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. K

    Messy Taylor polynomial question

    Homework Statement Find the Taylor polynomial approximation about the point ε = 1/2 for the following function: (x^1/2)(e^-x)The Attempt at a Solution I'm trying to get a taylor polynomial up to the second derivate i.e.: P2(×) = (×^1/2)(e^-x) + (x-ε) * [(e^-x)(1-2×)/2(×^1/2)] +...
  2. MarkFL

    MHB Zero's question at Yahoo Answers regarding polynomial fitting

    Here is the question: Here is a link to the question: Write a function for the polynomial that fits the following description.? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  3. D

    Lower and Upper bounds of Polynomial equations

    Recently I am studying about theorems regarding to polynomial equations and encounter the lower and upper bounds theorem. Which states that if a<0 and P(a) not equals 0, and dividing P(x) by (x-a) leads to coefficients that alternate signs, then a is a lower bound of all the roots of P(x)=0. The...
  4. MarkFL

    MHB Kkittiee's question at Yahoo Answers involving factoring a cubic polynomial

    Here is the question: Here is a link to the question: Math help: factoring? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  5. I

    MHB Proving Function Polynomial in Coordinates is Differentiable Everywhere

    The question is: Using the chain rule to prove that a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ which is polynomial in the coordinates is differentiable everywhere. (The chain rule is for the use under function composition circumstances, how to apply it here to prove that the function $f$...
  6. N

    Equation with logarithmic and polynomial terms

    This is not actually a homework question, but it seemed appropriate to put it here. In an old exam from 1921 I found the following problem. I never learned how to solve this type of thing and I haven't been able to figure it out, so: how does one solve this? Homework Statement Solve for...
  7. Z

    Taylor Polynomial approximation

    Homework Statement obtain the number r = √15 -3 as an approximation to the nonzero root of the equation x^2 = sinx by using the cubic Taylor polynomial approximation to sinxHomework Equations cubic taylor polynomial of sinx = x- x^3/3!The Attempt at a Solution Sinx = x-x^3/3! + E(x) x^2 =...
  8. W

    Understanding the Error of Taylor Polynomials in Approximating Functions

    the error of a taylor series of order(I think that's the right word) n is given by \frac{f^{n+1} (s)}{n!} (x-a)^n I think this is right. The error in a linear approximation would simply be \frac{f''(s)}{2} (x-a)^2 My question is what is s and how do I find it. Use linear...
  9. B

    Why does polynomial long division work?

    So I'm in a college algebra class and I know how to do polynomial long division. I'm curious as to why polynomial long division works. I've looked at some proofs, but they use scary symbols that I don't understand (I am quite dumb). Do I need very high-level math to comprehend why polynomial...
  10. S

    MATLAB MATLAB - Find the error on polynomial fit parameters of experimental data

    See attached PDF for details: How do I calculate errors on the fit parameters, p?
  11. phosgene

    Finding a cubic polynomial that attains a max/min value over an open interval

    Homework Statement Give an example of a cubic polynomial, defined on the open interval (-1,4), which reaches both its maximum and minimum values. Homework Equations - The Attempt at a Solution I can see that I would need a function such that there is some f(a) and f(b) in...
  12. S

    Linear Polynomial Transformation

    Homework Statement Let T:P_m(\mathbb{F}) \mapsto P_{m+2}(\mathbb{F}) such that Tp(z)=z^2 p(z) . Would a suitable basis for range T be (z^2, \dots, z^{m+2}) ?
  13. A

    The relation between span(In,A,A2, )and it's minimal polynomial

    Let A ∈ Mn×n(F ) Why dim span(In, A, A2, A3, . . .) = deg(mA)?? where mA is the minimal polynomial of A. For span (In,A,A2...) I can prove its dimension <= n by CH Theorem but what's the relation between dim span(In,A,A2...)and deg(mA)
  14. K

    A problem on polynomial fitting

    I encounter a problem on the fitting ability of a special class of multi-variable polynomials. To be specific, I need find whether a special class of multi-variable polynomials, denoted by p(m), where m is the number of variables, can universally and exactly fit all member in another special...
  15. P

    Polynomial with at most n-1 solutions.

    Hi, Homework Statement I am expected to show that the polynomial a1xb1 + a2xb2 + ... + anxbn = 0 has at most n-1 solutions in (0,infinity), where a1,a2,...,an are real numbers different than zero, and b1,b2,...,bn are real numbers so that bj is different than bk for j different than k...
  16. P

    Finite Difference (Interpolating Polynomial)

    Homework Statement http://puu.sh/1QFsA Homework Equations The Attempt at a Solution I'm actually not sure how to do this question. How do i find Δx^2. I kind of understand the question but I don't know how to prove it. I know that Δy becomes dy when the width becomes...
  17. J

    Finding roots to a recursively defined polynomial of degree n

    Hello all, I have a series of polynomials P(n), given by the recursive formula P(n)=xP(n-1)-P(n-2) with initial values P(0)=1 and P(1)=x. P(2)=xx-1=x2-1 P(3)=x(x2-1)-(x)=x3-2x ... I am confident that all of the roots of P(n) lie on the real line. So for P(n), I hope to find these roots. I...
  18. dkotschessaa

    Linear Algebra - Find a Polynomial

    Homework Statement Find all polynomials of the form a + bx + cx^2 that: Goes through the points (1,1) and (3,3) and such that f'(2) = 1Homework Equations a + bx + cx^2 f'(x) = x+2cx f'(2) = 2 + 4c polynomial through (1,1) = a + b1 + c1 = 1 polynomial through (3,3) = a + b3+ c3^2 = 3 The...
  19. P

    Roots of derivative of polynomial.

    Hi, Homework Statement I am asked to prove that given all roots of a polynomial P of order n>=2 are real, then all the roots of its derivative P' are necessarily real too. I am permitted to assume that a polynomial of order n cannot have more than n real roots. Homework Equations...
  20. B

    Taylor Polynomial. Understanding.

    Homework Statement Last exam in my school this exircise was given: From norweagen: " Decide the Taylor polynomial of second degree of x=0 of the function: f(x) = 3x^3 + 2x^2 + x + 1 I found the Taylor polynomial of second degree to be: 2X^2+X+1, which is correct. If I get an...
  21. A

    Puzzled about characteristic polynomial output from my calc

    Hello everyone, first time poster, long time reader here! I'm an ex-math major and while I'm no longer pursuing a degree anymore in mathematics, I still continue onwards in my spare time trying to learn as much as I can about it because it's always been something I've enjoyed partaking in and...
  22. H

    Is There a Positive Constant for a Polynomial Inequality with Two Variables?

    Hi Let p(x,y)≥0 be a polynomial of degree n such that p(x,y)=0 only for x=y=0.Does there exist a positive constant C such that the inequality p(x,y)≥C (IxI+IyI)^n (strong inequality!) holds for all -1≤x,y≤1? The simbol I I stands for absolute value.
  23. L

    Polynomial finite fields; ElGamal decryption

    Homework Statement Given some ElGamal private key, and an encrypted message, decrypt it. Homework Equations Public key (F_q, g, b) Private key a such that b=g^a Message m encrypted so that r=g^k, t=mb^k Decrypt: tr^-a = m The Attempt at a Solution My problem is finding r^-a...
  24. Square1

    Irreducibility of Polynomial part deux

    Ok I promise this time it is not a homework type question. If someone could direct with a name of a theorem here, then I'll go ahead and google it, otherwise have a look. I have a chunk of notes that I'm confused about. We're not shown any proofs here or any explanations, just what is seen. It...
  25. D

    Is this cublc polynomial function solvable?

    Here is a very difficult cubic polynomial. x^3 - x - 2 = 0 I am wondering whether it is solvable or not. Please think about it.
  26. R

    Given min. polynomial of a, find min. polynomial of 1/a

    Homework Statement Given that the minimal polynomial of a over rationals is x^4+x+8, find the minimal polynomial for 1/a over Q. Homework Equations I know there is a lot of work done out there for finding the min. polynomials of a^k for k>0, however I've never seen anything with a^k for...
  27. R

    Taylor Polynomial of Smallest Degree to approximation

    Hey, guys. Having problems with this question because I don't exactly know how to begin it. Homework Statement The problem states to: "Find the Taylor polynomial of smallest degree of an appropriate function about a suitable point to approximate the given number to within the indicated...
  28. C

    What is the Quadratic Maclaurin Polynomial for f(x)=x*sin(x)?

    Homework Statement I'm having a bit of trouble with this Maclaurin Series question. It should be simple enough but I can't get the answer which is given as x2. It's been a while since I've done series and my being rusty is a little annoying. Hopefully someone can help :) Consider...
  29. A

    MHB Polynomial and Rational Functions

    For the years 1998-2009, the number of applicants to US medical schools can be closely approximated by: A(t)= -6.7615t4+114.87t3-240.1t3-2129t2+40,966 where t is the number of years since 1998. a) graph the number of applicants on 0<= t <= 11 b) based on the graph in part a, during what...
  30. Square1

    What role do prime numbers play in proving the irreducibility of polynomials?

    Hi. There is a polynomial f = (x^3) + 2(x^2) + 1, f belongs to Q[x]. It will be shown that the polynomial is irreducible by contradiction. If it is reducible, (degree here is three) it must have a root in Q, of the form r/s where (r,s) = 1. Plugging in r/s for variable x will resolve to r^3 +...
  31. Square1

    Why Do Prime Numbers Play a Role in Proving the Irreducibility of Polynomials?

    Hi. There is a polynomial f = (x^3) + 2(x^2) + 1, f belongs to Q[x]. It will be shown that the polynomial is irreducible by contradiction. If it is reducible, (degree here is three) it must have a root in Q, of the form r/s where (r,s) = 1. Plugging in r/s for variable x will resolve to r^3 +...
  32. L

    How to factor 3rd degree polynomial with 3 terms

    -x^3+12x+16 Every single technique I read about online of how to factor 3rd degree polynomials, it says to group them. I don't think grouping works with this. I tried but it didn't work, since there's only 3 terms. Apparently I'm not supposed to have a cubic variable without a squared...
  33. camilus

    Product and intersection of ideals of polynomial ring

    Let k[x,y,z,t] be the polynomial ring in four variables and let I=<x,y>, J=<z, x-t> be ideals of the ring. I want to show that IJ=I \cap J and one direction is trivial. But proving I \cap J \subset IJ has stumped me so far. Anyone have any ideas?
  34. J

    Discriminant of Characteristic Polynomial > 0

    Homework Statement Show that the descriminant of the characteristic polynomial of K is greater than 0. K=\begin{pmatrix}-k_{01}-k_{21} & k_{12}\\ k_{21} & -k_{12} \end{pmatrix} And k_i > 0 Homework Equations b^2-4ac>0 The Attempt at a Solution I have tried the following...
  35. F

    What is the existence and value of the infimum of a polynomial function?

    Homework Statement Given the function "P" defined by: P(x) := x^2n + a2n-1*x^2n-1 + ... + a1x + a0; prove that there exists an x* in |R such that P(x*) = inf{P(x) : x belongs to | R} Also, prove that: |P(x*)| = inf{|P(x)| : x belongs to |R} The Attempt at a Solution As the...
  36. P

    Aproximating a morse potential using a taylor polynomial

    I am not going to post my question because I want to find out how to actually use the taylor polynomial and morse potential and then apply that to my problem. Say I have to approximate the morse potential using a taylor series expanding about some value. This will then find me the force...
  37. G

    Calculating the Minimal Polynomial for a Given Matrix A: A Guide

    I've been given a matrix A and calculated the characteristic polynomial. Which is (1-λ)5. Given this how does one calculate the minimal polynomial? Also just to check, is it correct that the minimal polynomial is the monic polynomial with lowest degree that satisfies M(A)=0 and that all the...
  38. V

    Roots of a third degree polynomial

    Homework Statement Knowing that the equation: X^n-px^2=q^m has three positive real roots a, b and c. Then what is log_q[abc(a^2+b^2+c^2)^{a+b+c}] equal to? Homework Equations a + b + c = -(coefficient \ of \ second \ highest \ degree \ term) = -k_2 abc = -(constant \ coefficient) =...
  39. D

    MHB What are the roots of this polynomial with a beta coefficient?

    $\beta m^5 + m^2 + 1 =0$ How do I find the roots?
  40. E

    Find the cubic polynomial satisfying f(0) = -5, f(1) = 0, f(2) = 15, f(3) = 52.?

    Here is the correct answer: 2x^3 - x^2 + 4x - 5 My attempt only gives me one cubed term and the other terms are also marginally off, any help on who can show me how to get the correct answer will be hugely appreciated
  41. K

    Legendre Polynomial and Rodrigues' Formula

    I am reading Jackson's electrodynamics book. When I went through the Legendre polynomial, I have a question. In the book, it stated that from the Rodrigues' formula we have Consider only the odd terms...
  42. V

    Roots of a fourth degree polynomial

    Homework Statement z^4 - z^2 + 1 = 0 is an equation in ℂ. Which of the following alternatives is the sum of two roots of this equation: (i) 2√3; (ii) -(√3)/2; (iii) (√3)/2; (iv) -i; (v) i/2 Homework EquationsThe Attempt at a Solution All I know is that the sum of all roots should equal 0...
  43. U

    Condition for this polynomial to be a perfect square

    Homework Statement The condition that x^4+ax^3+bx^2+cx+d is a perfect square, is Homework Equations The Attempt at a Solution If the above polynomial will be a perfect square then it can be represented as (x-\alpha)^2(x-\beta)^2 where α and β are the roots of it.This means that two...
  44. A

    Bernstein's Polynomials for f(x)=x and f(x)=x^2: Sequence and Formula

    Find the sequence (B_nf) of Bernstein's polynomials in a) f(x)=x and b) f(x)=x^2 Answers (from my textbook): a) B_nf(x) = x for all n. b) B_nf(x) = x^2 + \frac{1}{n} x (1-x) I know that the bernstein's polynomial is: B_nf(x) = \sum_{k=0}^n f (\frac{k}{n}) \binom{n}{k} x^k...
  45. 5

    Third degree Taylor polynomial in two variables

    Homework Statement Find the third degree Taylor polynomial about the origin of f(x,y) = \frac{\cos(x)}{1+xy} Homework Equations The Attempt at a Solution From my ventures on the Internet, this is my attempt: I see that \cos(x) = 1 - \frac{1}{2}x^2 + \frac{1}{4!}x^4 - \cdots...
  46. M

    Finding Roots of Bivariate Polynomial Surfaces: A Slice Technique Approach

    Is there a formula for finding the roots of a bivariate polynomial in x and y with the form: (a^2)xy+abx+acy+bc Where a, b, and c are constants, of course.
  47. J

    Finding roots of the derivative of a polynomial.

    hey i'm trying to figure out how to approach part b of this problem, http://imageshack.us/a/img850/6059/asdasdno.jpg so i can see that you can apply the mean value theorem to p'(x) so there exists some c between a and b such that f'(c) = [f(b) - f(a)] / (b-a)=0 so p'(x)...
  48. vrmuth

    Is there formula for zeres of a cubic polynomial

    is there any general formula to find out zeros of a cubic polynomial that will give you all the zeros ? if not please tell me what are the different methods to find out the zeros , guessing and trial and error , numerical etc. i want to see where are each methods useful and is there...
  49. C

    MHB Louis's Question from YahooAnswers:Fp1 Polynomial and roots question Help?

    Question: 1.Find the range of values of \(a\) for which \[(2-3a)x^2+(4-a)x+2=0\]has real roots.2. If the roots of the equation \(4x^3+7x^2-5x-1=0\) are \(\alpha\) , \(\beta\) and \( \gamma\),find the equation whose roots are: (a) \( \alpha+1,\beta+1\) and \(\gamma+1\) (b) \(\alpha^2 \beta^2\)...
  50. Z

    Find a polynomial p(t) of degree 6 which

    Find a polynomial p(t) of degree 6 which has a zero of multiplicity 2 at t = 1 and a zero of multiplicity 3 at t = 2, and also satisfying: p(0) = 2 and p`(0) = 1. What is the other root of p(t)? Attempt at solution: zero of multiplicity 2 at t =1 implies (t-1)^2 is a factor or p(1) = 0...
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