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

    Differentiating polynomial absolute value function

    For this problem, The solution is, My solution is, Where I solved these equations to find where the function is ##f(x) > 0## and ##f(x) < 0##. Using ##x^2(1 - x) \geq 0## and ##x^2(1 - x) < 0## First equation: ##x^2 \geq 0 \implies x \geq 0## and ## 1 \geq x## Second equation: ##x^2(1...
  2. M

    How did they rewrite this polynomial in this way?

    How can you rewrite polynomial in terms of (x-a) instead of x? One thing came to mind is rewrite each x as x-a+2 (So it is x-2+2 in our example) but this will take long time and a lot of algebra steps, how did they do it very fast in the attached picture? thanks
  3. chwala

    Determining the horizontal asymptote

    Consider, I am self-studying; My interest is on the horizontal asymptote, now considering the degree of polynomial and leading coefficients, i have ##y=\dfrac{2}{1} =2## Therefore ##y=2## is the horizontal asymptote. The part that i do not seem to get is (i already checked this on desmos)...
  4. P

    A Quintic and Higher Degree Polynomial Equations

    What's the root formula for fifth and higher degree polynomial equations, which have roots in radicals?
  5. V

    Roots of a polynomial mixed with a trigonometric function

    When I look at the left hand side of the equation in above question then I can see that the highest degree of x would be 6 after the denominators are eliminated. I know that a polynomial of degree n will have n roots, but this one is not a pure polynomial since there is also a trigonometric...
  6. chwala

    I Solve the problem involving Rings

    Its a bit clear; i can follow just to pick another polynomial say ##(x+1)^3## are we then going to have ##(2x-2)+ x+3##? or it has to be a polynomial with ##x^2+1## being evident? cheers...
  7. pawlo392

    A Differential equation and Appell polynomials

    Hello! Let $n$ be a natural number, $P_n(x)$ be a polynomial with rational coefficients, and $\deg P_n(x) = n$. Let $P_0(x)$ be a constant polynomial that is not equal to zero. We define the sequence ${P_n(x)}_{n \geq 0}$ as an Appell sequence if the following relation holds: \begin{equation}...
  8. chwala

    Show that ##f(x)=2',1',2'## in the irreducible Polynomial

    My interest is on the highlighted; my understanding is that, let ##f(x)=x^3+x^2+2^{'}## then ##f(1^{'})=1{'}+1{'}+2^{'}=4^{'} ## we know that in ##\mathbb{z_3} ## that ##\dfrac{4}{3}=1^{'}## ##f(2^{'})=8^{'}+4^{'}+2^{'}=14{'} ## we know that in ##\dfrac{14^{'}}{3}=1^{'}##... I hope...
  9. B

    I Questions about algebraic curves and homogeneous polynomial equations

    It is generally well-known that a plane algebraic curve is a curve in ##\mathcal{CP}^{2}## given by a homogeneous polynomial equation ##f(x,y)= \sum^{N}_{i+j=0}a_{i\,j}x^{i}y^{j}=0##, where ##i## and ##j## are nonnegative integers and not all coefficients ##a_{ij}## are zero~[1]. In addition, if...
  10. V9999

    I May I use set theory to define the number of solutions of polynomials?

    Let ##Q_{n}(x)## be the inverse of an nth-degree polynomial. Precisely, $$Q_{n}(x)=\displaystyle\frac{1}{P_{n}(x)}$$, It is of my interest to use the set notation to formally define a number, ##J_{n}## that provides the maximum number of solutions of ##Q_{n}(x)^{-1}=0##. Despite not knowing...
  11. chwala

    A Computing the Minimal polynomial - Ring Theory

    Am going through this notes...kindly let me know if there is a mistake on highlighted part. I think it ought to be; ##α^2=5+2\sqrt{6}##
  12. M

    Finding Integer Solutions to Polynomial Equations: Can it be Done Easily?

    Hello, Am re-studying math & calculus aiming to start pure math studying later. However, I got this problem in Stewart calculus. Typically, this is a straightforward IVT application. x = x^3 + 1, call f(x)= x^3 - x + 1 & apply IVT. However I have two things to discuss. First thing is simple...
  13. anemone

    POTW Prove that the roots of a polynomial cannot be all real

    Let ##a,\,b,\,c## and ##d## be any four real numbers but not all equal to zero. Prove that the roots of the polynomial ##f(x)=x^6+ax^3+bx^2+cx+d## cannot all be real.
  14. C

    A FEM basis polynomial order and the differential equation order

    Is there a good rubric on how to choose the order of polynomial basis in an Finite element method, let's say generic FEM, and the order of the differential equation? For example, I have the following equation to be solved ## \frac{\partial }{\partial x} \left ( \epsilon \frac{\partial u_{x}...
  15. chwala

    Find the values of a and b in the given polynomial

    My approach; Let ##(n+1)=k## ##(x-n+1)^3-(x-n)^3=3x^2+ax+b## ##(x-k)^3-(x-n)^3=3x^2+ax+b## ##(x^3-3x^2k+3xk^2-k^3)-(x^3-3x^2n+3xn^2-n^3)## ##=-3x^2k+3x^2n+3xk^2-3xn^2-k^3+n^3≡3x^2+ax+b## ##⇒1=n-k## ##⇒3(k^2-n^2)=a## ##⇒n^3-k^3=b## ##3(k+n)(k-n)=a## ##-3(k+n)=a## ##a=-3(n-1+n)## ##a=-3(2n-1)##...
  16. MathematicalPhysicist

    How to check if a polynomial is irreducible over the rationals

    I first checked for rational roots for this polynomial. The options are ##x=\pm 1/7##, both don't nullify the polynomial thus this polynomial doesn't have rational roots. Now, if it's reducible the only possible factors are: ##(ax^3+bx^2+cx+d)(Ax^3+Bx^2+Cx+D)=7x^6-35x^4+21x-1## or a product of...
  17. M

    B Why Is a Cubic Polynomial Called 'Third Degree'?

    Why is a third degree polynomial called a cubic polynomial? I just don’t see the connection between 3 and a cube.
  18. barryj

    Can a polynomial have an irrational coefficient?

    Is this a polynomial? y = x^2 + sqrt(5)x + 1 I was told NO, the coefficients had to be rational numbers. I this true? It seem to me this is an OK polynomial. I can graph it and use the quad formula to find the roots? so why or why not?
  19. H

    How to convince myself that I can take n=1 here?

    The Homework Statement reads the question. We have $$ \langle f,g \rangle = \sum_{k=0}^{n} f\left(\frac{k}{n}\right) ~g\left( \frac{k}{n} \right) $$ If ##f(t) = t##, we have degree of ##f## is ##1##, so, should I take ##n = 1## in the above inner product formula and proceed as follows $$...
  20. K

    I Proving properties of polynomial in K[x]

    We have Galois extension ## K \subset L ## and element ##\alpha \in L## and define polynomial $$f = \prod_{\sigma \in Gal(L/K)} (x - \sigma(\alpha))$$ Now we want to show that ## f \in K[x] ## which is relatively easy to see because we can take ##\phi(f)## for any ## \phi \in Gal(L/K) ## then...
  21. Semiclassical

    Quartic function of a non-ideal spring

    I'm stuck in a part of my problem where I need to find the roots of this function which represent turning points for a non-ideal spring.
  22. J

    MHB  Finding roots of this particular polynomial

    Hey guys, Nice to be on here. I have been banging my brain for the last two weeks trying to come up with an algebraic solution to the following question - to no avail. Any input would be MUCH appreciated! The problem is somewhat long but can be summarized as follows: Begin with the following...
  23. S

    I Finding a polynomial that has solution (root) as the sum of roots

    AIUI, an algebraic is defined as a number that can be the solution (root) of some integer polynomial, and is any number that can be constructed via any binary arithmetic operation or unary root operation with arguments that are themselves algebraic numbers. I have been able to prove this for...
  24. Y

    Strategies for Solving Polynomial Equations: An Engineer's Approach

    I have a very urgent question, this is the problem. I have no idea how to solve this. I don't even know where to start. This is urgent, please at least tell me what is the name of this kind of problem so I can look it up. Specifically, # 4, 6 and 8. I can guess #4 by dividing both sides by y...
  25. C

    Simple Induction Induction proof of Polynomial Division Theorem

    Theorem: Let ## f(x), g(x) \in \mathbb{F}[ x] ## by polynomials, s.t. the degree of ## g(x) ## is at least ## 1 ##. Then: there are polynomials ## q(x), r(x) \in \mathbb{F}[ x] ## s.t. 1. ## f(x)=q(x) \cdot g(x)+r(x) ## or 2. the degree of ## r(x) ## is less than the degree of ## g(x) ## Proof...
  26. J

    I Is this business graph an exponential or polynomial function?

    GRAPH WITH VALUES: Sorry I have a small dilema, I don't know if this is a exponential or polynomial function. I'd think its exponential but it doesn't have same change of factors.
  27. J

    Substitute PID Controls with a Polynomial Equation/Table?

    So, I had a discussion with a friend of mine, neither of us are in controls but I was curious about an answer here. In a PID controller, we essentially take in an error value, do a mathematical operation on it and determine the input (controller output signal B) needed to the actuator to produce...
  28. M

    C/C++ How to Implement Polynomial Operations in C++?

    Hey! :giggle: I am looking at the following: a) Create a class QuadraticPolyonym that describes a polynomial of second degree, i.e. of the form $P(x)=ax^2+bx+c, a\neq 0$.The coefficients have to be givenas arguments at the construction ofan instance of the class. Implement a method...
  29. appmathstudent

    I Rodrigues' Formula for Laguerre equation

    This is exercise 12.1.2 a from Arfken's Mathematical Methods for Physicists 7th edition :Starting from the Laguerre ODE, $$xy''+(1-x)y'+\lambda y =0$$ obtain the Rodrigues formula for its polynomial solutions $$L_n (x)$$ According to Arfken (equation 12.9 ,chapter 12) the Rodrigues formula...
  30. S

    MATLAB How to Plot a 4th Degree Polynomial in MATLAB: Step-by-Step Guide

    hey everyone . I want to plot a Grade 4 equation in MATLAB. but don't know how to do. Can anyone guide me? equation : f = 1.47*(x^4)-10^7*(x)+58.92*(10^6)
  31. M

    MHB How to show that T_k is the only polynomial of degree k with the specific properties?

    Hey! :giggle: Let $U\subset \mathbb{R}^n$ be an open set and $f:U\rightarrow \mathbb{R}$ is a $k$-times continusouly differentiable function. Let $x_0\in U$ be fixed. The $k$-th Taylor polynomial of $f$ in $x_0$ is $$T_k(x_1,\ldots ,x_n)=\sum_{m=0}^k\frac{1}{m!}\sum_{i_1=1}^n \ldots...
  32. PainterGuy

    Why are there only two roots of this cubic polynomial?

    Hi, I was trying to find roots of the following cubic polynomial and there are only two roots. I believe there should be three roots. Could you please guide me why there are only two roots? If you say that the "1" repeats itself as a root then I'd say the same could be said of "0.9". Thank...
  33. PainterGuy

    Cramer's rule and first degree polynomial curve fitting

    Hi, I did the first degree curve fitting in MATLAB. Please see below which also shows the output for each code line. But I wasn't able to get the same answer using Cramer's rule method presented below. I'm sure MATLAB answer is correct so where am I going wrong with the Cramer's rule method...
  34. anemone

    MHB Roots of Polynomial: Find $\frac{1}{A}+\frac{1}{B}+\frac{1}{C}$

    Let $p,\,q$ and $r$ be the distinct roots of the polynomial $x^3-22x^2+80x-67$. It is given that there exist real numbers $A,\,B$ and $C$ such that $\dfrac{1}{s^3-22s^2+80s-67}=\dfrac{A}{s-p}+\dfrac{B}{s-q}+\dfrac{C}{s-r}$ for all $s\not \in \{p,\,q,\,r\}$. What is...
  35. A

    I Determine if polynomial system has finite number of solutions

    Wish to determine when a system of polynomials has an infinite number of solutions, that is, is not zero-dimensional. The Wikipedia article : System of polynomial equations states: I interpret the quote to mean the system has an infinite number of solutions if the Grobner basis does not have...
  36. Monoxdifly

    MHB Solve Polynomial Division: -5a + 4b = x^2+1 Rem -A-2

    The remainder of p(x)=x^3+ax^2+4bx-1 divided by x^2+1 is –5a + 4b. If the remainder of p(x) divided by x + 1 is –a – 2, the value of 8ab is ... A. -\frac34 B. -\frac12 C. 0 D. 1 E. 3 Dividing p(x) by x^2+1 by x^2+1 with –5a + 4b as the remainder using long division, I got (4bx – 1) – ((a – 1)x...
  37. Eclair_de_XII

    If a polynomial is identically zero, then all its coefficients are 0

    Suppose ##a_0+a_1x+\ldots+a_nx^n=0## and restrict the domain of ##p## to the set of real numbers excluding the roots of ##p##. Note that: if ##a_0 == 0##: ##x=0## is a root of ##p## else: ##x=0## is not a root of ##p## Assume the latter. Subtract ##a_0## from both sides of the equation...
  38. L

    MHB Simplifying lagrange interpolation polynomial

    Now $\sum_{i=0}^{10}(x_{i}+1) L _{10,i}(5) = (x_{0}+1) L _{10,0}(5) + (x_{1}+1) L _{10,1}(5) + ... + (x_{10}+1) L _{10,10}(5)$ Which I can further decompose into $\frac{(x_{0}+1)(5-x_{1})(5-x_{2})...(5-x_{10})}{(x_{0}-x_{1})(x_{0}-x_{2})...(x_{0}-x_{10})} +...
  39. Eclair_de_XII

    Proving that a fifth-degree polynomial has a root using just the IVT

    I consider three cases, based on the sign of ##a_0##. if ##a_0 == 0##: Set ##x=0##. \begin{align*} f(0)&=&a_0+a_1\cdot 0+a_2\cdot 0^2+a_3\cdot0^3+a_4\cdot0^4+0^5\\ &=&a_0+0\\ &=&0+0\\ &=&0 \end{align*} elif ##a_0<0##: Define ##M=\max\{|a_i|:1\leq a_i\leq 5\}## and set ##x=5(M+1)\neq 0##...
  40. A

    B Methods to compute bounds on polynomial roots (not close yet)

    Consider an example polynomial: $$ \begin{align*} P_{16}(z)&=0.0687195 z^{16}+0.787411 z^{15}+4.58749 z^{14}+17.7271 z^{13}+50.5007 z^{12}\\ &+111.995 z^{11}+199.566 z^{10}+291.128 z^9+351.292 z^8+351.927 z^7+292.066 z^6\\ &+199.046 z^5+109.514 z^4+47.2156 z^3+15.1401 z^2+3.25759 z+0.362677...
  41. anemone

    MHB Prove Root of Polynomial $P(x)=x^{13}+x^7-x-1$ Has 1 Positive Zero

    Prove that the polynomial $P(x)=x^{13}+x^7-x-1$ has only one positive zero.
  42. M

    MHB What are Vieta's Formulas in Polynomial Functions?

    I say the answer is A.
  43. F

    A Problem calculating arbitrary Polynomial Chaos polynomials using SAMBA

    Hello everyone. I have recently read the following article (which title is SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos) since I have some data in the form of a histogram without knowing the probability distribution function of said data. I have been able to calculate...
  44. anemone

    MHB Can All Roots of a Quartic Polynomial Be Real?

    Let $a$ and $b$ be real numbers such that $a\ne 0$. Prove that not all the roots of $ax^4+bx^3+x^2+x+1=0$ can be real.
  45. greg_rack

    Horizontal inflection point of a parametric polynomial function

    For ##x=-1## to be an *horizontal* inflection point, the first derivative ##y'## in ##-1## must be zero; and this gives the first condition: ##a=\frac{2}{3}b##. Now, I believe I should "use" the second derivative to obtain the second condition to solve the two-variables-system, but how? Since...
  46. S

    I When swapping roots of a polynomial, how to prove discriminant loops?

    I was looking at this discussion of swapping roots of a polynomial causing the discriminant to loop around the origin. https://www.akalin.com/quintic-unsolvability Although it appears to be the case, has this mathematical fact ever been proven? It seems that the formula for the discriminant...
  47. greg_rack

    How do I solve this polynomial limit?

    I'll write my considerations which lead me to get stuck on the ##\infty-\infty## form. $$\lim_{x \to +\infty }\sqrt{x^{2}-2x}-x+1 \rightarrow |x|\sqrt{1-0}-x+1$$ And I have no idea on how to go on...
  48. greg_rack

    Doubt solving a polynomial inequality

    I got this function in a function analysis and got confused on how to solve its positivity; I rewrote it as: $$\sqrt{x^{2}-2x}>x-1 \rightarrow x^2-2x>x^2-2x+1$$ And therefore concluded it must've been impossible... but I'm certainly missing something stupid, since plotting the graphs of the two...
  49. LCSphysicist

    Find this sum involving a polynomial root

    if x_{I}, I = {1,2,...,2019} is a root of P(x) = ##x^{2019} +2019x - 1## Find the value of ##\sum_{1}^{2019}\frac{1}{1-\frac{1}{X_{I}}}## I am really confused: This polynomial jut have one root, and this root is x such that 0 < x < 1, so that each terms in the polynomial is negative. But the...
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