) show that the set B = {1,x,...x^M} b) consider the mapping T from X to X as defined as : f(x)=T...
  10. Somefantastik

    Basis functions for polynomial

    Homework Statement For I = [a,b], define: P3(I) = {v: v is a polynomial of degree ≤ 3 on I, i.e., v has the form v(x) = a3x3 + a2x2 + a1x + a0}. How to show v is uniquely determined by v(a), v'(a), v(b), v'(b). Homework Equations The Attempt at a Solution I'm not exactly sure...
  11. D

    Irreducible Polynomial (or not?)

    The polynomial is x^n + A1x^(n-1) + A2x^(n-2) + ... + A2x^2 + A1x + 1. Where An (integer) is not zero for all n and n is even. For example: x²+x+1; x^4+2x^3+3x^2+2x+1. I'm looking for a method to say if that kind of polynomial is irreducible over racionals... Or when it is. Thx!
  12. S

    Inverse of polynomial in dy^2 + ey + f form

    Hello, I'm trying to find the inverse polynomial of y = ax^2 + bx + c in the form of x = dy^2 + ey + f. I'm able to approximate this using Excel, but would prefer a more elegant solution. Any suggestions? Steve
  13. T

    Factoring a Polynomial Equation: Olympiad Question

    Homework Statement Factor the equation (without complex numbers) a^{10} + a^{5} + 1 This is a olympiad question The Attempt at a Solution I substituted a^{5} = x getting a quadratic eqation. But when I factored the quadratic equation I get complex roots and this is against the question...
  14. M

    Prove that f(x) is a polynomial

    Hi can anyone help me please or give me a strong hint? I have to prove this: If f(x) is a function from natural to natural Numbers and f(f(f(x))) is stricly increasing a polynomial than f(x) is also.
  15. Telemachus

    Approximation e number using taylors polynomial

    Homework Statement Well, this problem is quiet similar to the one I've posted before. It asks me to approximate to the e number using taylors polynomial, but in this case tells me that the error must be shorter than 0.0005 Homework Equations...
  16. 8

    Complex Numbers: 4th Degree Polynomial

    Homework Statement Solve the following equation: z^4+z^3+z^2+z+1 = 0 z is a complex number. 2. The attempt at a solution I was trying to factorize it to 1st degree polynomial multiplied by 3rd degree polynomial: (z+a)(z^3+bz^2+cz+1/a) = 0 I discovered that I need to solve 3rd...
  17. D

    Help Getting the roots of a polynomial

    Homework Statement I am trying to get the roots of: (x-a)^3 - (x-a)b^2 - 2b^3 -2b^2(x-a) which I know they are x = a-b and x = a+2 b the problem is, how can I reach that solution ? The Attempt at a Solution At first I thought of separating the independent term (the one...
  18. V

    Proving Polynomial Equation: a0+a1x+a2x2+...+anxn=0

    Homework Statement Prove that if a0+a1x+a2x2+a3x3+...+anxn=0 then a0=0, a1=0 ... an=0 Homework Equations none The Attempt at a Solution I think I can do this for n up to 2 in the following way (please tell me if you see any gaps in my logic here): f(x)=a0+a1x+a2x2=0 (from the...
  19. D

    Converting Exponential Decay to Polynomial: Solving for Y(0) and k

    Homework Statement Turn y = y(0) * e^(-kt) into a polynomial. Homework Equations The Attempt at a Solution I have no idea of how I would go about doing this. I know you can use taylor series to approximate it, but is there any other way? Thanks, Darthxepher
  20. D

    Solve for q: Polynomial Factors Homework

    Homework Statement If we divide f(x)= x^3+qx^2-x-2 by x+1, we get the same remainder as if we divide it by x-2. Determine the value of q Homework Equations f(x)= x^3+qx^2-x-2 The Attempt at a Solution I tried to plug in f(-1) into the equation, and then f(2) into the equation...
  21. K

    Associated Laguerre Polynomial

    Hello, (quick backgroun info) : I am a physics student who has gone through pre quantum type material and a little of quantum mechanics. I am working in a lab with fortan code based on Quantum field theory. Anyway I am working to change some pieces of this code to attempt to solve a...
  22. F

    Finding value of polynomial using the remainder theorem

    Homework Statement Find the indicated value of the polynomial using the Remainder Theorem p(x)=2x^3-2x^2+11x-100; find p(3) Homework Equations p(x)=2x^3-2x^2+11x-100 The Attempt at a Solution Synthetic division 3] 2 -2 11 -100 6 12 69 2 4 23 [-31 answer: p(3)=-31 im not...
  23. G

    Monic Generator (Minimal Polynomial)

    1. Homework Statement [/b] Let V be the space of all polynomials of degree less than or equal to 2 over the reals. Define the transformation, H, as a mapping from V to R[x] by (Hp)(x)=\int^x_{-1}p(t)dt\\. a) Find the monic generator, d, which generates the ideal, M, containing the range of H...
  24. C

    Graph a Harmonic Function and its Polynomial Match

    Hello I am appealing to the computer savvy bunch here and asking for a graph. I need the graph to show a function that is of the form (Acos(x)+Bsin(X)) it can be simple... just anything harmonic I dont' even need the function you can just draw it. and then I need a polynomial that has the...
  25. G

    Polynomial equation in several variables

    What is the most general solution to an equation of the form: a_1 p_1 + \ldots + a_n p_n =0 where p_i are given polynomials in several (N) variables with no common factor (i.e. their GCD is 1) and a_n are the polynomials we are looking for (again in the same N variables). Of course, I'm asking...
  26. S

    Linear Algebra Polynomial Vector Space

    Homework Statement Use the subspace theorem to decide which of the following are real vector spaces with the usual operations. a) The set of all real polynomials of any degree. b) The set of real polynomials of degree \leq n c) The set of real polynomails of degree exactly n...
  27. A

    Hermite Polynomial identity

    Does anyone know how to prove the following identity: \Sigma_{k=0}^{n}\left(\stackrel{n}{k}\right) H_{k}(x)H_{n-k}(y)=2^{n/2}H_{n}(2^{-1/2}(x+y)) where H_{i}(z)represents the Hermite polynomial?
  28. M

    What is the limit of a rational function as x goes to infinity?

    Hi Could someone see if I have done the following limit right please? By the way, where is the limit symbol in the latex reference as I couldn't find it :( Anyway the limit is as x-> infinity (I won't keep writing that out) of \frac{-x-1/2}{2x^4}
  29. P

    Complex numbers, solving polynomial, signs of i

    I'm revising complex numbers and having trouble with this question... Question: Verify that 2 of the roots of the equation: z^3 +2z^2 + z + 2 = 0 are i and -2. Find any remaining roots Attempt at solution: i^3 +2 i^2 + i + 2 = (-1)i + 2(-1) +i + 2 = -i -2 + i +2 =0...
  30. J

    MATLAB Constructing a Loop in Matlab to Solve Polynomial Equations

    Using Matlab, I have 4 polynomial equations which I can solve by substituting the starting values for the variables into the equations. However, I want to contruct a loop to then substitute the answer from the first round back into the equations to replace the orginal starting values, and do...
  31. J

    MATLAB Matlab Polynomial Equation Roots Approximations

    Is there a function in Matlab which gives a list of numerical approximations to the roots of a polynomial equation (eg. like NSolve in Mathematica?) Thanks!
  32. A

    Factoring a polynomial where factor theorem doesn't work

    Homework Statement Solve: x^3 - 9x^2 + 15x + 30 Homework Equations The Attempt at a Solution The factors of 30 are +-1, +-2, +-3, +-5, +-6, +-10, +-15, and +-30. I used my graphing calculator and got a zero close to -1. I plugged it into the original equation and got 5, not...
  33. S

    Decompossing a polynomial (keep getting the wrong result )

    decompossing a polynomial (keep getting the wrong result :() Homework Statement Given the polynomial f(x) = -x^2 - x + 1, decompose the polynomial into linear terms The Attempt at a Solution I get (x-(-(\frac{-\sqrt{5}+1}{2}))((x-(-\frac{-\sqrt{5}-1}{2})) I seem to be missing a...
  34. Char. Limit

    Convolution of a polynomial with itself

    Again, in my quest to learn things I won't use in a class for at least a year, I've been looking at convolutions. Specifically, after finishing the multiple choice section of an AP Chemistry test 50 minutes early, I looked at the convolution of a polynomial with itself. I'm confused about one...
  35. S

    Summation of the polynomial and division

    Homework Statement Let p(z) = \sum_{j=0}^{n} a_{n-j}z^j be a polynomial of at least degree 1 thus n \geq 1. Show that if z\neq 0 then 1/z is a root of the polynomial p. Homework Equations Fundamental theorem of Algebra The Attempt at a Solution If a expand the polynomial...
  36. A

    Solving a Polynomial: y=x^4/(x^2+1) and y=1/(x^2+1)

    Homework Statement The curves are: y = \frac{x^{4}}{x^{2}+1} and y = \frac{1}{x^{2}+1} The Attempt at a Solution So again I assume that: \frac {x^{4}}{x^{2}+1} = \frac {1}{x^{2}+1} and then cross multiply: (x^{2}+1) = x^{4}(x^{2}+1) not really sure at this point if i should...
  37. C

    Polynomial Division: Show g(x) Divides f(x)

    Homework Statement Show that g(x) = x^3 + 1 divides f(x) = x^{9999} +1. Homework Equations The Attempt at a Solution g(x) can obviously be factored into the irreducible polynomials (x+1)(x^2 - x + 1) in Z[x], and since f(-1) = (-1)^{9999} + 1 = 0, the factor theorem gives that (x+1) divides...
  38. G

    Criterion for Irreducibility of a polynomial in several variables?

    Is there any criterion for the irreducibility of a polynomial in several variables over an algebraically closed field (or specifically for the complex numbers)? For one variable, we know this is simply that only degree one polynomials are irreducible, is there anything similar for several variables?
  39. S

    Polynomial differential operators

    Homework Statement p(D) is a polynomial D operator of degree n>m. Suppose a is a m fold root of p(t)=0, but not a (m+1) fold root. Verify that \frac{1}{p(D)}e^{at}=\frac{1}{p^{(m)}(a)}t^me^{at} where p^{(m)}(t) is the m^{th} derivative of p(t).Homework Equations For this question, we were...
  40. S

    Finding the Taylor Polynomial f4 for sin(2x) at x=pi/4.

    Homework Statement find the taylor polynomial f4 for f(x)=sin(2x) and a=pi/4 Homework Equations sin(x)=((-1)^n)(x^(2n+1))/((2n+1)!) The Attempt at a Solution so replace x with 2x? you get ((-1)^n)(2x)^(2n+1)/(2n+1)!) is this right?
  41. J

    Polynomial approximation

    Homework Statement Obtain the Taylor polynomials Tnf(x) as indicated. In each case, it is understood that f(x) is defined for a11 x for which f(x) is meaningful. Problem one Tn = (a^x) = sigma from k = 0 to n of ((log a)^k)/k! x^k Problem two Tn = (1/(1+x)) = sigma from k = o...
  42. A

    Help proving polynomial identity

    Homework Statement Prove the following when p is a positive integer: b^p - a^p = (b-a)(b^{p-1}+b^{p-2}a+b^{p-3}a^2+...+ba^{p-2}+a^{p-1}) Hint: Use the telescoping property for sums. Homework Equations None The Attempt at a Solution I tried reducing it to, (b-a)\sum_{k=1}^p...
  43. C

    Simplest Interpolating Polynomial

    Homework Statement Consider the data in the following table for constant-pressure specific heat, C p (kJ/kg.K) at various temperatures T (K). Determine the simplest interpolating polynomial that is likely to predict Cp within 1% error over the specified range of temperature. T : 1000 1100...
  44. Mentallic

    Polynomial Factorization for Integers: Finding Real and Complex Roots

    Homework Statement In the polynomial x^5+22x^3-34x^2+117x-306 given that the roots on the real and imaginary plane are all integers, factorize the polynomial into real linear and quadratic factors. The Attempt at a Solution I was able to find the real factor, which is x=2 and then I...
  45. M

    Polynomial interpolation

    Let x_{0}, x_{1}, \cdots , x_{n} be distinct points in the interval [a,b] and f \in C^{1}[a,b]. We show that for any given \epsilon >0 there exists a polynomial p such that \left\| f-p \right\|_{\infty} < \epsilon and p(x_{i}) = f(x_{i}) for all i=1,2, \cdots , n I know \left\|...
  46. R

    Expanding taylor polynomial

    Homework Statement Derive a method for approximating f'''(x0) whose error term is of order h^{2} by expanding the function f in a fourth taylor polynomial about x0 and evaluating at x_{0} \pm h and x_{0} \pm 2h. Homework Equations The Attempt at a Solution I'm not sure where to...
  47. K

    P(x) be any polynomial of degree at least 2

    Homework Statement Let P(x) be any polynomial of degree at least 2, all of whose roots are real and distinct. Prove that all of the roots of P'(x) must be real. What happens if some of the roots of P are multiple roots? Homework Equations I think that question is related to the concept...
  48. H

    Taylor polynomial 1/(1-x^2)

    Homework Statement The question asks me to write out a taylor polynomial for 1/(1-x^2) of degree 2n+1 at 0. The Attempt at a Solution My answer was 1 + x^2 + x^4 + x^6 + ... + (x^4)/(1-x^2) which I just got from using hte geometric series formula. The textbook answer however...
  49. 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...
  50. M

    Symmetric polynomial algorithm?

    Let f, g \in \mathbb{Z}[x, y, z] be given as follows: f = x^8 + y^8 + z^6 and g = x^3 +y^3 + z^3. Express if possible f and g as a polynomial in elementary symmetric polynomials in x, y, z. Professor claims there is an algorithm we were supposed to know for this question on the midterm. I...
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