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. C

    Having trouble understanding minimal polynomial problems

    i understand how to find minimal poly. if a matrix is given. i am curious if you can find the matrix representation if minimal polynomial is given. i'm not exactly sure how you could since you can possibly lose repeated e-values when you write minimal polynomial. how can u create a n...
  2. A

    Determinant of Characterisitic Polynomial

    I'm doing a calculation which finds the characteristic polynomial of a matrix, HH, with rather complex entries and then determines the discriminant of that polynomial. For smaller matrices up to around 7x7 it finishes evaluating the Discriminant command within a few hours, but at a 10x10, which...
  3. B

    Polynomial roots & Mathematical induction

    hi i have this homework question and I am not sure if my thought process is valid. The Question: let a, b and c be roots of the polynomial equation: x^3+px+q=0 and S(n)=(a^n)+(b^n)+(c^n) now prove: that for S(n)= -p(S(n-2))-q(S(n-3)) for n>3my attempt: ------------- first off...
  4. J

    Complex number polynomial, with no root given

    Homework Statement z^3 + (-5+2i)z^2 + (11-5i)z -10+2i =0 has a real root, find all the solutions to this equation. The Attempt at a Solution I have only solved imaginary number polynomials with a given root, but this has no given root, how do I find the real solution? that I can then...
  5. S

    Complex roots of a quartic polynomial

    The polynomial z^4 + 2z^3 + 9z^2 - 52z + 200 = 0 has a root z=-3+4i. Find the other 3 roots. Since the given root is complex, one of the other roots must be the complex conjugate of the given root. So the 2nd root is z=-3-4i. To find the other roots, I divided the polynomial by z^2 + 6z +...
  6. D

    How Do You Tackle a Quartic Equation Like This?

    Homework Statement Solve the following equation: x^4 + 12x^3 + 46x^2 + 60x + 20 = 0 Homework Equations Well, I know how to solve simpler equations, in which the unknown dosen't appear at a power higher than 3. I tried to factor this polynom but I didin't suceed. The Attempt...
  7. bigfooted

    Intersection of straight line with (lagrange) polynomial

    Hi, To calculate the intersection of two straight lines the cross product of the line vectors can be used, i.e. when the lines start in points p and q, and have direction vectors r and s, then if the cross product r x s is nonzero, the intersection point is q+us, and can be found from...
  8. D

    Trigonometric interpolation polynomial

    Homework Statement Let t_j=j/100, a_j=j, b_j=-j, for j=0,1,...,99. Define f(t)=\sum\limits_{k=0}^{99} (a_k\cos(2\pi kt)+b_k\sin(2\pi kt)) Determine the values of c_l, d_m for l= 0,...5, m=1,...,4, so that P(t)=\frac{c_0}{2}+\sum\limits_{k=1}^4 (c_k\cos(2\pi kt)+d_k\sin(2\pi kt))+c_5\cos(10\pi...
  9. M

    Calculate 2nd order Taylor polynomial of a given function

    Homework Statement . Let ##f:\mathbb R^2 \to \mathbb R## / ##f \in C^2##, ##f(0,0)=0## and ##\nabla f(0,0)=(0,1)## ##Df(0,0)=\begin{pmatrix} 1 & 1\\ 1 & 2\\ \end{pmatrix}## Let ##g:\mathbb...
  10. A

    Tenth taylor polynomial for sinx

    Homework Statement So, we are supposed to find the tenth taylor polynomial of sinx. On wolfram, I get a final term with x^11 at the end. How does that make sense?! According to the formula, n=10 so the maximum degree should be 10... Homework Equations Taylor polynomial formula. The...
  11. G

    Finding Zeroes of a Complex Polynomial Inside |z| < 1

    Homework Statement Find the number of zeroes of p(z) = z^5 + 10z - 1 inside |z| < 1 Homework Equations The Attempt at a Solution Let f(z) = z^5 g(z) = 10z-1 On |z| = 1: 10|z| - 1< |10z-1| < 10|z| + 1 (is this true...?) 9 < |g(z)| < 11 |f(z)| = 1 So |g| >...
  12. Math Amateur

    MHB Polynomial Rings - Lemma 1.13 from Sharp: Steps in Commutative Algebra

    I am reading R.Y. Sharp: Steps in Commutative Algebra. Lemma 1.13 on page 7 (see attachment) reads as follows: -------------------------------------------------------------------------------------- 1.13 LEMMA. let R be a commutative ring, and let X be an indeterminate; let T be a commutative...
  13. PhizKid

    Length of a polynomial vector?

    Homework Statement S = {1, x, x^2} Find ||1||, ||x||, and ||x^2||.Homework Equations ##\sqrt{v \cdot v}##The Attempt at a Solution I don't know the components of each vector, so how can I perform the dot product?
  14. E

    How to determine Tchebysheff polynomial general expression

    Hello! Tchebysheff polynomials are often defined with trigonometric functions: T_m (x) = \begin{cases} \cos(m \arccos (x)) & -1 \le x \le 1\\ \mathrm{cosh} (m \mathrm{arccosh} (x)) & x > 1\\(-1)^m \mathrm{cosh} (m \mathrm{arccosh} |x|) & x < 1 \end{cases} But they are also polynomials, and...
  15. P

    Solving High-Degree Polynomial Functions: A Scientific Approach

    How would I go about solving an Nth degree polynomial function such that N>=5? Ax^{n}+Bx^{n-1}+...+Z = 0
  16. T

    MHB Is there a faster way to find the roots of a polynomial using Horners method?

    I have given polynomial: x^3-8x^2+19x-12 I know how to find the roots with Horners method,i am just wondering if there is an easier and quicker way to find them? Thank you!
  17. J

    Solution to polynomial of unknown degree

    Dear! Is possible to solution a polynomial of kind y = Ax^a + Bx^b ? Thx!
  18. lonewolf219

    What is the v(p) polynomial in radial wave function

    I think the solution to the radial schrodinger equation includes a form of the Laguerre polynomials, the polynomial v(ρ). Does anyone know what this v(ρ) polynomial is called? The only information my book gives is: "The polynomial v(ρ) is a function well known to applied mathematicians."...
  19. S

    Fundamental Polynomial Operations

    Hi guys! I'm kind of stuck in my review here in Quantitave on the Polynomial part :confused: Homework Statement A. The first problem I had is Problem 7 on the link I will provide which has this. (x + y)3 + (x-y)3 = ? B. The second problem I had is Problem 8 on the same link I will...
  20. X

    Taylor Polynomial of Ln (x)

    I need help understanding why the ln (x) taylor polynomial is (x-1)-1/2(x-1)^2... + etc. I cannot grasp the concept..
  21. T

    MHB Discovering Rational Roots for Simplifying Polynomials

    I wonder what are the tecniques,or what is the easiest way to simplify given polynomial: x^3-9x^2+27x-27 If possible,without Horners algorithm. Thank you!
  22. Mandelbroth

    Ring of Polynomials and Ring of Polynomial Functions

    Recently, I've developed a habit of trying to separate the idea of a function from the idea of the image of the function. This has mostly just confused me, but I am adamant about sticking to it. I think the two terms, "ring of polynomials" and "ring of polynomial functions," are not...
  23. ajayguhan

    Roots of a nth degree polynomial

    why does a nth degree polynomial has atleast one root and a maximum of n root...? In my book it's given, it's the fundamental theorem of algebra. Is there a proof...? Thank's for help. (In advance)
  24. Z

    Software to do polynomial calculation

    Hi, I'm looking for a program that will take [1 + (1+p)]*p and return the unchanged polynomial. i.e. 2p + p^2 I know that MATLAB allows u to do polynomial convolution but for my purposes a program that returns the polynomial in the above format will be much much easier to work with...
  25. paulmdrdo1

    MHB Factoring 4th degree polynomial.

    $\displaystyle x^4+2x^3-8x^2+18x+9$ this is what i tried, $x^4+2x^3-8x^2 = (x^2+4x)(x^2-2x)$ then, $a(x^2+4x)+b(x^2-2x)=18x$ where ab=9 did i set up my solution correctly? can you tell me where I'm wrong.
  26. anemone

    MHB Polynomial Challenge: Find $f(p)+f(q)+f(r)+f(s)$

    The roots of $x^4-x^3-x^2-1=0$ are $p, q, r, s$. Find $f(p)+f(q)+f(r)+f(s)$, where $f(x)=x^6-x^5-x^3-x^2-x$.
  27. Z

    Boolean polynomial and logical statement

    Homework Statement So my professor wanted us to convert the statement: [ (p -> q) & (q -> r) ] -> (p -> r) into a Boolean polynomial Homework Equations The Attempt at a Solution 1. (p -> q) & (q -> r) = (1 + p + pq)(1 + q + qr) = (1 + p + q + pq + qr) 2. [ (p...
  28. D

    MHB Simplifying a Legendre polynomial

    Given the following expression \[ \mathcal{P}_{n}(0) = \left.\frac{1}{2^{n}n!}\frac{d^{n}}{dx^{n}} \sum_{k = 0}^{n}\binom{n}{k}(x^2)^k(-1)^{n - k}\right|_{x = 0}, \qquad (*)...
  29. L

    Proving that an Odd Degree Polynomial Maps $\mathbb{R}$ to $\mathbb{R}$

    Homework Statement Show that if ##f:\mathbb{R}\to \mathbb{R}## is a polynomial function of odd degree, then ##f(\mathbb{R}) = \mathbb{R}##. The attempt at a solution How can I rigorously prove this? What is the most direct and concise way to prove this? What I have is the following...
  30. anemone

    MHB Finding $f(1)$ in a Polynomial of Integer Coefficients $\leq$ 4

    Given $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$, where $a_0, a_a,\cdots,a_n$ are all smaller than 4 and are integer, $a_n \in (0, 1, 2,\cdots)$. Given that $f(4)=2009$, find $f(1)$.
  31. A

    How to perform an integral with a polynomial and a radical?

    Homework Statement Homework Equations The Attempt at a Solution How did they go from the first step in the blue to the second step in the blue? I tried integration by parts but that didn't work.
  32. T

    MHB Find Polynomial Roots: x4-2x3-25x2+50x

    I have to find all solutions for X when: x4-2x3-25x2+50x I have done it so,but I am not sure if this is ok: x(x3-2x2-25x+50) = x(x2(x-2)-25(x+2) = x(x2-25)(x-2) =x(x-5)(x+5)(x-2) Now i see that root/zeroes are +5,-5 and 2. I know that this polynomial has another zero that is 0,but how do i...
  33. F

    Approximating Distance with a Second-Degree Taylor Polynomial

    Homework Statement A car is moving with speed 10m/s and acceleration 1 m/s^2 at the given instant. Using a second-degree Taylor polynomial, estimate how far the car moves in the next second. Homework Equations The Attempt at a Solution I don't get how you're supposed to apply...
  34. J

    MHB No. of real solutions in polynomial equation

    The no. of real solution of the equation $\displaystyle 1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!} = 0$
  35. Seydlitz

    Techniques for Factoring Large Polynomial

    Hello guys, I'd like to ask you how to efficiently factorize complicated polynomial like this one for example: $$\frac{128t^4+128t^3+192t^2+32t+40}{(4t^2+1)^4}$$ I've spend more than a hour trying to decrypt and decompose the polynomial, but to no avail. For simple cubic polynomial I know...
  36. MarkFL

    MHB Ray's question at Yahoo Answers regarding polynomial division

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  37. E

    What are the solutions to x^3+3x^2-4=0?

    Hi guys, somehow after a couple years of not doing math I got a bit rusty... How do I solve x^3+3x^2-4=0 ? I'm kinda stuck? I figured factorizing but I can't seem to find any good factors :/
  38. H

    Factoring 3rd degree polynomial for eigenvalues

    Homework Statement Was given a matrix To find the eigenvalues I set up the characteristic equation [-1-x | 7 | -5 ] [-4 | 11-x | -6 ] [-4 | 8 | -3-x] With some dirty work I got this bad boy out, which I'm having trouble factoring -x3+7x2-15x+9Homework Equations...
  39. Sudharaka

    MHB Minimal Polynomial Finding Algorithm

    Hi everyone, :) This is one of the thoughts that I got after thinking about finding the minimal polynomial of a matrix. I know that the minimal polynomial is easy to find when a matrix is diagonalizable. Then the minimal polynomial only consist all the distinct linear factors of the...
  40. B

    MHB Solve Polynomial: Factoring Tips & Tricks

    any hints on how to start this problem? $12x^4+19x^3-26x^2-61x-28$
  41. Sudharaka

    MHB Eigenvalues and Eigenvectors over a Polynomial Ring

    Hi everyone, :) Here's another question that I solved. Let me know if you see any mistakes or if you have any other comments. Thanks very much. :) Problem: Prove that the eigenvector \(v\) of \(f:V\rightarrow V\) over a field \(F\), with eigenvalue \(\lambda\), is an eigenvector of \(P(f)\)...
  42. paulmdrdo1

    MHB Factoring Polynomials: Start Here

    how would i start factoring this $m^8-n^8-2m^6n^2+2n^6m^2$
  43. paulmdrdo1

    MHB Solving Complex Polynomial Equations

    this stunned me. i stare at these problem for a while and tried my best to factor it but no success. please help me solve this. 1. $x^3+x^2y^2+x^2+xy+y^3+4y-3xy^2-12x$ 2. $3x^2+7xy-3xz-2yz+4y^2-6z^2$
  44. B

    MHB Need help factoring these polynomials?

    can you guys help me factor this polynomial. $\displaystyle 2x^2-4xy+2y^2+5x-3-5y$ $6x^2-xy+23x-2y^2-6y+20$ by the way this is what i tried in prob 1 $2(x-y)^2+5(x-y)-3$ -->> I'm stuck here. and in prob 2 i have no idea where and how to start. thanks!
  45. C

    Can a polynomial model any continuous function?

    If I could use any polynomial up to degree ∞, then can I get a close fit to any continuous function? I know that with a 4th degree polynomial you can get a pretty close fit to the sine function between 0 and 2pi...
  46. C

    Find Minimal Polynomial for Matrix: Solution Help

    Homework Statement Given the matrix 2 0 0 0 0 0 0 1 2 0 0 0 0 0 0 1 2 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 2 What is the minimal polynomial? Homework Equations - The Attempt at a Solution This is the Jordan form, so I guess the solution is just...
  47. O

    Challenge 3b: What's in a polynomial?

    In order to challenge a broader section of the forum, there is a part a and a part b to this challenge - if you feel that part b is an appropriate challenge, then I request you do not post a solution to part a as part a is a strictly easier question than part b. The new challenge: Are there any...
  48. O

    Challenge 3a: What's in a polynomial?

    In order to challenge a broader section of the forum, there is a part a and a part b to this challenge - if you feel that part b is an appropriate challenge, then I request you do not post a solution to part a as part a is a strictly easier question than part b. The challenge: Prove that the...
  49. F

    Determining the least possible degree of a polynomial function

    Homework Statement Determine the least possible degree of the function corresponding to the graph shown below. Justify your answer. Homework Equations The graph is attached. I remade the graph using google grapher, but the graph I got in the test have exactly the same x-intercepts (-2 of order...
  50. C

    MHB Simplifying polynomial fraction

    so I was reading my textbook and was showing steps on applying the quotient rule to the function: y=ex/(1+x2) it went from (1+x2)(ex)-(ex)(2x)/(1+x2)2 to ex(1-x)2/(1+x2)2 I understand the first step, but don't get how they got to ex(1-x)2 in the numerator. can someone please explain the...
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