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. YoungPhysicist

    B Is This a Valid Proof for the Number of Roots in a nth Degree Polynomial?

    Recently I came up with a proof of “ for a nth degree polynomial, there will be n roots” Since the derivative of a point will only be 0 on the vertex of that function,and a nth degree function, suppose ##f(x)##has n-1 vertexes, ##f’(x)## must have n-1 roots. Is the proof valid?
  2. M

    Solving a Trigonometry Problem: Find u(x,t) Polynomial

    <Moderator's note: Moved from a technical forum and thus no template.> Task: http://snk066.tk/math/Task.png My solution: http://snk066.tk/math/my_solution.jpg What you need to? I need an answer in the form: u (x,t) = (some polynomial) The solution is not really necessary, if someone will...
  3. YoungPhysicist

    B A rookie question for integrals of polynomial functions

    $$\int x^2+3 = \frac{x^3}{3}+3x+C$$ I can get the front two part by power rule, but what is the C doing there? Wolframalpha suggested it should be a constant, but what value should it be? Sorry for asking rookie questions:-p
  4. C

    Factoring Combinatorial Functions

    Homework Statement Define {x \choose n}=\frac{x(x-1)(x-2)...(x-n+1)}{n!} for positive integer n. For what values of positive integers n and m is g(x)={{{x+1} \choose n} \choose {m}}-{{{x} \choose n} \choose {m}} a factor of f(x)={{{x+1} \choose n} \choose {m}}? Homework Equations The idea...
  5. K

    Limit of Taylor Polynomial for Tn(x) as n Approaches Infinity

    Homework Statement Let Tn(x)=1+2x+3x^2+...+nx^(n-1) Find the value of the limit lim n->infinity Tn(1/8).The Attempt at a Solution How do I solve this? I know how to write the polynomial as a series, but not sure how if this is the best way of finding the limit.
  6. K

    Taylor polynomial, approximative solution of this equation

    Homework Statement The equation 4x = (1/3)*cos(3x) has a solution on the interval [0,1]. Find an approximative solution by replacing the right hand side with a Taylor polynomial of degree 2 around 0. Homework EquationsThe Attempt at a Solution So as I understand the task we should find a...
  7. M

    MHB Irreducible factors of polynomial

    Hey! :o Let $K$ be a normal extension of $F$ and $f\in F[x]$ be irreducible over $F$. Let $g_1, g_2$ be irreducible factors of $f$ in the ring $K[x]$. Show that there exists $\sigma \in G(K/F)$ such that $g_2=\sigma (g_1)$. If $f$ is reducible over $K$, show that all its irreducible...
  8. S

    How do I factor this cubic polynomial?

    Homework Statement -x^3 - 6x^2 -12x -8 Homework EquationsThe Attempt at a Solution I don't know, I just know the roots are -2 with multiplicity 3.
  9. M

    Finding the coefficients of a polynomial given some restriction

    Homework Statement Find all ##a,b,c\in\mathbb{R}## for which the zeros of the polynomial ##az^3+z^2+bz+c=0## are in this relation $$z_1^3+z_2^3+z_3^3=3z_1z_2z_3$$ Homework Equations we know that if we have a polynomial of degree 3 the zeroes have relation in this case ##z_1+z_2+z_3=-1/a##...
  10. M

    Fractional polynomial addition

    Homework Statement Determine whether there exist ##A## and ##B## such that: $$\frac{1}{3x^2-5x-2} = \frac{A}{3x+1} + \frac{B}{x-2}$$Homework Equations None The Attempt at a Solution [/B] First I divided the polynomial ##3x^2-5x-2## by ##3x+1## and got ##x-2## as a result without a...
  11. danielFiuza

    A Solution?: Quintic Equation from Physical System

    First time in this forum, so greetings to everyone! I am currently working with some physical models in the field of natural ventilation and I came across the following 5th order polynomial equation (quintic function): $X^{5}+ C X - C =0$ This is the steady state solution of a physical system...
  12. J

    Using binomial coefficients to find sum of roots

    Homework Statement >Find the sum of the roots, real and non-real, of the equation x^{2001}+\left(\frac 12-x\right)^{2001}=0, given that there are no multiple roots. While trying to solve the above problem (AIME 2001, Problem 3), I came across three solutions on...
  13. Eclair_de_XII

    How to show that a 5-th degree polynomial has a root?

    Homework Statement "Show that for some ##x\in ℝ##, that ##x^5+2x^4+3x^3+2x^2+x=1##." Homework EquationsThe Attempt at a Solution Okay, so I know from Descartes' rule of sign that the function ##f(x)=x^5+2x^4+3x^3+2x^2+x-1## has exactly one positive root, since the sign of the coefficients...
  14. M

    MHB The polynomial is irreducible over Q(i)

    Hey! :o I want to show that the polynomial $x^4-2\in \mathbb{Q}[x]$ remains irreducible in the ring $\mathbb{Q}(i)[x]$. I have done the following: The polynomial is irreducible in $\mathbb{Q}[x]$ by Eisenstein's criterion with $p=2$. Then if $a$ is a root of $x^4-2$ then the degree of the...
  15. A

    MHB How to integral legendre polynomial

    Question \int_{-1}^{1} cos(x) P_{n}(x)\,dx ____________________________________________________________________________________________ my think (maybe incorrect) \int_{-1}^{1} cos(x) P_{n}(x)\,dx \frac{1}{2^nn!}\int_{-1}^{1} cos(x) \frac{d^n}{dx^n}(x^2-1)^n\,dx This is rodrigues formula by...
  16. W

    I Polynomial Ideals: Struggling with Ring Ideals

    This time my struggle is with ring ideals. Book still won't provide examples, so I'm again trying to come up with some of my own. I figured {0,2} might fit the definition as an ideal of ##\mathbb{Z/4Z}## since it is an additive subgroup and ##\forall x \in I, \forall r \in R: x\cdot r, r\cdot x...
  17. evinda

    MHB Show uniqueness of polynomial

    Hello! (Wave) Let $\mathbb{R}[x]_{ \leq n}$ be the vector space of the real polynomials of degree $\leq n$, where $n$ a natural number. I want to show that there is a unique $q(x) \in \mathbb{R}[x]_{\leq n}$, with the property that $\int_{-1}^1 p(x) e^x dx=\int_0^1 p(x) q(x) dx$, for each $p(x)...
  18. evinda

    MHB Minimal polynomial of matrices

    Hello! (Wave) If the matrix $A \in M_n(\mathbb{C})$ has $m_A(x)=(x^2+1)(x^2-1)$ as its minimal polynomial, then I want to find the minimal polynomials of the matrices $A^2$ and $A^3$. ($M_n(k)$=the $n \times n$ matrices with elements over the field $k=\mathbb{R}$ or $k=\mathbb{C}$) Is there a...
  19. Monoxdifly

    MHB Find Polynomial Given Remainder After Division

    11. Given a polynomial with the degree 3. If it is divided by x^2+2x-3, the remainder is 2x + 1. If it is divided by x^2+2x, the remainder is 3x - 2. The polynomial is ... A. \frac23x^3+\frac43x^2+3x-2 B. \frac23x^3+\frac43x^2+3x+2 C. \frac23x^3+\frac43x^2-3x+2 D. x^3+2x^2+3x-2 E. 2x^3+4x^2+3x+2...
  20. S

    I Is quintic only polynomial that needs to be proven imposs?

    I've been trying to prove the impossibility of the quintic "on the cheap" without having to go through a graduate course in abstract algebra (I haven't even done the undergraduate course, although I've been reading up on it a little bit at a time). I understand Bezout's Lemma, with a practical...
  21. M

    MHB Zeroes of cubic polynomial

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  22. K

    I RL circuit with polynomial forcing function solution (units)

    Hello. This is the differential equation. $$ i \cdot \dfrac{R}{L} + \dfrac{di}{dt} = t $$ My solution path: Homogenous solution: $$ r + \dfrac{R}{L} = 0 \\ \\ r = -\dfrac{R}{L} \\ \\ i_{h}(t) = C_{0} e^{ -\dfrac{R}{L} t} \\ \\ $$ Particular solution: Try $$ y_{p} = at + b \\ \\ y_{p}...
  23. opus

    Help in factorization of a third degree polynomial

    Homework Statement Looking to factor ##-2x^3-3## and having an issue. To my understanding, the Fundamental Theorem of Algebra tells us that it is at least theoretically possible to factor any polynomial of degree n. Homework EquationsThe Attempt at a Solution So my first step to factor this...
  24. R

    MHB Solving a Polynomial x^6 – 7x^3 + 12 by Factoring.

    Hello, I have been going through the Wisconsin Placement Exam sample test. I'm trying to figure out how to find the solution set for x6 – 7x3 + 12. I have tried having u = x3 and solving for u2-7u+12, but I'm unsure what to do once I get (u - 4)(u - 3). Would someone help me figure out how to...
  25. Mr Davis 97

    I Integral of polynomial times exp(-x^2)

    I have the integral ##\int_{-\infty}^{\infty} x^2 e^{-x^2} ~dx##. Is there any simple way to integrate this, given that that I already know that the value of the Gaussian integral is ##\sqrt{\pi}##?
  26. I

    B Polynomial degree and root relationship

    Why is it that for a 7th degree polynomial, the number of real roots is either 1, 3, 5, or 7?
  27. Mr Davis 97

    Factoring a quartic polynomial

    Homework Statement Factor ##x^4-3x^2+9## over the reals Homework EquationsThe Attempt at a Solution I am factoring this polynomial over the reals. So there are two options. It will either split into two linear factors and an irreducible quadratic, or two irreducible quadratics. I'm really not...
  28. Mr Davis 97

    I Factoring a quartic polynomial over the reals

    I have the simple quartic polynomial ##x^4+1##. How in general do I determine whether this is factorable over the reals or not? Since it has no real roots, it could only factor into two quadratic polynomials, but I am not sure what I can do to determine whether this is possible or not.
  29. T

    MHB Can you help me find the third zero of this complex polynomial?

    Hey, first off, I'm not sure if this is the right section. If another section is better, please let me know and I'll be more careful next time. So, my problem is with a degree 3 complex polynomial. I'm given one zero of the equation, but since it is a complex zero, I can use the conjugate too...
  30. B

    Find roots of cubic polynomial with complex coefficient

    Homework Statement Find roots of $$ -\lambda ^3 +(2+2i)\lambda^2-3i\lambda-(1-i) = 0 $$ Homework EquationsThe Attempt at a Solution I tried my old trick I tried to separating the 4 terms into 2 pairs and try to find a common factor in the form of ##\lambda + z## between them, $$ -\lambda ^2...
  31. L

    Find the smallest value for the polynomial

    The graph below shows a portion of the curve defined by the quartic polynomial P(x) = x^4 + ax^3 + bx^2 + cx + d. Which of the following is the smallest?https://imgur.com/a/1VuGSiA(A) P(-1) (B) The product of the zeros of P (C) The product of the non-real zeros of P (D) The sum of the...
  32. Mr Davis 97

    Showing that one polynomial divides another

    Homework Statement Show that ##\displaystyle \sum_{i=0}^{100} {100\choose i}{200-i\choose 198-i}x^i## is divisible ##(x+1)^{98}##. Homework EquationsThe Attempt at a Solution I am pretty stumped, but I have a few general. I think that the the binomial theorem will be involved. That is, I think...
  33. A

    MHB Irreducible Polynomial g = X^4 + X + 1 over F2

    I am really struggling on the following Algebra question: Consider the Irreducible Polynomial g = X^4 + X + 1 over 𝔽2 and let E be the extension of 𝔽2 = {0,1} with root α of g. (a) How many elements does E have? (b) Is every non-zero element of E of the form α^n with n ϵ N (natural numbers)...
  34. R

    When is the minimum polynomial of a scalar matrix kI equal to t-k?

    Homework Statement Show that A is a scalar matrix kI if and only if the minimum polynomial of A is m(t) = t-k Homework EquationsThe Attempt at a Solution f(A) is monic f(A) = 0 since A = kI Next we must show that deg(g) < deg(f) I guess I'm not sure where g comes from. Is it merely an...
  35. R

    From a given basis, express a polynomial

    Homework Statement Express a polynomial in terms of the basis vectors. {x2 + x, x + 1, 2} Homework Equations 3. The Attempt at a Solution [/B] I think the answer is: (x2+x)^2 + (x + 1) + 2 = 0 simplified to become: x4 + 2x3 + x2 + x + 3 = 0
  36. J

    MHB Polynomial Proof: Verification & Correction

    I would like to have verification if the following attached proof is correct. If it is not correct, what can be done to make it correct? Thanks.
  37. lfdahl

    MHB Prove Polynomial Congruence $(x+y)^n \equiv x^n + y^n$ (mod $p$)

    Given a prime number $p$, prove that the polynomial congruence $(x + y)^n \equiv x^n + y^n$ (mod $p$) is true if and only if $n$ is a power of $p$.
  38. Greg Bernhardt

    B Super basic polynomial and exponent definition help

    Please bare with me. Most of you know I actually don't have a great math background. In any case I'm going way back and filling in some very basic math that I have long forgot. I have some questions about terms in a polynomial. Here is an example $$3x^5+7x^3-5$$ 1. From my book 3 and 7 are...
  39. C

    Polynomial expansion from Python to Mathematica

    Hi everybody. In Python there is a library called chaospy. One useful command is cp.orth_ttr which generates a polynomial expansion, e. g. a series of orthogonal polynomials or orders zero, one, two... for a random variable e.g normal, uniform... For more information see...
  40. opus

    B Behavior of polynomial functions at their zeros

    Just a general question here. So for a polynomial function, the behavior of the graph at the zeros is determined by the evenness or oddness of the magnitude of the zeros. If the magnitude is odd, the graph will cross the zero. If the magnitude is even, it will bounce at the zero. Why is this...
  41. Monoxdifly

    MHB [ASK] polynomial f(x) divided by (x - 1)

    A polynomial f(x) = 2x^3-5x^2+ax+18 is divisible by (x - 3). The result of that polynomial f(x) divided by (x - 1) is ... A. 2x^2-7x+2 B. 2x^2+7x-2 C. 2x^2-7x-2 D. x^2-6x-2 E. x^2-6x+3 I got a + 3 = -6 and so a = -9 and f(x) = 2x^3-5x^2-9x+18, but when I divided it with x - 1 I got x^2-3x+6...
  42. D

    MHB Find Real Solutions to x4-2x3+kx2+px+36 = 0

    One of the solutions to x4-2x3+kx2+px+36 = 0 is x = 3i Prove that this polynomial has no real solutions (roots) and find the real values of k and p. ------------------------------------------------------------------------------------------------------------------- So far the only progress...
  43. chwala

    Solve three simultaneous polynomial equations in three variables....

    Homework Statement ## x^2 +xy + y^2 = 3## ## y^2+yz+z^2=1## ##z^2+zx+x^2=4##Homework EquationsThe Attempt at a Solution ##yz^2-xz^2+yx^2-xy^2=4y-x## ##z^2(y-x)-xy(y-x)=4(y-x)## thus ##z^2-xy=4## or ##z^2=4+xy## ......on working out i end up with ## z^4-8z+16+(z^2-4)y^2+y^4= 3y^2## and ##...
  44. M

    MHB How is the remainder for Taylor polynomials calculated?

    Hey! :o I want to calculate the Taylor polynomial of order $n$ for the funktion $ f(x) = \frac{1}{ 1−x}$ for $x_0=0$ and $0 < x < 1$ and the remainder $R_n$. We have that \begin{equation*}f^{(k)}(x)=\frac{k!}{(1-x)^{k+1}}\end{equation*} I have calculated that...
  45. M

    MATLAB Associated Legendre Polynomial of 1st and 2nd kind

    Hi PF! In MATLAB I'm trying to use associated Legendre polynomials of the 1st and second kind, widely regarded as ##P_i^j## and ##Q_i^j##, where ##j=0## reduces these to simply the Legendre polynomials of the 1st and second kind (not associated). Does anyone here know if MATLAB has a built in...
  46. C

    Coefficient Matching for different series

    Homework Statement Hello, I have a general question regarding to coefficient matching when spanning some function, say , f(x) as a linear combination of some other basis functions belonging to real Hilbert space. Homework Equations - Knowledge of power series, polynomials, Legenedre...
  47. StevenScott

    Airy Stress Func. Polynomial order to satisfy the biharmonic equation

    Hello, When choosing a polynomial stress function Φ to satisfy the biharmonic equation, how does once decide on which order of the polynomial to choose? For example, is it based upon the number of boundary conditions, like a 3rd order polynomial would satisfy 3 boundary conditions?
  48. castor28

    MHB Polynomial challenge: Show that not all the coefficients of f(x) are integers.

    $f(x)$ is a degree 10 polynomial such that $f(p)=q$, $f(q)=r$, $f(r)=p$, where $p$, $q$, $r$ are integers with $p<q<r$. Show that not all the coefficients of $f(x)$ are integers.
  49. S

    B Remainder of polynomial division

    Is this true? If the remainder of f(x) / g(x) is a (where a is constant), then the remainder of (f(x))n / g(x) is an I don't know how to be sure whether it is correct or wrong. I just did several examples and it works. Thanks
  50. E

    Python Polynomial Regression with Scikit-learn

    Hello, I followed an example in a book that compares polynomial regression with linear regression. We have one feature or explanatory variable. The code is the following: import numpy as np import matplotlib.pyplot as plt from sklearn.linear_model import LinearRegression from...
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