Polynomial Definition and 45 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. V9999

    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...
  2. M

    B Why cubic?

    Why is a third degree polynomial called a cubic polynomial? I just don’t see the connection between 3 and a cube.
  3. H

    How to convince myself that I can take n=1 here?

    The Homework Statement reads the question. We have $$ \langle f,g \rangle = \sum_{k=0}^{n} f\left(\frac{k}{n}\right) ~g\left( \frac{k}{n} \right) $$ If ##f(t) = t##, we have degree of ##f## is ##1##, so, should I take ##n = 1## in the above inner product formula and proceed as follows $$...
  4. Semiclassical

    Quartic function of a non-ideal spring

    I'm stuck in a part of my problem where I need to find the roots of this function which represent turning points for a non-ideal spring.
  5. appmathstudent

    I Rodrigues' Formula for Laguerre equation

    This is exercise 12.1.2 a from Arfken's Mathematical Methods for Physicists 7th edition : Starting from the Laguerre ODE, $$xy''+(1-x)y'+\lambda y =0$$ obtain the Rodrigues formula for its polynomial solutions $$L_n (x)$$ According to Arfken (equation 12.9 ,chapter 12) the Rodrigues...
  6. G

    How to expand this ratio of polynomials?

    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...
  7. Lauren1234

    Rook polynomial

    This is my solution however I feel like the number is far too big can anyone see what I’ve done wrong
  8. C

    Python How can I evaluate a Chebishev polynomial in python?

    Hello everyone. I need to construct in python a function which returns the evaluation of a Chebishev polynomial of order k evaluated in x. I have tested the function chebval form these documents, but it doesn't provide what I look for, since I have tested the third one, 4t^3-3t and import numpy...
  9. R

    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...
  10. R

    B Roots of polynomials

    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...
  11. SamRoss

    B Trouble with polynomial long division

    I'm reading a book where the author gives the long division solution of ##\frac 1 {1+y^2}## as ##1-y^2+y^4-y^6...##. I'm having trouble duplicating this result and even online calculators such as Symbolab are not helpful. Can anyone explain how to get it?
  12. YoungPhysicist

    Find the function that matches the equation

    Homework Statement ##3f(x)+2f(\frac{1}{x}) = x##, solve ##f(x)## Homework Equations Not sure.Maybe the ones of inverse functions. The Attempt at a Solution The only thing that I came up so far is that the function’s highest order term is ##x## because if there are higher orders,it will show...
  13. YoungPhysicist

    B Is this a valid proof?

    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?
  14. 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
  15. 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...
  16. 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##...
  17. 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...
  18. 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...
  19. 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...
  20. N

    I Problem when evaluating bounds....Is the result 1 or 0^0?

    Consider the summation ∑,i=0,n (t^(n-i))*e^(-st) evaluated from zero to infinity. You could break down the sum into: (t^(n))*e + (t^(n-1))*e + (t^(n-1))*e + ... + (t^(n-n))*e ; where e = e^(-st) To evaluate this, notice that all terms will go to zero when evaluated at infinity However, when...
  21. B

    Rescaling Curves

    All variables and given/known data and Relevant equations: So I got the functions for a bottle design (one side with the bottle lying horizontally): 1. y=-1/343x^3+3/98x^2 + 2.5 ; 0<x<7 2. y=3; 7<x<15 3. y=-1/98x^2+15/49x+69/98; 15<x<22 Combined they give the volume of 570.2mL using the volume...
  22. B

    Water Bottle Design Using Polynomials

    Homework Statement [/B] I am to design a 600mL water bottle by drawing one side (bottle lying horizontally). Three types of functions must be included (different orders). The cross-sectional view would be centred about the x-axis, and the y-axis would represent the radius of that particular...
  23. D

    I Need help solving for X in third order polynomial

    Hello I have a third order polynomial, for example y(x) = -60000x^3 - 260x^2 + 780x + 0.6 I need to know what is x at y = 28 and/or y= 32. I can goto MATLAB and find the roots ( x = - .1158, -.0007, and .1122 ) or I can go to http://www.wolframalpha.com and it also finds the roots and...
  24. pairofstrings

    B What are the applications of roots of a polynomial?

    Hello. Assume that I have two polynomials of degree 2, i.e., Quadratic Equations. 1. Assume that the Quadratic Equation is: x2 + 7x + 12 = 0 The roots of the Quadratic Equation is -3 and -4. 2. Assume that there is another Quadratic Equation: x2 + 8x + 12 = 0 The roots of the Quadratic...
  25. M

    I Polynomial division

    Hello everyone. Iam working on a course in digital control systems and by reading my textbook I stumbled over this expression. C(z) = 0.3678z + 0.2644 : z^2 − 1.3678z + 0.3678 = 0.3678z^−1 + 0.7675z^−2 + 0.9145z^−3 + ... Now Iam wondering how the result of the polynomial division is...
  26. CynicusRex

    Prove: polynomial is uniquely defined by three of its values

    Homework Statement Algebra - I.M. Gelfand, Problem 164. Prove that a polynomial of degree not exceeding 2 is defined uniquely by three of its values. This means that if P(x) and Q(x) are polynomials of degree not exceeding 2 and P(x1) = Q(x1), P(x2) = Q(x2), P(x3) = Q(x3) for three different...
  27. S

    Verifying subspaces

    Homework Statement Determine if the following is a subspace of ##P_3##. All polynomials ##a_0+a_1x+a_2x^2+a_3x^3## for which ##a_0+a_1+a_2+a_3=0## Homework Equations use closure of addition and scalar multiplication The Attempt at a Solution Let ##P=a_0+a_1x+a_2x^2+a_3x^3## and...
  28. doktorwho

    Solve a polynomial function

    Homework Statement Given the polynomial function ##x^4+x^3+2x^2+4=0## solve it if you know that it has at least one complex zero whose real part equals the complex part. Homework Equations 3. The Attempt at a Solution [/B] My guess is that if this function has one complex zero it must have a...
  29. T

    A How is this 'root stability' differential equation derived?

    I'm currently studying the sensitivity of polynomial roots as a function of coefficient errors. Essentially, small coefficient errors of high order polynomials can lead to dramatic errors in root locations. Referring to the Wilkinson polynomial wikipedia page right...
  30. J

    Can't find all the zeroes of a polynomial

    Homework Statement Help i have a homework quiz done and i simply can't find out how to do the 3rd problem as we haven't even learned how to do it or maybe my notes aren't good or something , however I am close to an A in the class and this would help bring it closer. It asks me: "Find all the...
  31. caters

    Complex zeros to polynomial

    Homework Statement Form a polynomial whose zeros and degree are given below. You don't need to expand it completely but you shouldn't have radical or complex terms. Degree 4: No real zeros, complex zeros of 1+i and 2-3i Homework Equations (-b±√b^2-4ac)/2a The Attempt at a Solution I want...
  32. S

    I If pair of polynomials have Greatest Common Factor as 1 ....

    NOTE: presume real coefficients If a pair of polynomials have the Greatest Common Factor (GCF) as 1, it would seem that any root of one of the pair cannot possibly be a root of the other, and vice-versa, since as per the Fundamental Theorem of Algebra, any polynomial can be decomposed into a...
  33. I

    Solution of "polynomial" with integer and fractional powers

    Hello, I have a question regarding "polynomials" that have terms with interger and fractional powers. Homework Statement I want to solve: $$ x+a(x^2-b)^{1/2}+c=0$$ Homework Equations The Attempt at a Solution My approach is to make a change of variable x=f(y) to get a true polynomial (integer...
  34. T

    Proving theorem for polynomials

    Homework Statement Prove the following statement: Let f be a polynomial, which can be written in the form fix) = a(n)X^(n) + a(n-1)X^(n-1) + • • • + a0 and also in the form fix) = b(n)X^(n) + b(n-1)X^(n-1) + • • • + b0 Prove that a(i)=b(i) for all i=0,1,2,...,n-1,n Homework Equations 3. The...
  35. V

    A problem from polynomials

    Homework Statement [/B] Th value of 'a' for which the equation x3+ax+1=0 and x4+ax+1=0 have a common root is? Homework Equations The Attempt at a Solution i initially thought of subtracting both the equations and then finding x and substituting back in the equation but it did not work.
  36. M

    Confusion about eigenvalues of an operator

    Suppose ##V## is a complex vector space of dimension ##n## and ##T## an operator in it. Furthermore, suppose ##v\in V##. Then I form a list of vectors in ##V##, ##(v,Tv,T^2v,\ldots,T^mv)## where ##m>n##. Due to the last inequality, the vectors in that list must be linearly dependent. This...
  37. Stephanus

    Polynomial 3 degrees

    Dear PF Forum, As we know in polynomial 2 degrees AX2 + BX + C = 0, there's a formula for solving it. What about 3 degrees for example: AX3 + BX2 + CX + D = 0, there's is really no formula for solving it? The only way to solve it is by hand? I have several methods in my head, at least...
  38. stungheld

    Polynomial division homework

    Homework Statement How many pairs of solutions make x^4 + px^2 + q = 0 divisable by x^2 + px + q = 0 Homework Equations x1 + x2 = -p x1*x2= q[/B] The Attempt at a Solution I tried making z = x^2 and replacing but got nowhere. I figure 0,1,-1 are 3 numbers that fit but I am not sure what's...
  39. aikismos

    Question about Terminology

    Just to double check, but if one wanted to, like in partial fraction decomposition, associate literal coefficients of polynomials with corresponding unknowns on the other side of the equation, the justification for this action is the definition of equality of polynomials? EDIT: I know this...
  40. F

    Python How can I input a polynomial equation of infinite terms in P

    I have been given a task to create an interpolating/extrapolating programme. I have completed the programme for linear interpolation (2 points) but now must make it usable for 3 or more points, ie a polynomial of n points. I think I have the equation in general for a polynomial as it is an...
  41. S

    Can a cubic polynomial be solved without arccos?

    I was reviewing the Cardano's method formula for a real cubic polynomial having 3 real roots. It seems that to do so, the arccos (or another arc*) of a term involving the p & q parameters of the reduced cubes must be done, and then followed by cos & sin of 1/3 of the result from that arccos -...
  42. T

    Can't remember how to solve equation with two variables

    Umm from memory I used to use...that triangle: 1 1 1 1 2 1 1 3 3 1 Fibonachii was it? Pathetic I can't even remember the name. To factorise...or was it expand...polynomials...anyway, I don't think that's elevant here. My question is; I had an...
  43. avikarto

    Question on polynomial orders

    I am trying to use a numerical polynomial root finding method, but I am unsure of the order of an expression. For example, if I have something that looks like x2+5x √(x2+3)+x+1=0 what is the coefficient of the second order (and potentially even the first order) term? Is the entire 5x√... term...
  44. N

    Complex Polynomial of nth degree

    Homework Statement Show that if P(z)=a_0+a_1z+\cdots+a_nz^n is a polynomial of degree n where n\geq1 then there exists some positive number R such that |P(z)|>\frac{|a_n||z|^n}{2} for each value of z such that |z|>R Homework Equations Not sure. The Attempt at a Solution I've tried dividing...
  45. D

    Degree of Polynomial

    Can someone just confirm my answers to this easy polynomial question, State the degree and dominant term to f(x)=2x(x-3)^3(x-1)(4x-2) I am working on this online and there is nothing on working on equations like this in the lesson. I believe the degree to be either 2 or 6, as the functions end...