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. M

    When do roots of a polynomial form a group?

    I've been studying for my final exam, and came across this homework problem (that has already been solved, and graded.): "Show that the Galois group of ##f(x)=x^3-1## over ℚ, is cyclic of order 2." I had a question related to this problem, but not about this problem exactly. What follows is...
  2. Fantini

    MHB Legendre Polynomial and Legendre Equation

    Given $f(x) = (x^2-1)^l$ we know it satisfies the ordinary differential equation $$(x^2-1)f'(x) -2lx f(x) = 0.$$ The book defines the Legendre polynomial $P_l(x)$ on $\mathbb{R}$ by Rodrigues's formula $$P_l(x) = \frac{1}{2^l l!} \left( \frac{d}{dx} \right)^l (x^2-1)^l.$$ I'm asked to prove by...
  3. A

    Calculating DFT for specific polynomial

    Hello everyone, This seems like a simple problem but I get the impression that I'm missing something. 1. Homework Statement Given the values ## v_1,v_2,...,v_n ## such that DFTn ## (P(x)) = (v_1, v_2, \ldots, v_n) ## and ##deg(P(x)) < n##, find DFT2n## P(x^2)## Homework EquationsThe Attempt...
  4. MisterH

    Curve extrapolation: polynomial or Bézier?

    On a stationary, non-periodic signal (black) a smooth causal filter is calculated (green/red). It is sampled discretely (every distance unit of 1 on the X-axis). My goal is to find which "path" it is "travelling" on so I can extrapolate the current shape until it is completed (reaches a...
  5. K

    MHB Find Min Polynomial of $\alpha$ Over $\mathbb{Q} | Solution Included

    I started by setting $\alpha= e^{2\pi i/3} + \sqrt[3]{2}.$ Then I obtained $f(x) = x^9 - 9x^6 - 27x^3 - 27$ has $\alpha$ as a root. How can I proceed to find the minimal polynomial of $\alpha$ over $\mathbb{Q},$ and identify its other roots?
  6. 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...
  7. C

    Critical points and of polynomial functions

    Homework Statement A rectangular region of 125,000 sq ft is fenced off. A type of fencing costing $20 per foot was used along the back and front of the region. A fence costing $10 per foot was used for the other sides. What were the dimensions of the region that minimized the cost of the...
  8. K

    MHB Irreducible polynomial of ζ_6, ζ_8, ζ_9 over the field Q(ζ_3).

    How can I get started on finding the irreducible polynomial of $\zeta_6, \zeta_8, \zeta_9$ over the field over $\mathbb{Q}(\zeta_3)?$ Should I construct field extensions and then use the degrees of the extensions? This question has been crossposted here: abstract algebra - Finding the...
  9. E

    MHB Find the minimal polynomial of some value a over Q

    I'm trying find the minimal polynomial of a=3^{1/3}+9^{1/3} over the rational numbers. I am currently going about this by trying to construct a polynomial from a (using what I intuitively feel would be a sufficiently small number of operations). Then I'd show it's irreducible by decomposing it...
  10. N

    MHB Can a Non-Integer Exponent be Used to Solve a Polynomial Equation?

    Not sure if this is the right place but could somebody help me solve the following equation B.x^b - x - A =0 wher A, B and b are constants. Thanks
  11. D

    Confirm Degree & Dominant Term of Polynomial Equation

    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...
  12. M

    Orthogonality of Associated Laguerre Polynomial

    I have a problem when trying to proof orthogonality of associated Laguerre polynomial. I substitute Rodrigue's form of associated Laguerre polynomial : to mutual orthogonality equation : and set, first for and second for . But after some step, I get trouble with this stuff : I've...
  13. evinda

    MHB Two different algorithms for valuation of polynomial

    Hello! (Wave) The following part of code implements the Horner's method for the valuation of a polynomial. $$q(x)=\sum_{i=0}^m a_i x^i=a_0+x(a_1+x(a_2+ \dots + x(a_{m-1}+xa_m) \dots ))$$ where the coefficients $a_0, a_1, \dots , a_m$ and a value of $x$ are given: 1.k<-0 2.j<-m 3.while...
  14. S

    MHB Is |H_n(x)| Always Less Than or Equal to |H_n(ix)| for Hermite Polynomials?

    How to prove that |H_n(x)|<=|H_n(ix)| where H_n(x) is the Hermite polynomial?
  15. T

    Estimate number of terms needed for taylor polynomial

    Homework Statement For ln(.8) estimate the number of terms needed in a Taylor polynomial to guarantee an accuracy of 10^-10 using the Taylor inequality theorem. Homework Equations |Rn(x)|<[M(|x-a|)^n+1]/(n+1)! for |x-a|<d. The Attempt at a Solution All I've done so far is take a couple...
  16. B

    Polynomial fractions simplest form?

    I was taught that when you have a polynomial fraction where the denominator is of a higher degree than the numerator, it can't be reduced any further. This seems wrong to me for a couple of reasons. 1. If the denominator can be factored some of the terms may cancel out 2. Say you have the...
  17. I

    Understanding square root of a polynomial

    Hello This is not exactly a homework problem. I was browsing through an old book, "Elementary Algebra for Schools" by Hall and Knight, first published in England in 1885. The book can be found online at https://archive.org/details/elementaryalgeb00kniggoog . I was studying the process of...
  18. anemone

    MHB Polynomial Challenge: Show $f(5y^2)=P(y)Q(y)$

    Given that $f(x)=x^4+x^3+x^2+x+1$. Show that there exist polynomials $P(y)$ and $Q(y)$ of positive degrees, with integer coefficients, such that $f(5y^2)=P(y)\cdot Q(y)$ for all $y$.
  19. PsychonautQQ

    Finding inverse in polynomial factor ring

    Homework Statement find the inverse of r in R = F[x]/<h>. r = 1 + t - t^2 F = Z_7 (integers modulo 7), h = x^3 + x^2 -1 Homework Equations None The Attempt at a Solution The polynomial on bottom is of degree 3, so R will look like: R = {a + bt + ct^2 | a,b,c are elements of z_7 and x^3 = 1 -...
  20. M

    Polynomial Long Division for Limit Calculation

    Homework Statement \frac{x^5-a^5}{x^2-a^2}, where a is some constant. Homework EquationsThe Attempt at a Solution I can't figure out how to do this with long division. With synthetic, I can get to \frac{a^4+a^3 x+a^2 x^2+a x^3+x^4}{a+x} x^3+xa^2+...
  21. anemone

    MHB Solving Cubic Polynomial: Prove Two Distinct Roots

    Let $p,\,q,\,r,\,s,\,t$ be any real numbers and $s\ne 0$. Prove that the equation $x^3+(p+q+r)x^2+(pq+qr+rp-s^2)x+t=0$ has at least two distinct roots.
  22. anemone

    MHB Challenge for Polynomial with Complex Coefficients

    Let $ax^2+bx+c$ be a quadratic polynomial with complex coefficients such that $a$ and $b$ are non-zero. Prove that the roots of this quadratic polynomial lie in the region $|x|\le\left|\dfrac{b}{a}\right|+\left|\dfrac{c}{b}\right|$.
  23. 22990atinesh

    Highest degree of a given polynomial is

    Homework Statement A polynomial p(x) is such that p(0)=5, p(1)=4, p(2)=9 and p(3)=20. the minimum degree it can have a) 1 b) 2 c) 3 d) 4 Homework EquationsThe Attempt at a Solution a) Not Possible can't connect these points using straight line b) Not even possible to connect these points using...
  24. M

    MHB How Does Polynomial Splitting Occur in Finite Field Extensions?

    Hello :o The extension $\mathbb{Z}_p \leq \mathbb{F}_{p^n}$ is normal, as the splitting field of the polynomial $f(x)=x^{p^n}-x$ ($\mathbb{Z}_p$ is a perfect field therefore each polynomial is separable). So, if $a \in \mathbb{F}_{p^n}$, then $q(x)=Irr(a,\mathbb{Z}_p)$ can be splitted over...
  25. R

    Real Solutions of 4th Degree Polynomial Equation

    Homework Statement To find number of real solutions of: ##\frac{1}{x-1}## ##+\frac{1}{x-2}## + ##\frac{1}{x-3}## + ##\frac{1}{x-4}## =2[/B] Homework Equations It will form a 4th degree polynomial equation. The Attempt at a Solution The real solutions could be 0 or 2 or 4 as complex...
  26. anemone

    MHB Is the Remainder of Polynomial $f(x)$ the Same for Two Different Divisors?

    Show that the remainder of the polynomial $f(x)=2008+2007x+2006x^2+\cdots+3x^{2005}+2x^{2006}+x^{2007}$ is the same upon division by $x(x+1)$ as upon division by $x(x+1)^2$.
  27. I

    Factoring a third degree polynomial

    Homework Statement Factor out the polynomial and find its solutions x^3-5x^2+7x-12[/B]Homework EquationsThe Attempt at a Solution I tried to factor it, but I'm stuck in this step x^2(x-5)+7(x-5)+23= 0. I graphed the equation, and I know there is two imaginary solutions and one real positive...
  28. evinda

    MHB Showing $XF_{X}+YF_{Y}+ZF_{Z}=nF$ with a Homogeneous Polynomial

    Hi! (Smile) Let $F(X,Y,Z) \in \mathbb{C}[X,Y,Z]$ a homogeneous polynomial of degree $n$. Could you give me a hint how we could show the following? (Thinking) $$XF_{X}+YF_{Y}+ZF_{Z}=nF$$
  29. RJLiberator

    Evaluating the remainder of a Taylor Series Polynomial

    Homework Statement The goal of this problem is to approximate the value of ln 2. We will use two different approaches: (a) First, we use the Taylor polynomial pn(x) of the function f(x) = lnx centered at a = 1. Write the general expression for the nth Taylor polynomial pn(x) for f(x) = lnx...
  30. RJLiberator

    Basic Taylor Polynomial Question involving e^(-x)^2

    Homework Statement Consider:[/B] F(x) = \int_0^x e^{-x^2} \, dx Find the Taylor polynomial p3(x) for the function F(x) centered at a = 0. Homework Equations Tabulated Taylor polynomial value for standard e^x The Attempt at a Solution [/B] I started out by using the tabulated value for Taylor...
  31. anemone

    MHB Find the smallest possible degree of a polynomial

    Let $h(x)$ be a nonzero polynomial of degree less than 1992 having no non-constant factor in common with $x^3-x$. Let $\dfrac{d^{1992}}{dx^{1992}}\left(\dfrac{h(x)}{x^3-x}\right)=\dfrac{m(x)}{n(x)}$ for polynomials $m(x)$ and $n(x)$. Find the smallest possible degree of $m(x)$.
  32. MartinJH

    Integration of a polynomial problem

    Hi, I'm using KA Stroud 6th edition (for anyone with the same book, P407) and there is a example question where I just can't seem to get the answer they have suggested: Homework Statement [/B] Question: Determine the value of I = ∫(4x3-6x2-16x+4) dx when x = -2, given that at x = 3, I = -13...
  33. ch3cooh

    Polynomial approximation: Chebyshev and Legendre

    Chebyshev polynomials and Legendre polynomials are both orthogonal polynomials for determining the least square approximation of a function. Aren't they supposed to give the same result for a given function? I tried mathematica but the I didn't get the same answer :( Is this precision problem or...
  34. J

    I don't understand polynomial division

    At first he shows 2x+4 / 2 and you just divide both 2x and 4 by 2. But then in the next example he is dividing x^2+3x+6 by x+1 and he doesn't divide x^2 by x+1, 3x by x+1 and 6 by x+1. I do not understand how he does the problem.
  35. T

    Taylor Polynomial of 3rd order in 0 to f(x) = sin(arctan (x))

    The problem is as the title says. This is an example we went through during the lecture and therefore I have the solution. However there is a particular step in the solution which I do not understand. Using the Taylor series we will write sin(x) as: sin(x) = x - (x^3)/6 + (x^5)B(x) and...
  36. M

    MHB The polynomial is irreducible iff the condition is satisfied

    Hey! :o I need some help at the following exercise: Show that the polynomial $f(x)=x^n+1 \in \mathbb{Q}[x]$ is irreducible if and only if $n=2^k$ for some integer $k \geq 0$. Could you give me some hints what I could do?? (Wondering)
  37. C

    Find the constant polynomial g closest to f

    Homework Statement In the real linear space C(1, 3) with inner product (f,g) = integral (1 to 3) f(x)g(x)dx, let f(x) = 1/x and show that the constant polynomial g nearest to f is g = (1/2)log3. Homework EquationsThe Attempt at a Solution I seem to be able to get g = log 3 but I do not know...
  38. E

    MHB Solving Polynomial Inequalities

    Solve the following inequality: 6e) $(x - 3)(x + 1) + (x - 3)(x + 2) \ge 0$ So, I created an interval table with the zeros x-3, x+1, x-3 and x+2 but I keep getting the wrong answer. Could someone help? (this is grade 12 math - so please don't be too complicated). Thanks.
  39. C

    Transform 10 to 1000 Points on x^9 to x^2 Polynomial

    In the above title 10 and 1000 are arbitrary numbers I will use them below to signify the concept of a smaller and larger number. I know that n points are described by at most an x^(n-1) polynomial. What I really mean to ask is: Is it possible to take a "smaller" amount of points say 10, go...
  40. M

    How Do You Formulate a Polynomial for Volume in This Prism Problem?

    Homework Statement A package sent by a courier has the shape of a square prism. The sum of the length of the prism and the perimeter of its base is 100cm. Write a polynomial function to represent the volume V of the package in terms of x. width and height are in x centimeters, length is in y...
  41. datafiend

    MHB Find zeros of polynomial and factor it out, find the reals and complex numbers

    Hi all, f(x) = 3x^2+2x+10 I recognized that this a quadratic and used the quadratic formula. I came up with -1/3+-\sqrt{29}/3. But the answer has a i for imaginary. When I was under the \sqrt{116}, I broke that down, but didn't realize there would be an i Can someone explain that one to me...
  42. anemone

    MHB Find Polynomial Q(x): Remainder -1 & 1

    Determine a real polynomial $Q(x)$ of degree at most 5 which leaves remainders $-1$ and 1 upon division by $(x-1)^3$ and $(x+1)^3$ respectively.
  43. G

    How to find a polynomial from an algebraic number?

    Given some algebraic number, let's say, √2+√3+√5, or 2^(1/3)+√2, is there some way to find the polynomial that will give 0 when that number is substituted in? I know that there are methods to find the polynomial for some of the simpler numbers like √2+√3, but I have no clue where to begin for...
  44. datafiend

    MHB Determine if a function is a polynomial

    I'm going through polynomials and the the problem: g\left(x\right)= (4+x^3)/3 IS NOT A POLYNOMIAL FUNCTION. I don't get it. The answer says x\ne0, it's not a polynomial. How did you deduce that? Going down the rabbit hole...and it's the third week.
  45. O

    MHB Can a polynomial ever just have 2 terms?

    Or does it always have to have MORE THAN 2 like x^2 +x^2 -4a polynomial can never be x^2 - x-^3 Right?
  46. A

    Convert a polynomial to hypergeometric function

    i want to write a hypergeometric function (2F1(a,b;c,x)) as function of n that generate polynomials below n=0 → 1 n=1 → y n=2 → 4(ω+1)y^2-1 n=3 → y(2(2ω+3)y^2-3) n=4 → 8(ω+2)(2ω+3)y^4-6(6+4ω)y^2+3 ... → ... 2F1(a,b;c,x)=1+(ab)/(c)x+(a(a+1)b(b+1))/(c(c+1))x^2/2!+... the...
  47. R

    Calculating P(2013) of Polynomial P(x) of Degree 2012

    P(x) is polynomial of degree 2012, P(k)=2^k, k=0,1,...,2012. Find P(2013)
  48. anemone

    MHB My TOP Favorite Polynomial Challenge

    Like I mentioned in the title, this is probably one of the greatest challenge problems (I've seen so far) that designed for, hmm, well, for a challenge!:o Let $x_1$ be the largest solution to the equation $\dfrac{6}{x-6}+ \dfrac{8}{x-8}+\dfrac{20}{x-20}+\dfrac{22}{x-22}=x^2-14x-4$ Find the...
  49. caffeinemachine

    MHB Theorem: If Polynomials Converge, Roots Also Converge

    Proposition 5.2.1 in Artin states that: THEOREM. Let $p_k(t)\in \mathbf C[t]$ be a sequence of monic polynomials of degree $\leq n$, and let $p(t)\in \mathbf C[t]$ be another monic polynomial of degree $n$. Let $\alpha_{k,1},\ldots,\alpha_{k,n}$ and $\alpha_1,\ldots,\alpha_n$ be the roots...
  50. J

    Irreducible Polynomial of Degree 3

    Homework Statement If p(x) ∈F[x] is of degree 3, and p(x)=a0+a1∗x+a2∗x2+a3∗x3, show that p(x) is irreducible over F if there is no element r∈F such that a0+a1∗r+a2∗r2+a3∗r3 =0. Homework Equations The Attempt at a Solution Is this approach correct? If p(x) is reducible, then there...
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