# 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. ### B Finding polynomials with given roots

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?
2. ### I Analogy question for algebraists

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...
3. ### A Coefficients of Chebyshev polynomials

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...
4. ### I Antisymmetrizing a Factorized Polynomial Vanishes?

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...
5. ### I Want to understand how to express the derivative as a matrix

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...
6. ### I Resolve the Recursion of Dickson polynomials

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...
7. ### A Differential equation and Appell polynomials

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: ...
8. ### I A doubt about the multiplicity of polynomials in two variables

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...
9. ### I Questions about algebraic curves and homogeneous polynomial equations

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...
10. ### I May I use set theory to define the number of solutions of polynomials?

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...
11. ### I Polynomials can be used to generate a finite string of primes....

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?
12. ### Proof: Integer Divisibility by 3 via 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...
13. ### How Do You Simplify and Analyze Taylor Polynomials for Higher Degree Functions?

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...
14. ### MHB Can you factor the following two polynomials?

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
15. ### Potential of a charged ring in terms of Legendre polynomials

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...
16. ### MHB Program to calculate the sum of polynomials

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...
17. ### Check that the polynomials form a basis of R3[x]

I put it in echelon form but don't know where to go from there.
18. ### I Irreducible polynomials and prime elements

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 :)

47. ### Vector space - polynomials vs. functions

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...
48. ### B Properties of roots of polynomials

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...
49. ### B Roots of Polynomials: Understanding Mathematical Methods

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...
50. ### MHB Open neighbourhoods and equating coefficients of polynomials

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...