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. J

    Proving Polynomial Property: Get Hint Here

    Homework Statement Prove the following For each real a, the function p given by p(x) = f(x+a) is a polynomial of degree n. Homework Equations Can I have a hint I have hard time starting.Thanks The Attempt at a Solution
  2. Math Amateur

    MHB Polynomial Rings - Irreducibility - Proof of Eisenstein's Criteria

    I am reading Dummit and Foote Section 9.4 Irreducibility Criteria. In particular I am struggling to follow the proof of Eisenstein's Criteria (pages 309-310 - see attached). Eisenstein's Criterion is stated in Dummit and Foote as follows: (see attachment) Proposition 13 (Eisenstein's...
  3. D

    Is there a quick analytic way to solve this polynomial?

    ny = x^(n-1) + x^(n-2) + ... + x + 1 for a certain y and n (>10000) with y!=1 and ny > 1. Is there an analytic way to solve this? Thank you.
  4. J

    Characteristics of a certain degree polynomial.

    I have asked a similar question but it wasn't answered fully so here it goes. Why is it that you only need two points to find a linear equation, three for a quadratic and so on. What determines this and is this determined because of the characteristics of these different degree polynomials?
  5. J

    Finding Polynomial Equations with Points

    Hey guys, I know what polynomials are but what I really don't understand is the way you are able to find the equation to a set amount of points. I don't understand why you have to have a certain amount of points to find different degrees of functions. For example, why are only three points...
  6. E

    Polynomial Division 2: Step-by-Step Solution for (x^3-3x^2+12x-5)/(x-2)"

    Homework Statement Show by polynomial division that \frac{x^3-3x^2+12x-5}{x-2}=(x^2-x+10)+\frac{15}{x-2} The Attempt at a Solution Please see attachment
  7. E

    P.S. I did it on paperPolynomial Division: 3x^3-5x^2+10x+4 / 3x+1

    Homework Statement Use polynomial long division to determine the quotient when 3x^3-5x^2+10x+4 divided by 3x+1 The Attempt at a Solution Please see attachment as I wasn't quite sure how to write my answer here :shy:
  8. Math Amateur

    MHB Polynomial Rings and UFDs - Dummit and Foote pages 303-304

    I am reading Dummit and Foote Section 9.3 Polynomial Rings That are Unique Factorization Domains (see attachment Section 9.3 pages 303 -304) I am working through (beginning, anyway) the proof of Theorem 7 which states the following: "R is a Unique Factorization Domain if and only if R[x] is a...
  9. T

    Fit to (orthgonal?) polynomial function

    Hi all. I need some advice in a project I'm into. I have some experimental (simulation) data and i need to find a function that fits to it. The experimental data behaviour change when I modify some parameters I have. My goal is, from that single function, been able to predict how the...
  10. Q

    Why is (2x^2)/(x^2+1) not dividing evenly?

    I've looked examples up online and I just can't figure out what to do exactly when I have (2x^2)/(x^2+1), for some stupid reason that was probably the work of satan, EVERY problem on the internet only has the lead coefficient of the numerator equal to or less than that in the denominator and...
  11. Math Amateur

    MHB Proving Integer Coefficients in Polynomial Rings w/ Gauss Lemma

    Prove that if f(x) and g(x) are polynomials with rational co-efficients whose product f(x)g(x) has integer co-efficients, then the product of any co-efficient of g(x) with any coefficient of f(x) is an integer. My initial thoughts on this are that the exercise seems to be set up for an...
  12. Math Amateur

    MHB Is Gauss' Lemma the Key to Non-UFDs in Polynomial Rings?

    Exercise 1, Section 9.3 in Dummit and Foote, Abstract Algebra, reads as follows: Let R be an integral domain with quotient field F and let p(x) \in R[x] be monic. Suppose p(x) factors non-trivially as a product of monic polynomials in F[x], say p(x) = a(x)b(x) , and that a(x) \notin R[x] ...
  13. Petrus

    MHB Calc MacLaurin Polynom Grade 3 for \cos(\ln(1+2x-3x^2))

    Calculate MacLaurin-polynom of grade 3 to function \cos(\ln(1+2x-3x^2))if i make Taylor expansion in that ln first is this correct \ln(1+2x-3x^2)=2x-3x^2-\frac{(2x-3x^2)^2}{2}+\frac{(2x-3x^2)^3}{3}... Is that correct? Regards, |\pi\rangle
  14. T

    Proof: Complex entire function bounded by a monomial is a polynomial

    A little explanation here. My professor assigned a homework question without attempting the problem herself. When we were assigned this problem, we were forbidden to use the notion of a Taylor series in our proof (at least not without proving Taylor's Theorem on our own) as we had not covered...
  15. B

    Functions for which f(nx) is a polynomial of f(x).

    What are some examples of functions such that f(nx) = a_{k}f(x)^{k}+...+a_{1}f(x)+a_{0} for some integers n, k, and integer coefficients in the polynomial? The only example I can think of is cos(x), for which \cos(2x) = 2\cos(x)^{2}-1 and there are similar relations for n = 3, 4...
  16. anemone

    MHB Solving Higher Degree Polynomial For Real Solution(s).

    Find real solution(s) to the equation (x^2-9x-1)^{10}+99x^{10}=10x^9(x^2-1)
  17. Math Amateur

    MHB Polynomial Rings, UFDs and Fields of Fractions

    [This item has also been simultaneously posted on MHF] Polynomial Rings, UFDs and Fields of Fractions In Dummit and Foote Section 9.3 Polynomial Rings that are Unique Factorization Domains, Corollary 6, reads as follows...
  18. D

    Help Solving a Quartic Polynomial

    I am trying to solve an equation: arccos(Y) = arctan(Y), where Y = 1/x This turns into a quartic equation: x4 - x2 - 1 = 0 It looks simple enough to simplify further; however, I must be having a brain-fart because I can't do it. I'd like to avoid resorting to a numerical solution...
  19. B

    Volume beneath a two-dimensional polynomial

    Hello Firstly apologies for what seems like an extremely fundamental question, it's been a while since I've done any calculus! I'm currently using a program to fit data with a two dimensional 3 degree polynomial curve( which outputs the fit in the following format) with the aim of...
  20. Petrus

    MHB Solve Polynomial Equation z^6-2z^3+2=0

    Hello MHB, Find all roots to z^6-2z^3+2=0 I can se we there will be 6 roots. I start with subsitute z^3=t so we got t^2-2t+2=0 and we get t_1=1+i and t_2=1-i what shall I do next? Shall I go to polar form? Regards,
  21. J

    Is X^M-N the Minimal Polynomial of Irrational Root \sqrt[M]{N} in \mathbb{Q}[X]?

    Assume that \sqrt[M]{N} is irrational where N,M are positive integers. I'm under belief that X^M-N is the minimal polynomial of \sqrt[M]{N} in \mathbb{Q}[X], but I cannot figure out the proof. Assume as an antithesis that p(X)\in\mathbb{Q}[X] is the minimal polynomial such that \partial p <...
  22. J

    Distinct zeros of irreducible polynomial

    This claim is supposed to be true. Assume that p\in\mathbb{F}[X] is an irreducible polynomial over a field \mathbb{F}\subset\mathbb{C}. Also assume that p(X)=(X-z_1)\cdots (X-z_N) holds with some z_1,\ldots, z_N\in\mathbb{C}. Now all z_1,\ldots, z_N are distinct. Why is this claim true...
  23. I

    Finding the Coefficients of a Taylor Polynomial: A Tricky Integration Question

    Homework Statement Here's a screenshot of the problem: http://puu.sh/2Bta5 Homework Equations The Attempt at a Solution As can be seen by the screenshot, the answer's already given, but I'm not sure how to go about getting it. This one has me stumped since e-4x and sin(5x) are...
  24. 1

    Finding the roots of a high degree polynomial equation

    Homework Statement y(6) - 3y(4) + 3y''-y = 0 Homework Equations The Attempt at a Solution The characteristic equation of that differential equation is: r^6 - 3r^4 + 3r^2 - r = 0 But how am I expected to solve such a high degree polynomial (and thus the DE?)
  25. C

    Abstract Algebra- Finding the Minimal Polynomial

    Homework Statement Given field extension C of Q, Find the minimal polynomial of a=sqrt( 5 + sqrt(23) ) (element of C).Homework Equations The Attempt at a Solution I may be complicating things, but let me know if you see something missing. Doing the appropriate algebra, I manipulated the above...
  26. L

    Compute the second order Taylor polynomial centered at 2 for ln(x)

    Homework Statement a. Compute the second order Taylor polynomial centered at 2, P2(x), for the function ln(x). b. Estimate the maximum error of the answer to part a for x in the interval [1,2].Homework EquationsThe Attempt at a Solution For part a, I'm thinking that when it says "second...
  27. C

    Characteristic polynomial and polynomial vector space

    Homework Statement Let V:= ℝ_{2}[t] V \in f: v \mapsto f(v) \in V, \forall v \in V (f(v))(t) := v(2-t) a) Check that f \in End(V) b) Calculate the characteristic polynomial of f. Homework Equations The Attempt at a Solution a) Is it sufficient to check that (f+g)(t)=f(t)+g(t) ...
  28. Math Amateur

    MHB Polynomial Rings - Gauss's Lemma

    I am trying to understand the proof of Gauss's Lemma as given in Dummit and Foote Section 9.3 pages 303-304 (see attached) On page 304, part way through the proof, D&F write: "Assume d is not a unit (in R) and write d as a product of irreducibles in R, say d = p_1p_2 ... p_n . Since p_1...
  29. D

    How to Determine if a Polynomial Has Six Nonreal Zeros?

    Homework Statement Show that the polynomial function: P(x)=x6+2x4+3x2+4 has six nonreal zeros Homework Equations -none- The Attempt at a Solution I tried using synthetic division with all the possible values it could have(p/q) but none of them worked. I was just wondering...
  30. Math Amateur

    MHB What are the ideals of the ring Z[x]/<2, x^3 + 1>?

    I would like help to get started on the following problem: Determine all the ideals of the ring \mathbb{Z}[x]/<2, x^3 + 1> Appreciate some guidance. Peter
  31. M

    A polynomial of degree ≤ 2 ? what does this mean.

    A polynomial of degree ≤ 2 ? what does this mean. Would it just be a + bt + c t^2 = f(t) Or at^2 + bt + c = f(t) Is there even a difference between the two equations considering the fact that a,b, and c are unknown?
  32. Math Amateur

    MHB Polynomial Rings - Z[x]/(x^2) and Z[x^2 + 1>

    I am trying to get a good understanding of the structure of the rings \mathbb{Z}[x]/<x^2> and \mathbb{Z}[x]/<x^2 +1> . I tried to first deal with the rings \mathbb{R}[x]/<x^2> and \mathbb{R}[x]/<x^2 +1> as they seemed easier to deal with ... my thinking ... and my problems are as...
  33. MarkFL

    MHB Solve Quartic Polynomial: Find Coefficients & Values of q - Yahoo! Answers

    Here is the question: Here is a link to the question: Let P(x) = x4 + ax3 + bx2 + cx + d. Need to find the values of the variable that satisfy the guidelines below? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  34. Math Amateur

    MHB What Does Reducing Z[x] Modulo the Prime Ideal (p) in Polynomial Rings Mean?

    I am reading Dummit and Foote Section 9.2: Polynomial Rings Over Fields I I am having some trouble understanding Example 3 on page 300 (see attached) My problem is mainly with understanding the notation and terminology. The start of Example 3 reads as follows. "If p is a prime, the ring...
  35. MarkFL

    MHB Gabriel's question at Yahoo Answers regarding polynomial division and remainders

    Here is the question: Here is a link to the question: The remainder of f(x)/(x^2+x+1) and f(x)/[(x+1)^2] are x+5 and x-1 respectively.? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  36. B

    Polynomial to represent a linear rectangle element

    Folks, I have attached a picture illustrating the labelling of the linear reactangle element which can be represented by the following equation ##u(x,y)=c_1+c_2 x +c_3 y +c_4 xy## (1) ##u_1=u(0,0)=c_1## ##u_2=u(a,0)=c_1+c_2a## ##u_3=u(a,b)=c_1+c_2a+c_3b+c_4ab##...
  37. N

    Polynomial Linear Transformation

    Let V be the linear space of all real polynomials p(x) of degree < n. If p ∈ V, define q = T(p) to mean that q(t) = p(t + 1) for all real t. Prove that T has only the eigenvalue 1. What are the eigenfunctions belonging to this eigenvalue? What I did was T(p)= (lamda) p = q (Lamda) p(t+1) =...
  38. F

    Factoring Polynomials: Help for a Mathematically Challenged Young Man

    Hey, This isn't really a homework question, per se, as I am relearning some pre calculus for kicks. But I figured his would be the place to ask. For whatever reason, this very simply factoring issue has got my head spinning. I'm not exactly sure what I am doing wrong to factor this equation...
  39. C

    MHB Factor Polynomial: x^3-27, Find A, B, C

    Hello, for homework I was given x^{3}-27 to factor and my answer can be written this way: (x - A)(x^{2} + Bx + C) I am being asked to find the value of A, B and C. I am sure this is simpler than I think but it's perhaps the Bx that is confusing me as I haven't practiced a lot of these examples...
  40. MarkFL

    MHB C.c.'s question at Yahoo Answers regarding factoring a polynomial

    Here is the question: Here is a link to the question: Help with math question please!? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  41. P

    MHB Integral closure of rational polynomial ring

    Consider the ring $\mathbb{Q}[X]$ of polynomials in $X$ with coefficients in the field of rational numbers. Consider the quotient field $\mathbb{Q}(X)$ and let $K$ be the finite extension of $\mathbb{Q}(X)$ given by $K:=\mathbb{Q}(X)[Y]$, where $Y^2-X=0$.Let $O_{K}$ be the integral closure of...
  42. P

    Is There a Polynomial That Meets the Given Criteria?

    Could anyone help me out with this: Which of the following statements are true and which are false? Justify your answers. iii) There exists a polynomial P such that |P(x) - \cos(x)| \leq 10^{-6} I've tried to thinking about it, and it seems as though it is false, because |cos(x)|...
  43. A

    Cubic polynomial unable to be factored nicely

    Found this on a test for an integrated algebra 2 high school math class! Factor completely. 6x3 - 3x2 + 12 The Attempt at a Solution 3( 2x3 - x2 + 4) eq.1 At this point I checked for rational roots using the rational roots theorem and synthetically dividing. I got nothing...
  44. C

    Help needed, rearranging polynomial for inverse equation

    Hi, I need to rearrange an equation: y = ax^2 + bx + c to the form of: x = ? I'm not entirely sure how to go about this and the examples I've found require the equation to be in a different form. Any tips or a point in the right direction would be great! Thanks in advance
  45. A

    Need help solving 3rd order polynomial

    hi i need some help solving this equation for x y=0.0001x^3-0.0409x^2+4.6716x+280.32 can someone please help me, i really need to solve this for x! full layout would be great many thanks!
  46. S

    Taylor polynomial remainder term

    Homework Statement Consider the followign function f(x) = x^-5 a=1 n=2 0.8 \leq x \leq 1.2 a) Approximate f with a tayloy polynomial of nth degree at the number a = 1 b) use taylor's inequality to estimate the accuracy of approximation f(x) ≈ T_{n}(x) when x lies in the interval...
  47. A

    Maximizing Planck's law using Taylor polynomial for e^x

    Homework Statement The energy density of electromagnetic radiation at wavelength λ from a black body at temperature T (degrees Kelvin) is given by Planck's law of black body radiation: f(λ) = \frac{8πhc}{λ^{5}(e^{hc/λkT} - 1)} where h is Planck's constant, c is the speed of light, and...
  48. Z

    How to Shift a Polynomial Curve by a Fixed Distance?

    How to "Offset" a polynomial Suppose I have a function for a curve, for example y=x2. I want to find a function to "offsets" it by 2 units. That is, I want a larger parabola that is exactly 2 units away from my original parabola. What I have in mind is the offset command in AutoCAD. Is...
  49. S

    Dual vector space - Lagrange Interpolating Polynomial

    I think I solved it a week ago, but I didn't write down all the things and I want to be sure of doing the things right, plus the excersise of writing it here in latex helps me a loot (I wrote about 3 threads and didn't submited it because writing it here clarified me enough to find the answer...
  50. R

    How approximate a sextic polynomial to a lower degree polynomial

    Hi all, I have been stopped by a sextic (6th degree) polynomial in my research. I need to find the biggest positive root for this polynomial symbolically, and since its impassible in general, I came up with this idea, maybe there is a way to approximate this polynomial by a lower degree...
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