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. P

    Unbounded Entire Function must be Polynomial

    Homework Statement Let f be entire. Then if lim_{z\rightarrow \infty}|f(z)|=\infty then f must be a non-constant polynomial.Homework Equations The Attempt at a Solution So we know f is entire. Thus I suppose it makes sense to go ahead and expand it as a power series centered at zero. Thus...
  2. S

    Proof of Polynomial Countability

    Homework Statement Let P(n) be the set of all polynomial of degree n with integer coefficients. Prove that P(n) is countable, then show that all polynomials with integer coefficients is a countable set. 2. The attempt at a solution For this problem the book gives me a hint that using...
  3. F

    Determine if this polynomial has a repeated factor

    f(t) = t4 - 23 + 3t2 - 2t + 1 in Q[t] Am i right in thinking I just show by the rational root theorem that the only possible roots are +-1 f(+-1) =/= 0 so there are no repeated factors? Seems too easy..
  4. D

    Simplifying Equations with Polynomial denominators/numerators

    Homework Statement I uploaded a picture of the question because I didn't want to confuse people from the way it looks because it is pretty long. http://tinypic.com/r/2itpm39/5 Homework Equations -None- The Attempt at a Solution I went from the original equation and took an 'x'...
  5. Q

    Can you have a formula for every degree of polynomial?

    I have some math people who say you can, and some who say you can't beyond quintic because of the Abel-Ruffini theorem. Which is it? Can I generalize all polynomials? Or at least can I manually make a formula for each individual degree?
  6. L

    Long Division of cubic polynomial

    Homework Statement \frac{x^3+x^2-5x+3}{x^3-3x+2} Homework Equations The Attempt at a Solution well I'm drawing that long division house with x^3-3x+2 on the outside and x^3+x^2-5x+3 on the inside. I'm seeing that x^3 goes into x^3 one time, so i put a 1 on top of the...
  7. T

    Help Finding Roots of Polynomial

    Homework Statement First find all rational zeros of f, then use the depressed equation to find all roots of the equation f(x) = 0. f(x) = x^3 + 5x^2 - 8x + 2Homework Equations The Attempt at a Solution Possible rational zeros: 2, -2, 1, -1 Synthetic division: 1 | 1 5 -8 2 _____1 6 -2...
  8. C

    MHB What is the proof for this polynomial inequality problem?

    I may have posted this back in the Old Country, but: let the polynomial: \[P(x)=x^n+a_1X^{n-1}+ ... + a_{n-1}x+1 \] have non-negative coeficients and \(n\) real roots. Prove that \(P(2)\ge 3^n \) CB
  9. G

    Finding the Value of a in a Polynomial Function Using Remainder Theorem

    Polynomial functions... find "a" Homework Statement When ax3 - 4x2 + 5x - 3 is divided by (x+2) and (x-1), the remainders are equal. Find a. Don't know where to start. A little help? Hints?
  10. J

    Polynomial Rings: Finding 8 Elements with r^2=r

    Homework Statement Find eight elements r \in \mathbb{Q}[x]/(x^4-16) such that r^2=r. Homework Equations N/A The Attempt at a Solution The elements 0+(x^4-16) and 1+(x^4-16) clearly satisfy the desired properties, but I still need six more elements. Can anyone help me figure out a...
  11. J

    Find the inverse of the polynomial.

    Homework Statement Find (f^{-1})'(a) of: f(x)=\sqrt{x^{3}+x^{2}+x+22} ; a=5. Homework Equations (f^{-1})'(a)=\frac{1}{f'((f^{-1})(a))} The Attempt at a Solution Well, I know to find an inverse: I need to set the equation equal to y, solve for x, then swap x and y. But I don't...
  12. J

    How to find the roots of polynomial of a 5.th order

    Homework Statement I have a polynimal equation as this - 0.00000000000049125*T^4 + 0.00000000021358333333333333333333333333333*T^3 + 0.00000290233125*T^2 - 0.032444109375*T + 19.891472013020833333333333333333 Homework Equations The Attempt at a Solution I insert those...
  13. J

    Computing the Galois Group of a Univariate Polynomial (Irreducible or reducible)

    Is it possible to compute the Galois Group of a polynomial manually (without a computer)? If so, can someone please explain how? I can't seem to find any information (aside from computer algorithms) on how to find a Galois Group or how to factor a polynomial modulo a prime. If it helps to...
  14. C

    What are the Inverses of y=x^{2}+4x-6?

    Homework Statement The function y=x^{2}+4x-6 has two inverses. What are they and which domains lead to these inverses? Homework Equations The Attempt at a Solution y=x^{2}+4x-6 x=y^{2}+4y-6 y(y+4)=x+6 Not really sure where to go from here.
  15. R

    Factoring a 3rd order polynomial

    Factoring a 4th order polynomial Homework Statement Example: (jw)^{3}+6(jw)^{2}+5jw+30=0 can be re-written into 6(5-w^{2})+jw(5-w^{2}). The fact that there are two identical (5-w^{2}) is a desirable outcome. Imaginary number j=\sqrt{-1} becomes -1 when raised to the power of 2. Homework...
  16. G

    Question about the permutations of roots as polynomial coefficients

    Ok, so obviously, given some polynomial P(x) of degree r, it has r roots in the complex numbers by the FTOA, and if these roots are u_1, u_2,... it can be written as \begin{array}{l} P(x) = (x - {u_1})(x - {u_2})(x - {u_3}) \cdots \\ P(x) = {x^r} - ({u_1} + {u_2} + {u_3} + \ldots ){x^{r - 1}}...
  17. K

    Exponential grows faster than polynomial

    Homework Statement If k is a natural number, prove that 2^{n} > n^{k} for all n \geq k^{2} + 1.Homework Equations We need to use a proof by induction.The Attempt at a Solution Let's do the case when k = 4. We check the base case directly: 2^{17} = 131072 > 83521 = 17^4 Suppose 2^{n} > n^{4} for...
  18. sankalpmittal

    Problem on quadratic polynomial.

    Homework Statement The graph of the quadratic polynomial , y=ax2+bx+c is as shown below in the figure : http://postimage.org/image/nvkxv74yd/ Then : (A) b2-4ac<0 (B) c<0 (C) a<0 (D) b<0Homework Equations y=ax2+bx+c If y=0 , then ax2+bx+c=0 Then , x = {-b+-sqrt(b^2-4ac)}/2a The Attempt at...
  19. E

    Factoring a third degree polynomial as part of a limits problem

    Homework Statement I have a limit problem, however I do know how to work limits, I guess what I need is more of a refresher on how to work third degree polynomials. The polynomial(s) I am trying to work with are the following: x3-2x2+2x-15 -and- x3-5x2+10x-12 The limit is a limit where x...
  20. J

    Solving a polynomial congruence?

    Homework Statement Find one solution (or prove no solutions exist) to the equation x100 = 3 mod 83, where "=" means "congruent to" Homework Equations Possibly Fermat's theorem: If p is prime and p does not divide a, then ap-1 = 1 mod p. The Attempt at a Solution 83 is...
  21. R

    A Why Question:Taylor Polynomial of e^x over x?

    So I was just wondering why when you approximate using the Taylor Polynomials for something like e^x/x at x = 0 you can just find the approximation for e^x and make it all over x, could you do the same for like e^x/x^2 or e^x/x^3? I hope my question makes sense... thanks
  22. R

    Chebyshev polynomial - induction problem

    Homework Statement Let Tn(x)=cos(narccosx) where x is real and belongs to [-1,1] and n E Z+ Find T1(x). Show that T2(x)=2x^2 - 1. Show that Tn+1 (x) + Tn-1 (x) = 2xTn(x) Hence, prove by induction that Tn(x) is a polynomial of degree n. The Attempt at a Solution Since cosθ=x and arccosx=θ we...
  23. M

    Find the MacLaurin polynomial of degree 5 for F(x).

    Homework Statement Homework Equations The Attempt at a Solution I keep getting the answer 1- 96/4!*x^4. Can't find where I'm going wrong.
  24. A

    Quaternion Polynomial Equation

    I'm working on the following: "Prove that x^2 - 1=0" has infinitely many solutions in the division ring Q of quaternions." The Quaternions are presented in my book in the representation as two-by-two square matrices over ℂ. The book gives that for a quaternion (sorry for the terrible...
  25. A

    Interpolating polynomial for sin([itex]\pi{x}[/itex])

    Homework Statement Consider the function sin(\pix) on [-1,1] and its approximations by interpolating polynomials. For integer n\geq1, let x_{n,j}=-1+\frac{2j}{n} for j=0,1,...,n, and let p_{n}(x) be the nth-degree polynomial interpolating sin(\pix) at the nodes x_{n,0},...,x_{n,n}. Prove that...
  26. J

    Linear Algebra - Linear Transformation of a polynomial

    Homework Statement Let h: \mathbb{P_2} \rightarrow \mathbb{P_2} represent the transformation h(p(x)) = xp'(x) + p(1-x) for every polynomial p(x) \in \mathbb{P_2}. Find the matrix of h with respect to the standard basis \{1, x, x^2\} Homework Equations Matrix A of transformation: {\bf A}...
  27. M

    Appropriateness of Constrained Segmented Univariate Polynomial Regression Model

    Hi all, I've learned that in unconstrained polynomial regression, the optimal order can be determined using two F tests : one to test for the significance of the overall regression, the other to test for the significance of the higher coefficients (assuming the first test passed of course)...
  28. J

    Mathematica [Mathematica] Sorting polynomial terms

    Hi. Can someone explain to me how to sort a Series so that the terms are in increasing powers of the exponent? For example the code: myseries = Normal[ Series[Sqrt[1 - w], {w, 0, 5}]] /. w -> 1/z produces 1-\frac{7}{256 z^5}-\frac{5}{128 z^4}-\frac{1}{16 z^3}-\frac{1}{8 z^2}-\frac{1}{2...
  29. A

    Minimal Polynomial and Jordan Form

    Homework Statement Suppose that A is a 6x6 matrix with real values and has a min. poly of p(s) = s^3. a) Find the Characteristic polynomial of A b) What are the possibilities for the Jordan form of A? c) What are the possibilities of the rank of A? Homework Equations See below...
  30. L

    Factoring polynomial over rings

    Homework Statement Factor x^3+3x+6 over Z5, Z10, Z, Q and ℝ. 2. The attempt at a solution For Z5, I have the roots of x^3+3x+1 in Z5, x=2=-3 and x=3=-4, so x+3 and x+4 are factors. By long division of x^3+3x+1, it is found that x+3 is a repeated factor, so x^3+3x+6=(x+4)(x+3)^2. For Z and...
  31. G

    What Is the Correct Solution to the Differential Equation xy'^2 + yy' = 0?

    xy'^2 + yy' = 0 where y' = dy/dx The answer is C1 = y and C2 = xy but I get this: y'(xy' + y) = 0 where y' = 0 and thus y = C1 For the other solution: xy' + y = 0 y' = -y/x y = -y ln x + C2 C2 = y + y ln x Full question is here: http://www.cramster.com/solution/solution/640396
  32. R

    Polynomial transformation of random variable

    Homework Statement Given a random variable X with a known distribution (e.g. a beta distribution), find the distribution of f(X) = X^2 + X The Attempt at a Solution I've tried the normal approaches: the standard transformation theorem; conditioning on X; Laplace transformation, etc...
  33. D

    Proving Polynomial Irreducibility over Z

    Homework Statement Prove that the polynomial $(x-1)(x-2)...(x-n) + 1$ is irreducible over Z for n\geq 1 and n \neq 4Homework Equations N/AThe Attempt at a Solution Let $f(x) = (x-1)(x-2) \cdots (x-n) + 1$ and suppose $f(x) = h(x)g(x)$ for some $h,g \in \mathbb{Z}[x]$ where $\deg(h), \deg(g) <...
  34. A

    Does this polynomial produce only prime numbers?

    I'm trying to find a general formula for an algebraic equation, I'm studying the behavior of ∏_{i=2}^n(1-\frac{1}{i^m}) for m=3 and so far I've seen that I can find a general formula if n^2 + n + 1 produces only prime numbers. if not, it would get way harder to find a general formula for it by...
  35. D

    Differentiation of a polynomial function

    Hi guys I'm getting into a little trouble when differentiating polynomial functions. How do you differentiate f(x)=ax+b/cx+d ? Is there other ways of calculating this apart from the chain rule ? Thanks for any help.
  36. Peeter

    How to arrive at Bessel function solution to 1D polynomial potential

    My quantum text, leading up to the connection formulas for WKB and the Bohr-Sommerfeld quantization condition states that for \begin{align}u'' + c x^n u = 0 \end{align} one finds that one solution is \begin{align}u &= A \sqrt{\eta k} J_{\pm m}(\eta) \\ m &= \frac{1}{{n + 2}} \\ k^2 &=...
  37. Shackleford

    Analysis: Taylor Polynomial Approximation

    For #4, I'm mostly confident I did it correctly. In determining the error, we're supposed to find the maximum absolute value on an interval I. I set I = (0,2pi). Is that right? http://i111.photobucket.com/albums/n149/camarolt4z28/4-1.png For #5...
  38. S

    Plot with polynomial degree of 14 in Mathematica

    Homework Statement I have list( Depth vs. Deflection) of data which requires plotting with the polynomial degree of 14. Homework Equations I need help to figure out how to plot with polynomial degree. The Attempt at a Solution
  39. G

    Polynomial Subspace Dimension & Basis Calculation

    Homework Statement Let M be a subspace of the vector space \mathbb{R}_2[t] generated by p_1(T)=t^2+t+1 and p_2(T)=1-t^2, and N be a subspace generated by q_1(T)=t^2+2t+3 and q_2(T)=t^2-t+1. Show the dimension of the following subspaces: M+N, M \cap N, and give a basis for each...
  40. M

    Continuity of Polynomial Functions in their Domain

    The first hypothesis is that f is continuous on [a,b]... Is there a more concise mathematical way of saying... "because the function f is a polynomial it is continuous in its domain."? Because I rather not write that on my test it looks sloppy and non professional...
  41. L

    Did I Divide This Polynomial Correctly?

    As part of solving a DE I need to make this friendlier to integrate: \frac{2u}{1+u^{-2}} I figured trying to divide it couldn't hurt. I got: u^{-1}+u^{-3} I can't type out all the steps easily, I'm on a mobile device at the moment. That answer looks suspect, did I do it correctly?
  42. S

    Hermite Polynomial Reccurence Relation

    Homework Statement Prove, using Rodrigues form, that Hn+1=2xHn -2nHn-1 Homework Equations The Rodrigues form for Hermite polynomials is the following: Hn = (-1)nex2\frac{d^n}{dx^n}(e-x2)The Attempt at a Solution Hn+1 = (-1)n+1ex2\frac{d^(n+1)}{dx^(n+1)}(e-x2) where...
  43. M

    Solving Polynomial Functions: Find x in 0<|x|≤1/2

    Homework Statement f:ℝ→ℝ x:→sqrt(x^2+1)-lxl Homework Equations calculate f^-1(]0,1]) The Attempt at a Solution well i chose a y from ]0,1] and tried to find an x that solves the problem like the following. 0<sqrt(x^2+1)-lxl≤1 at the end i got the following...
  44. K

    Finding quadratic maclaurin polynomial

    Homework Statement the question asks to find the quadratic maclaurin polynomial for f(x) Given f(x) = x sin(x)The Attempt at a Solution i know that a maclaurin series is when a=0 in a taylor series. i did the 1st-5th derivatives of f(x) and then used the formula for taylor polynomial and set...
  45. K

    Using Intermediate Value Theorem to prove # of polynomial roots

    I've heard there's a proof out there of this, basically that (I think) you can use the intermediate value theorem to prove that an Nth-degree polynomial has no more than N roots. I'm not in school anymore, just an interested engineer. Does anyone know where I can find this proof or any...
  46. S

    Eigenvalues,charateristic polynomial

    Homework Statement http://img213.imageshack.us/img213/7867/problemeigen.png Uploaded with ImageShack.us Homework Equations http://img339.imageshack.us/img339/771/equationh.jpg Uploaded with ImageShack.usThe Attempt at a Solution so i tried doing the problem i going to use L for lambda i...
  47. M

    Another polynomial function

    Homework Statement f:[1,+∞[→ℝ x→sqrt(x+2)-sqrt(x-1)Homework Equations show that f([1,+∞[)=]0,sqrt(3)]The Attempt at a Solution Any tips on how to start it.
  48. M

    Is the Function f(x) = 2x/(x-1) Both Injective and Surjective?

    Homework Statement f:]1,+∞[→]2,+∞[ x→ 2x/(x-1) Homework Equations Prove that f is injective and serjective. The Attempt at a Solution I already proved that it's injective by stating the injectivity law: for every (a,b)ε]1,+∞[: f(a)=f(b) implies a=b so: 2a/(a-1)=2b/(b-1)...
  49. S

    Expressing a root of a cubic polynomial as a series

    Is there a good way to do this? I have an equation, say x^3 - 4*x^2 + 2, so a=1, b=-4, c=0, d=2. Is there an easy way to express the largest root of such an equation? In this case, the roots are: 3.8661982625090223 -0.65544238154983 0.7892441190408067 But I am trying to find an...
  50. J

    Taylor Polynomial with Remainder Question

    Homework Statement What is the minimal degree Taylor polynomial about x=0 that you need to calculate sin(1) to 3 decimal places? 6 decimal places? Homework Equations R_nx = f^(n+1)(c)(x-a)^(n+1)/(n+1)(factorial) The Attempt at a Solution I have attached my attempt. I am stuck on the...
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