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.
Say we have the following conditions:
For an any degree polynomial with integer coefficients, the root of the polynomial is n. There should be infinite polynomials that satisfy this condition. What is the general way to generate one of the polynomial?
An "analogy question":
Polynomials with one variable and coefficients in the field K are to finite dimensional K vector spaces as polynomials in several variables over the field K are to ....?
As a teenager, I recall taking tests that had "analogy questions" on them. The format was: Thing A...
Not long ago, I derived the formula for Chebyshev polynomials
$$T_{n}\left( x\right)= \sum_{k=0}^{\lfloor \frac{n}{2} \rfloor}{n \choose 2k}x^{n-2k}\left( x^2-1\right)^{k}$$
How to extract the coefficients of this polynomial of degree n ?
I tried using Newton's binomial but got a double sum...
Hi all,
I am having trouble understanding the argument below equation (3.5) in https://arxiv.org/pdf/cond-mat/9605145.pdf where they claim that "Upon antisymmetrization, however, a term with ##k## factors of ##(z_{i}-z_{j})## would have to antisymmetrize ##2k## variables with a polynomial that...
In https://www.math.drexel.edu/~tolya/derivative, the author selects a domain P_2 = the set of all coefficients (a,b,c) (I'm writing horizontally instead off vertically) of second degree polynomials ax^2+bx+c, then defines the operator as matrix
to correspond to the d/dx linear transformation...
I am trying to prove the expression for Dickson polynomials:
$$D_n(x, a)=\sum_{i=0}^{\lfloor \frac{n}{2}\rfloor}d_{n,i}x^{n-2i}, \quad \text{where} \quad d_{n,i}=\frac{n}{n-i}{n-i\choose i}(-a)^i$$
I am supposed to use the recurrence relation:
$$D_n(x,a)=xD_{n-1}(x,a)-aD_{n-2}(x,a)$$
I have...
Hello!
Let $n$ be a natural number, $P_n(x)$ be a polynomial with rational coefficients, and $\deg P_n(x) = n$. Let $P_0(x)$ be a constant polynomial that is not equal to zero. We define the sequence ${P_n(x)}_{n \geq 0}$ as an Appell sequence if the following relation holds:
\begin{equation}...
Let ##P(x,y)## be a multivariable polynomial equation given by
$$P(x,y)=52+50x^{2}-20x(1+12y)+8y(31+61y)+(1+2y)(-120+124+488y)=0,$$
which is zero at ##q=\left(-1, -\frac{1}{2}\right)##. That is to say,
$$ P(q)=P\left(-1, -\frac{1}{2}\right)=0.$$
My doubts relie on the multiplicity of this point...
It is generally well-known that a plane algebraic curve is a curve in ##\mathcal{CP}^{2}## given by a homogeneous polynomial equation ##f(x,y)= \sum^{N}_{i+j=0}a_{i\,j}x^{i}y^{j}=0##, where ##i## and ##j## are nonnegative integers and not all coefficients ##a_{ij}## are zero~[1].
In addition, if...
Let ##Q_{n}(x)## be the inverse of an nth-degree polynomial. Precisely,
$$Q_{n}(x)=\displaystyle\frac{1}{P_{n}(x)}$$,
It is of my interest to use the set notation to formally define a number, ##J_{n}## that provides the maximum number of solutions of ##Q_{n}(x)^{-1}=0##. Despite not knowing...
F(n)=##n^2 −n+41## generates primes for all n<41.
Questions:
(1) Are there polynomials that have longer lists?
(2) Is such a list of polynomials finite (yes, no, unknown)?
(3) Same questions for quadratic polynomials?
Proof:
Let ## P(x)= \Sigma^{m}_{k=0} a_{k} x^{k} ## be a polynomial function.
Then ## N=a_{m}10^{m}+a_{m-1}10^{m-1}+\dotsb +a_{1}10+a_{0} ## for ## 0\leq a_{k}\leq 9 ##.
Since ## 10\equiv 1\pmod {3} ##, it follows that ## P(10)\equiv P(1)\pmod {3} ##.
Note that ## N\equiv (a_{m}+a_{m-1}+\dotsb...
f(x) = 4 + 5x - 6x^2 + 11x^3 - 19x^4 + x^5
question a almost seems too easy as I end up 'removing' the x^4 and x^5 terms
a.
T_{2} (x) = 4 + 5x - 6x^2
b.
= R_{2} (x) = 11x^3 - 19x^4 + x^5
c.
i don't understand what i need to do here. To find the maximum value of a function, we...
Can you factor the following polynomials over integers?
x^4 + 4
x^4 + 3 ~x^2~y^2 + 2 ~y^4 + 4 ~x^2 + 5 ~y^2 + 3
If not, you can get help from the following free math tutoring YouTube channel "Math Tutoring by Dr. Liang"
https://www.youtube.com/channel/UCWvb3TYCbleZjfzz8HEDcQQ
hi guys
I am trying to calculate the the potential at any point P due to a charged ring with a radius = a, but my answer didn't match the one on the textbook, I tried by using
$$
V = \int\frac{\lambda ad\phi}{|\vec{r}-\vec{r'}|}
$$
by evaluating the integral and expanding denominator in terms of...
Hey! 😊
A polynomial can be represented by a dictionary by setting the powers as keys and the coefficients as values. For example $x^12+4x^5-7x^2-1$ can be represented by the dictionary as $\{0 : -1, 2 : -7, 5 : 4, 12 : 1\}$. Write a function in Python that has as arguments two polynomials in...
let p∈Z a prime how can I show that p is a prime element of Z[√3] if and only if the polynomial x^2−3 is irreducible in Fp[x]? ideas or everything is well accepted :)
Using the generating function for the legendre polynomial: $$ \sum_{n=0}^{\infty} P_{n}(x) t^{n}=\frac{1}{\sqrt{1-2 x t+t^{2}}} $$ It's possible to expand the coulomb potential in a basis of legendre polynomials (or even spherical harmonic ) like this: $$ \begin{aligned} &\frac{1}{\left.\mid...
The problem has a hint about finding a relationship between ##\int_{-1}^1 (P^{(k+1)}(x))^2 f(x) dx## and ##\int_{-1}^1 (P^{(k)}(x))^2 g(x) dx## for suitable ##f, g##. It looks they're the weighting functions in the Sturm-Liouville theory and we may be able to make use of Parseval's identity...
How do we prove that every polynomial (with coefficients from C) of degree n has exactly n roots in C?
This is not a homework (I wish I was young enough to have homework) I guess this is covered in every typical undergraduate introductory algebra course but for the time being I can't find my...
Summary: countability, topological vector spaces, continuity of linear maps, polynomials, finite fields, function theory, calculus1. Let ##(X,\rho)## be a metric space, and suppose that there exists a sequence ##(f_i)_i## of real-valued continuous functions on ##X## with the property that a...
Prove that in the following product
$P=(1-x+x^2-x^3+\cdots-x^{99}+x^{100})(1+x+x^2+x^3+\cdots+x^{99}+x^{100})$
after multiplying and collecting like terms, there does not appear a term in $x$ of odd degree.
Hello everyone,
I'm currently doing some research about feedback systems in engineering and right now I'm playing around with special types of feedback matrices. In the process, I stumbled upon a potentially interesting polynomial, which is actually the characteristic polynomial of the system...
[Mentor Note -- Multiple threads merged. @issue -- please do not cross-post your threads]
Hi, everyone
It is known that binomial distribution can also be solved by polynomials. i add document with a question I can not solve.
Glad to get for help
Thanks to all the respondents
I'm not sure if I should post this here or in the mathematics section.
I'm trying to find a way to implement a mapping of a larger finite field such as GF(2^64) to a composite field GF((2^32)^2). Let f(x) be a primitive polynomial for GF(2^64), with 1 bit coefficients. If the coefficients of...
Hello everyone. I have recently read the following article (which title is SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos) since I have some data in the form of a histogram without knowing the probability distribution function of said data. I have been able to calculate...
Hello everyone,
I need some help with this solution. I'm trying to obtain a set of orthogonal polynomials up to the 7th term. I think i got it up to the 6th term, but the integration is getting more complex. I'm not sure if I'm on the right track. Please help
0
Let T: P2 (R) → P2 (R) be the linear map defined by T(p(x)) = p''(x) - 5p'(x). Is T invertible ? P2 (R) is the vector space of polynomials of degree 2 or less
How could I show that this limit:
##\lim_{N\to\infty}\frac{\sum_{p=1}^N T_{4N} \left(u_0(N)\cdot \cos\frac{p\pi}{2N+1}\right)}{N}##
is equal to 0?
In the expression above ##T_{4N}## is the Chebyshev polynomials of order ##4N##, ##u_0(N)\geq 1## is a number such that ##T_{4N}(u_0)=b##, with...
Is it possible (read: reasonably easy) to find global minima of an nth degree polynomial of the general form $$a_nx^n + a_{n-1}x^{n-1} ... a_2x^2 +a_1x + a_0 = 0$$ It seems to have applications in computational chemistry as I have a "hunch" that polynomial regression could be used to somewhat...
If you had legendre polynomials defined in ##L^2([-1,1])##, with ##||Pn_2||^2=\frac{2}{2n+1}##, show that for any polynomial with p a set of ##L^2([-1,1])##, with degree less than n, we have the inner product of ##P_n## and p = 0. Find the polynomials ##P_0,... P_4##
Tried to use the integral...
Let $k\ne 0$ be an integer. Find all polynomials $P(x)$ with real coefficients such that $(x^3-kx^2+1)P(x+1)+(x^3+kx^2+1)P(x-1)=2(x^3-kx+1)P(x)$ for all real number $x$.
Suppose all roots of the polynomial ##x^n+a_{n−1}x^{n−1}+\cdots+a_0## are real. Then the roots are contained in the interval with the endpoints
$$
-\dfrac{a_{n-1}}{n} \pm \dfrac{n-1}{n}\sqrt{a_{n-1}^2-\dfrac{2n}{n-1}a_{n-2}}\,.
$$
Hint: Use the inequality of Cauchy-Schwarz.
Suppose p = a + bx + cx².
I am trying to orthogonalize the basis {1,x,x²}
I finished finding {1,x,x²-(1/3)}, but this seems different from the second legendre polynomial.
What is the problem here? I thought could be the a problem about orthonormalization, but check and is not.
Hey! 😊
Let $\mathbb{K}$ be a field and $1\leq n\in \mathbb{N}$.
For a polynom $\displaystyle{\sum_{i=0}^mc_it^i\in \mathbb{K}[t]}$ and a matrix $a\in M_n(\mathbb{K})$ the $f(a)\in M_n(\mathbb{K})$ is defined by \begin{equation*}f(a):=\sum_{i=0}^mc_ia^i=c_ma^m+c_{m-1}a^{m-1}+\ldots...
I could simplify the expressions in the numerator and denominator to (1+x^n)/(1+x) as they are in geometric series and I used the geometric sum formula to reduce it. Now for what value of n will it be a polynomial?
I do get the idea for some value of n the simplified numerator will contain the...
Hi, Can someone please help me in understanding few parameters of cubic equation formula for solving polynomial of degree three. I attached the formula in the screenshot.
My questions are:
(1) what is ". " dot in the end of the formula and what does it mean?
(2) I want to use it only for real...
I'm not at all C++ literate, but I need to understand what a C++ code is doing for a math problem regarding the Stieltjes polynomials.
Especially, I want to know what is happening to the "num" and "den" in the code; the code is in this link:
Stieltjes
I am looking for a recurrence relation and/or defining expression for the Stieltjes polynomials with regard to the Legendre polynomials. I found an article about it here: Legendre-Stieltjes but they do not offer a formula.
For example a recurrence relation for the Gegenbauer polynomials is...
Hello everyone.
I am studying this article since I am interested in optimization. The article makes use of Clenshaw–Curtis quadrature scheme to discretize the integral part of the cost function to a finite sum using Chebyshev polynomials.
The article differentiates between the case of odd...
Hello everyone. I am trying to construct an optimization problem using Chebyshev pseudospectral method as described in this article. For that, I need to calculate the zeros of the Chebyshev polynomial of any order. In the article is sugested to do it as
tk=cos(πk/N) k=0, ..., N...
From Griffiths E&M 4th edition. He went over solving a PDE using separation of variables. It got to this ODE
\frac{d}{d\theta}\left(\sin\theta\frac{d\Theta}{d\theta}\right)= -l(l+1)\sin \theta \Theta
Griffths states that this ODE has the solution
\Theta = P_l(\cos\theta)
Where $$P_l =...
As per source # 1 ( link below), when treating polynomials as vectors, we use their coefficients as vector elements, similar to what we do when we create matrices to represent simultaneous equations.
However, what I noticed in Source #2 was that, when functions are represented as vectors, the...
i have some doubts from chapter 1 of the book Mathematical methods for physics and engineering.
i have attached 2 screenshots to exactly point my doubts.
in the first screenshot...could you tell me why exactly the 3 values of f(x) are equal.
the first is the very definition of polynomials...but...
I was reading this book - " mathematical methods for physics and engineering"
in it in chapter 1 its says
"F(x) = A(x - α1)(x - α2) · · · (x - αr),"
this makes sense to me but then it also said
We next note that the condition f(αk) = 0 for k = 1, 2, . . . , r, could also be met
if (1.8) were...
Hi all,
I am trying to understand some examples given to me by my supervisor but am struggling with some bits. The part I don't understand is: if the equation
$$ax+b\lambda=\bar{a}x-\bar{d}y$$
holds for any $x,y\in V$, an open neighbourhood of the origin, and $\lambda$ is a mapping from $V$ to...