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.

View More On Wikipedia.org
Recent contents Top users
  1. K

    Power series absolute convergence/ Taylor polynomial

    1. What if absolute convergence test gives the result of 'inconclusive' for a given power series? We need to use other tests to check convergence/divergence of the powerr series but the matter is even if comparison or integral test confirms the convergence of the power series, we don't know...
  2. Math Amateur

    MHB Ideals in Polynomial rings - Knapp - page 146

    I am reading Anthony W. Knapp's book, Basic Algebra. On page 146 in the section of Part IV (which is mainly on groups and group actions) which digresses onto rings and fields, we find the following text on the nature of ideals in the polynomial rings \mathbb{Q} [X] , \mathbb{R} [X] , \mathbb{C}...
  3. anemone

    MHB Polynomial Challenge VIII

    A polynomial $P(x)=x^3+mx^2+nx+k$ is such that $n<0$ and $mn=9k$. Prove that the polynomial has three distinct real roots.
  4. anemone

    MHB Can You Crack the Polynomial Challenge VII? Prove 4 Distinct Real Solutions!

    Let $p,\,q,\,r,\,s,\,t$ be distinct real numbers. Prove that the equation $(x-p)(x-q)(x-r)(x-s)+(x-p)(x-q)(x-r)(x-t)+(x-p)(x-q)(x-s)(x-t)+(x-p)(x-r)(x-s)(x-t)+(x-q)(x-r)(x-s)(x-t)=0$ has 4 distinct real solutions.
  5. anemone

    MHB What is the value for $a+b$ in the Polynomial Challenge VI?

    If $a,\,b$ are the two largest real roots of the polynomial $f(x)=3x^3-17x+5\sqrt{6}$, and their sum can be expressed as $\dfrac{\sqrt{m}+\sqrt{n}}{k}$ for positive integers $m,\,n,\,k$, find the value for $a+b$.
  6. TheSodesa

    Polynomial long division -- How does it work?

    Let's say I wanted to do the following calculation: (x^2 + 2x + 1) / (x+1) I've scrolled through some online guides, and they all show how to do it, but not the principle behind it. I'm specifically having trouble with the fact, that instead of dividing the largest degree term with the entire...
  7. anemone

    MHB Polynomial Challenge V: Real Solution Implies $p^2+q^2\ge 8$

    Show that if $x^4+px^3+2x^2+qx+1$ has a real solution, then $p^2+q^2\ge 8$.
  8. H

    Associated Legendre Polynomial Identity

    Does anyone know how to prove this identity? I don't quote understand why the associated Legendre function is allowed to have arguments where |x|>1. h_n(kr)P_n^m(\cos\theta)=\frac{(-i)^{n+1}}{\pi}\int_{-\infty}^\infty e^{ikzt}K_m(k\rho\gamma(t))P_n^m(t)\,dt where \gamma(t)=\begin{cases}...
  9. N

    Lambert W function with rational polynomial

    Hi all, During my research i ran into the following general type of equation: \exp(ax+b)=\frac{cx+d}{ex+f} does anyone have an idea how to go about solving this equation? thx in advance
  10. S

    MHB Doing polynomial long division

    rewrite using polynomial long division \frac{x^3 + 4x^2 + 3}{x+4} so I did x+4 \sqrt{x^3 + 4x^2 + 0x + 3} and got x^2 + 1 - \frac{1}{x+4} What am i doing wrong? How is that incorrect?
  11. anemone

    MHB Can the Polynomial $x^7-2x^5+10x^2-1$ Have a Root Greater Than 1?

    Prove that $x^7-2x^5+10x^2-1$ has no root greater than 1. This is one of my all time favorite challenge problems! :o
  12. anemone

    MHB Polynomial Challenge: Find $k$ Integral Values

    Find all integral values of $k$ such that $q(a)=a^3+2a+k$ divides $p(a)=a^{12}-a^{11}+3a^{10}+11a^3-a^2+23a+30$.
  13. D

    Solve Polynomial Riddle: Find Coefficients & Rank

    Hi guys, My boss gave me a riddle. He says that you have a "black box" with a polynomial inside it like f(x)=a0+a1x+a2x^2+a3x^3 ... you don't know the rank of it or the coefficients a0, a1, a2 ... You do know: all of the coefficients are positive you get to input two x numbers and...
  14. J

    What are Symmetric Functions for Polynomial Roots?

    My question is hard of answer and the partial answer is in the wikipedia, but maybe someone known some article that already approach this topic and the answer is explicited. So, my question is: given: ##A = x_1 + x_2## ##B = x_1 x_2## reverse the relanship: ##x_1 = \frac{A +...
  15. M

    Minimal polynomial and diagonalization of a block matrix

    Homework Statement . Let ##X:=\{A \in \mathbb C^{n\times n} : rank(A)=1\}##. Determine a representative for each equivalence class, for the equivalence relation "similarity" in ##X##. The attempt at a solution. I am a pretty lost with this problem: I know that, thinking in terms of...
  16. T

    Quick Tips for Factoring Polynomials: Solving 7x^2-9X-6 Efficiently

    hi all I am stuyding how to factor equations such as :- 7x^2-9X-6 The problem i have is that it takes me too long to find 2 numbers whose sum is D and the same numbers whoes product is E. Is there any way/tips/guide on how i can achieve this quickly?
  17. K

    Is this Polynomial an SOS Polynomial?

    F = -0.2662*x^6 + 48.19*x^5 - 3424.2*x^4 + 121708*x^3 - 2*e^6*x^2 + 2*e^7*x - 6*e^7;
  18. S

    What Are the Automorphisms of Z[x]?

    Question: What are the automorphisms of Z[x]? I know there are two automorphisms, one of which is the identity map, ø(f(x)) = f(x). What is the other one? ø(f(x)) = -f(x) for all f(x) in Z[x]? Or does it have something to do with the degree or factorization of the polynomials? Please...
  19. J

    Exploring Different Forms of Polynomials in Two Variables

    If a polynomial of 1 variable, for example: P(x) = ax²+bx+c, can be written as P(x) = a(x-x1)(x-x2), so a polynomial of 2 variables like: Q(x,y) = ax²+bxy+cy²+dx+ey+f can be written of another form?
  20. evinda

    MHB Why can we show that an other polynomial is irreducible?

    Hello! :) I am looking at the exercise: Prove that $f(x)=10x^4-18x^3+4x^2+7x+16 \in \mathbb{Z}[x]$ is irreducible in $\mathbb{Q}[x]$. According to my notes,a way to do this is the following: We know that $\forall m>1 \exists $ ring homomorphism $\widetilde{ \phi }: \mathbb{Z}[x] \to...
  21. Saitama

    MHB Show that the polynomial has no real roots

    Problem: Show that the polynomial $x^8-x^7+x^2-x+15$ has no real root. Attempt: I am not sure what should be the best way to approach the problem. I thought of defining $f(x)=x^8-x^7+x^2-x$ because $f(x)+15$ is basically a shifted version of $f(x)$ along the y-axis. So if $15$ is greater than...
  22. Q

    Non Polynomial Hamiltonian Constraint

    1. Is the root(det(q)) term in the Hamiltonian Constraint what makes it non polynomial 2. Is the motivation for Ashtekar Variables to remove the non polynomial terms by replacing the Hamiltonian with a densitised Hamiltonian
  23. U

    Complex numbers polynomial divisibility proof

    I'm not sure whether this should go in this forum or another. feel free to move it if needed Homework Statement Suppose that z_0 \in \mathbb{C}. A polynomial P(z) is said to be dvisible by z-z_0 if there is another polynomial Q(z) such that P(z)=(z-z_0)Q(z). Show that for...
  24. MarkFL

    MHB Synthetic and polynomial long division

    Until now, I have avoided trying to display techniques of division using $\LaTeX$ because there just didn't seem to be a nice way to carry it out. However, we may use the array environment for the display of synthetic and polynomial long division methods very nicely. I will demonstrate how to...
  25. anemone

    MHB Finding $q$ in a Polynomial with Negative Integer Roots

    If $P(x)=x^4+mx^3+nx^2+px+q$ is a polynomial whose roots are all negative integers, and given that $m+n+p+q=2009$, find $q$.
  26. I

    Hey where do i begin with this complex polynomial question?

    Homework Statement Let f(x), g(x), and h(x) be polynomials in x with real coefficients. Show that if (f(x))^2 −x(g(x))^2 =x(h(x))^2, then f(x) = g(x) = h(x) = 0. Find an example where this is not the case when we use polynomials with complex coefficients. i have no idea where to start...
  27. A

    Roots of a squared polynomial ( complex numbers)

    Homework Statement problem in a pic attached Homework Equations The Attempt at a Solution i solved i and ii a , when it came to b , i just said that every one of the 3 roots will be squared having 2 roots 1 + and 1 - but then i read the marking schemes ( also attached) , and i got...
  28. S

    Several difficult problems on polynomial remainder/factor theorems

    Homework Statement I am currently working through a chapter on Polynomial Remainder and Factor Theorems in my book, Singapore College Math, Syllabus C. There were a few problems which I got stuck on: 25) The positive or zero integer ##r## is the remainder when the positive integer...
  29. J

    Is x^2+1 Irreducible Over Finite Field F_2?

    Homework Statement Is f(x)=x^2+1 irreducible in \mathbb{F}_2[x] If not then factorise the polynomial. The Attempt at a Solution \mathbb{F}_2[x]=\{0,1\} f(0)=1 f(1)=1+1=0 Hence the polynomial is not irreducible
  30. Maxo

    Polynomial approximation to find function values

    Homework Statement If we have the following data T = [296 301 306 309 320 333 341 349 353]; R = [143.1 116.3 98.5 88.9 62.5 43.7 35.1 29.2 27.2]; (where T = Temperature (K) and R = Reistance (Ω) and each temperature value corresponds to the resistor value at the same position) Homework...
  31. S

    Difficult polynomial question involving factor and remainder theorems

    Homework Statement Prove that ##(a-b)## is a factor of ##a^5-b^5##, and find the other factor. Homework Equations Remainder theorem : remainder polynomial ##p(x)## divided by ##(x-a)## is equal to ##p(a)## Factor theorem : if remainder = 0, then divisor was a factor of dividend...
  32. D

    Is S a Subspace of P_3 and Does q(x) Belong in S?

    Could someone help me with this question? Because I'm stuck and have no idea how to solve it & it's due tomorrow :( Let S be the following subset of the vector space P_3 of all real polynomials p of degree at most 3: S={p∈ P_3 p(1)=0, p' (1)=0} where p' is the derivative of p...
  33. N

    Solving a 4-th degree polynomial

    Homework Statement This is part of a linear algebra problem. I've found the characteristic polynomial of a matrix however it's a degree 4 polynomial and I'm having trouble solving it Homework Equations λ4+λ3-3λ2-λ+2 = 0 The Attempt at a Solution I replaced λ2 with a and did...
  34. JJBladester

    Find a message given a CRC and generating polynomial

    I am working on a circuit that inputs a 31-bit pseudo-random binary string into a CCIT CRC-16 block which generates a 16-bit CRC output. I know that M(x)/G(x) = Q(x) + R(x) and the transmitted code will be R(x) appended to M(x). When I simulated the circuit, I got a CRC of 1 0 1 0 1 0 0 1...
  35. P

    MHB Find the zeros of the polynomial function and state the multiplicity of each

    f(x)= x (x+2)^2 (x-1)^4Zeros would be: 0, -2, 1 Multipicity of : 1 2 4Then for y- intercept: f(0)=0 And don't know how to graph it...
  36. anemone

    MHB What is the value of $\dfrac{f(-5)+f(9)}{4}$ in the Polynomial Challenge III?

    Let $f(x)=x^4+px^3+qx^2+rx+s$, where $p,\,q,\,r,\,s$ are real constants. Suppose $f(3)=2481$, $f(2)=1654$, $f(1)=827$. Determine the value of $\dfrac{f(-5)+f(9)}{4}$.
  37. Illuvitar

    Finding a polynomial function given zeros

    Hey guys I am having a little bit of trouble with using and understanding the linear factorization theorem to find the polynomial function. Homework Statement Find an nth degree polynomial function with real coefficents satisfying the given conditions. n=3; -5 and 4+3i are zeros...
  38. C

    Product of the conjugates of a polynomial

    Two days ago, I was absolutely certain to have proved the theorem hereafter. But then, micromass pointed out that another theorem the truth of which I was also certain was in fact false. It seems that in mathematics, never be certain until other mathematicians are. This is the reason why I...
  39. anemone

    MHB What is the equation that guarantees a non-real root for every real number p?

    Show that the equation $8x^4-16x^3+16x^2-8x+p=0$ has at least one non-real root for every real number $p$ and find the sum of all the non-real roots of the equation.
  40. V

    Therefore, D is diagonalizable if and only if P_n(R) = ker(D).

    Please see attached. From part (a), we know that kernel (D) is a constant function, i.e f(x)=c, say From part (d), we know that eigenvalue of D is zero My question: For part (e), Is it correct to say that D is diagonalizable if and only if P_n (R) = ker (D) ?? so the only solution is...
  41. S

    MHB Polynomial Long Division w Integrals

    I do not understand how I would do this with long division since there is only 2 terms. I can't remember the trick. Here is what I have so far. \int \frac{3x^2 - 2}{x^2 - 2x - 8} dx so I got \int 3 + \frac{x^2 - 2}{(x - 4)(x + 2)} I'm not sure if that's right? I just factored it out instead...
  42. M

    General polynomial transformation (transformation matrices).

    Homework Statement A polynomial of degree two or less can be written on the form p(x) = a0 + a1x + a2x2. In standard basis {1, x, x2} the coordinates becomes p(x) = a0 + a1x + a2x2 equivalent to ##[p(x)]_s=\begin{pmatrix}a0\\ a1\\ a2 \end{pmatrix}##. Part a) If we replace x with...
  43. C

    Polynomial Algebra: Show Alpha is Power of Prime p

    Homework Statement Let f(x) = anxn + an-1xn-1 + ... + a1x + a0 be a polynomial where the coefficients an, an-1, ... , a1, a0 are integers. Suppose a0 is a positive power of a prime number p. Show that if \alpha is an integer for which f( \alpha ) = 0, \alpha is also a power of p. Homework...
  44. P

    MHB Polynomial including Sigma Notation

    Hello everyone! I have this polynomial: $p(x) =$ 1 + \sum_{k=1}^{13}\frac{(-1)^k}{k^2}x^k - I'm supposed to show that this polynomial must have at least one positive real root. - I'm supposed to show that this polynomial has no negative real roots. - And I'm supposed to show that if $z$ is...
  45. evinda

    MHB Finding Minimum Value of Polynomial Function f

    Hi! I have also an other question (Blush) Knowing that $f$ is a polynomial function,how can I show that there is a $y \in \mathbb{R}$,such that $|f(y)|\leq |f(x)| \forall x \in \mathbb{R}$ ?
  46. P

    MHB Polynomial that includes complex numbers

    Hello everyone, I'm new to this forum. I have this Linear Algebra question that I have no clue how to solve. Any help would be much appreciated. :) The question goes as follows: The polynomial p(x) = x3 + kx + (3 - 2i) where k is an unknown complex number. It is given to you that if p(x) is...
  47. evinda

    MHB No problem! Always happy to help with math questions.

    Hello! :) The interpolating polynomial that interpolates at the following data: $f(5)=?,f(8)=14,f(12)=214$ is $4,125x^{2}-32,5x+10$. The corresponding interpolating polynomial in the Newton form is $p_{2}(x)=a_{0}+a_{1}(x-5)+a_{2}(x-5)(x-3)$.Which is the value of $a_{2}$? Is it 4,125 because...
  48. S

    Proving the Inequality: x^4+x^3y+x^2y^2+xy^3+y^4 > 0 for x,y>0

    Prove that if x and y are not both , then x^4+x^3y+x^2y^2+xy^3+y^4 > 0 I have no idea how to start this proof, can anyone give me an idea?
  49. E

    Approximating function by trigonometric polynomial

    Hi! Say that we wish to approximate a function f(x), \, x\in [0, 2\pi] by a trigonometric polynomial such that f(x) \approx \sum_{|n|\leq N} a_n e^{inx} \qquad (1) The best approximation theorem says that in a function space equipped with the inner product (f,g) = \frac{1}{2...
  50. C

    Linear algebra find the minimal polynomial

    Homework Statement M: V -> V linear operator st M^2 + 1_v = 0 find the POSSIBILITIES for min. pol. of M^3+2M^2+M+3I_v Homework Equations The Attempt at a Solution using M^2 = -1_v, i rewrote the operator(?) as M^3 + M + I_v i don't know what to do. i guessed min poly to...
Back
Top