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. J

    MHB Complex Variables - Legendre Polynomial

    We define the Legendre polynomial $P_n$ by $$P_n (z)=\frac{1}{2^nn!}\frac{d^n}{dz^n}(z^2-1)^n$$ Let $\omega$ be a smooth simple closed curve around z. Show that $$P_n (z)=\frac{1}{2i\pi}\frac{1}{2^n}\int_\omega\frac{(w^2-1)^n}{(w-z)^{n+1}}dw$$ What I have: We know $(w^2-1)^n$ is analytic on...
  2. N

    I Problem when evaluating bounds....Is the result 1 or 0^0?

    Consider the summation ∑,i=0,n (t^(n-i))*e^(-st) evaluated from zero to infinity. You could break down the sum into: (t^(n))*e + (t^(n-1))*e + (t^(n-1))*e + ... + (t^(n-n))*e ; where e = e^(-st) To evaluate this, notice that all terms will go to zero when evaluated at infinity However, when...
  3. lfdahl

    MHB Polynomial with five roots: determine the roots of the equation x^5+ax^4+bx^3+cx^2+dx+e=0 as functions of a,d and e

    I am so sorry for having posted this challenge/puzzle with a serious typo: The roots of the equation should be functions of $a, d$ and $e$. In my old version I wrote $a, b$ and $e$. I will see to, that future challenges are properly debugged before posting.For $e \ne 0$, determine the roots...
  4. M

    MHB What is the Polynomial P(x) Given a Specific Quotient and Remainder?

    I can't seem to figure this out. When a polynomial P(x) is divided by (2x+1) the quotient is x^2-x+2 and the remainder is 5. What is P(x)?
  5. Hawksteinman

    Is 18 a Polynomial? Understanding Polynomials and Degrees

    Homework Statement Is the following a polynomial or not: 18 Homework EquationsThe Attempt at a Solution Not a polynomial (computer says it is)
  6. karush

    MHB Solve Polynomial Scale: Find a,b,c,d

    what is a b c and d so that all values of s are true \begin{align}\displaystyle &f_{15}=\\ &-17d+11s^2-4s+10as^3=(b+2)s+90s^3+(3c-1)s^2+85\\ &-17d+11s^2-6s+10as^3=bs+90s^3+3cs^2-s^2+85\\ &(s=0)\\ &-17d=85 \therefore d=-5\\ &11s^2-6s+10as^3=bs+90s^3+3cs^2-s^2\\ &(s=-1)\\...
  7. topsquark

    MHB First order corrections to a polynomial equation

    I've seen this done in a video but I can no longer find the video! :( What I would like to do is approximate the solutions to a polynomial equation in terms of a small perturbation. For example, say we have y = f(x) and know the corresponding zeros exactly. How would I go about finding first...
  8. Hawksteinman

    I How do I factorise polynomials using a new method?

    As part of my degree in physics with Astrophysics, I have to do some maths modules. In the maths lectures, the lecturer just goes though a giant 212 page work booklet explaining everything as she goes along. Me and a friend only do the work booklet in the lectures and we're already on page 33...
  9. B

    Irreducibility of a Polynomial

    Homework Statement Let ##f(x) = x^2 + x + 1 \in \Bbb{F}_2[x]##. Prove that ##f(x)## is irreducible and that ##f(x)## has a root ##\alpha \in \Bbb{F}_4##. Use the construction of ##\Bbb{F}_4## to display ##\alpha## explicitly Homework Equations Definition: An element ##p## in a domain ##R## is...
  10. B

    Function generation using Chebyshev polynomial

    In case of a four bar linkage if we have a function y=f(x) does it mean that we have to design a four bar linkage such that the input crank angle and the output rocker angle will satisfy the equation? Moreover after finding the precision points assuming linear relations wherever required how do...
  11. S

    Chebyshev polynomial approximation

    Homework Statement Find the quadratic least squares Chebyshev polynomial approximation of: g(z) = 15π/8 (3-z^2)√(4-z^2) on z ∈ [-2,2] Homework Equations ϕ2(t) = c0/2 T0(t) +c1T1(t)+c2T2(t) T0(t)=1 T1(t)=t T2(t)=2t2-1 Cj = 2/π ∫ f(t) Tj(t) / (√(1-t2) dt where the bounds for the integration...
  12. lfdahl

    MHB Polynomial in n variables: Prove the identity

    Suppose $f$ is a polynomial in $n$ variables, of degree $ \le n − 1$, ($n = 2, 3, 4 ...$ ).Prove the identity: \[\sum (-1)^{\epsilon_1+\epsilon_2+\epsilon_3+ ...+\epsilon_n}f(\epsilon_1,\epsilon_2,\epsilon_3,...,\epsilon_n) = 0\;\;\;\;\; (1)\] where $\epsilon_i$ is either $0$ or $1$, and the...
  13. F

    Proof of No Solution for x^2 - 3xy + 2y^2 = 10 Conjecture | Polynomial Homework

    Homework Statement Prove or refute the following conjecture: There are no positive integers x and y such that ##x^2 - 3xy + 2y^2 = 10## Homework Equations ##10 = 5*2## ##10 = 10*1## The Attempt at a Solution I graphed it using a graphing calculator, so I know this is true. Proof: This will...
  14. lfdahl

    MHB Prove a Polynomial has no real roots

    Prove that polynomials of the form:\[P_n(x)=x^{2n}-2x^{2n-1}+3x^{2n-2}-...-2nx+2n+1, \: \: n = 1,2,...\]- have no real roots.
  15. B

    Can I Rescale These Bottle Design Functions for Half the Volume?

    All variables and given/known data and Relevant equations: So I got the functions for a bottle design (one side with the bottle lying horizontally): 1. y=-1/343x^3+3/98x^2 + 2.5 ; 0<x<7 2. y=3; 7<x<15 3. y=-1/98x^2+15/49x+69/98; 15<x<22 Combined they give the volume of 570.2mL using the volume...
  16. W

    Justification for upper bound in Taylor polynomial

    Homework Statement I've been reviewing some Taylor polynomial material, and looking over the results and examples here. https://math.dartmouth.edu/archive/m8w10/public_html/m8l02.pdf I'm referring to Example 3 on the page 12 (page numbering at top-left of each page). The question is asking...
  17. B

    Water Bottle Design Using Polynomials

    Homework Statement [/B] I am to design a 600mL water bottle by drawing one side (bottle lying horizontally). Three types of functions must be included (different orders). The cross-sectional view would be centred about the x-axis, and the y-axis would represent the radius of that particular...
  18. D

    I Need help solving for X in third order polynomial

    Hello I have a third order polynomial, for example y(x) = -60000x^3 - 260x^2 + 780x + 0.6 I need to know what is x at y = 28 and/or y= 32. I can goto MATLAB and find the roots ( x = - .1158, -.0007, and .1122 ) or I can go to http://www.wolframalpha.com and it also finds the roots and...
  19. M

    Legendre Polynomial Integration

    Homework Statement Simplify $$\int_{-1}^1\left( (1-x^2)P_i''-2xP'_i+2P_i\right)P_j\,dx$$ where ##P_i## is the ##i^{th}## Legendre Polynomial, a function of ##x##. Homework Equations The Attempt at a Solution Integration by parts is likely useful?? Also I know the Legendre Polynomials are...
  20. pairofstrings

    B What are the applications of roots of a polynomial?

    Hello. Assume that I have two polynomials of degree 2, i.e., Quadratic Equations. 1. Assume that the Quadratic Equation is: x2 + 7x + 12 = 0 The roots of the Quadratic Equation is -3 and -4. 2. Assume that there is another Quadratic Equation: x2 + 8x + 12 = 0 The roots of the Quadratic...
  21. 1

    I Finding Critical Points of a Quartic Function: A Scientific Approach

    I need this solved for x: y' = 4ax^3 + 3bx^2 + 2cx + d = 0 This is to say, I need the formula for the "critical points" of a Quartic function. Wikipedia says: "The derivative of a quartic function is a cubic function." https://en.wikipedia.org/wiki/Quartic_function And I found the above...
  22. T

    Can this polynomial be factored into two integer products

    Homework Statement Homework Equations none The Attempt at a Solution i assumed it can be factored into the form ## (x^2 + m_1 x + m_0)(x + n_0) ## by comparison of coefficients ## m_0 n_0 = -abc -1\\ m_1 + n_0 = -a -b -c\\ m_0 + m_1 n_0 = ab +ac + bc\\ ## the only other information i have is...
  23. M

    I Polynomial Division: Solve with Mike's Help!

    Hello everyone. Iam working on a course in digital control systems and by reading my textbook I stumbled over this expression. C(z) = 0.3678z + 0.2644 : z^2 − 1.3678z + 0.3678 = 0.3678z^−1 + 0.7675z^−2 + 0.9145z^−3 + ... Now Iam wondering how the result of the polynomial division is...
  24. T

    Proving a polynomial cannot be factored with integer coefficients

    Homework Statement [/B]Homework EquationsThe Attempt at a Solution i tried to do it by writing it as ## a_{1999} x^{1999} + a_{1998} x^{1998} ... a_0 \pm1 = 0 ## for 1999 different integer values of x i am thinking of writing it as ## a_{1999} x^{1999} = -a_{1998} x^{1998} - a_{1997} x ^...
  25. A

    Integration of an inverse polynomial

    Hello, I want to integrate this expression : ∫ (x5 + ax4 + bx3 + cx2 + dx)-1 between xmin>0 and xmax>0 a is positive but b, c and d can be positive or negative. I have no idea to integrate this expression... Do you have methods to do this ? Thanks in advance !
  26. Math Amateur

    MHB Example of an Inseparable Polynomial .... Lovett, Page 371 ....

    I am reading "Abstract Algebra: Structures and Applications" by Stephen Lovett ... I am currently focused on Chapter 7: Field Extensions ... ... I need help with Example 7.7.4 on page 371 ...Example 7.7.4 reads as follows: In the above text from Lovett we read the following:" ... ... The...
  27. Math Amateur

    I Example of an Inseparable Polynomial .... Lovett, Page 371 ...

    I am reading "Abstract Algebra: Structures and Applications" by Stephen Lovett ... I am currently focused on Chapter 7: Field Extensions ... ... I need help with Example 7.7.4 on page 371 ...Example 7.7.4 reads as follows: In the above text from Lovett we read the following:" ... ... The...
  28. T

    Proving a polynomial has no integer solution

    Homework Statement let p(x) be a polynomial with integer coefficients satisfying p(0) = p(1) = 1999 show that p has no integer zeros Homework EquationsThe Attempt at a Solution ## p(x) = \sum_{i= 0}^{n}{a_i x^i} ##[/B] using the given information a0 = 1999( a prime number) and ## a_n +...
  29. Alaguraja

    B How to Apply Hermite Polynomial for Physics Problems

    I have doubt since a long time, that is How we apply the Hermite polynomial for a physics problem. And I don't know weather everyone known about how the analyze a physics problem and how do they apply a correct mathematical methods?
  30. M

    MHB What is the Factorization of p^3 - q^3 -p(p^2 - q^2) + q(p - q)^2?

    Factor. p^3 - q^3 -p(p^2 - q^2) + q(p - q)^2 Solution: p^3 - q^3 - p^3 + pq^2 + q(p^2 -2pq + q^2) p^3 - q^3 - p^3 + pq^2 + qp^2 -2pq^2 + q^3 pq^2 + qp^2 - 2pq^2 -pq^2 + qp^2 qp^2 - pq^2 pq(p - q) Correct?
  31. M

    MHB Why is x^2 + 1 an Irreducible Polynomial?

    Why is x^2 + 1 irreducible?
  32. M

    MHB How do I factor this expression: 3(x + 5)^3 + 2(x + 5)^2?

    Precalculus by David Cohen, 3rd Edition Chapter 1, Section 1.3. Question 46a. Factor the expression. 3(x + 5)^3 + 2(x + 5)^2 (x + 5)^2[3(x + 5) + 2] (x + 5)^2[3x + 15 + 2] (x + 5)^2[3x + 17] Correct?
  33. Mr Davis 97

    Computing the order in a polynomial quotient ring

    Homework Statement Consider the quotient ring ##F = \mathbb{Z}_3 [x] / \langle x^2 + 1 \rangle##. Compute the order of the coset ##(x+1) + \langle x^2 + 1 \rangle## in the group of units ##F*##. Homework EquationsThe Attempt at a Solution I was thinking that I just continually compute powers...
  34. Mayan Fung

    I Determining the coefficient of the legendre polynomial

    We know that the solution to the Legendre equation: $$ (1-x^2)\frac{d^2 y}{dx^2} - 2x \frac{dy}{dx} + n(n+1) = 0 $$ is the Legendre polynomial $$ y(x) = a_n P_n (x)$$ However, this is a second order differential equation. I am wondering why there is only one leading coefficient. We need two...
  35. J

    Cryptographic attacks as minimization of degree 4 polynomial

    Cryptography is based on reason-result chains like hash functions: which are inexpensive to propagate in the intended direction, but seem hard to reverse. However, decomposing them into satisfaction of simple (direction-agnostic) relations like 3-SAT clauses, may bring a danger of existence of...
  36. L

    Given the zhegalkin polynomial, find the boolean function

    Homework Statement the given zhegalkin polynomial is y=x1+x2. find the corresponding boolean functio The Attempt at a Solution zhegalkin polynomial for 2 variables is: f(x) = c12*x1*x2 + c1*x1 + c2*x2 + c => c12=0, c1=1, c2=1, c=0. therefore, f(0,0) = c = 0, f(0,1) = c2+c = 1+0 = 1 f(1,0) =...
  37. lfdahl

    MHB Expressing a polynomial P(x)=(x−a)^2(x−b)^2+1 by two other polynomials

    Let $a$ and $b$ be two integer numbers, $a \ne b$. Prove, that the polynomial: $$P(x) = (x-a)^2(x-b)^2 + 1$$ cannot be expressed as a product of two nonconstant polynomials with integer coefficients.
  38. L

    MHB Finding a third degree polynomial... stumped

    Problem: Find a third degree polynomial with rational coefficients if two of its zeros are 6 and – 𝑖 and it passes through the point (2, -10)So far, I have came up with this: (x-6)(x^2+1) however, instead of passing through (2,-10), it passes through (2,-20) Anyone know how to come up with a...
  39. karush

    MHB 10.8.3 Find the Taylor polynomial

    $\textrm{10.8.{7} Find the Taylor polynomial of orders $0, 1, 2$, and $3$ generated by $f$ at $a$.}$ \begin{align*} \displaystyle f(x)&=\sin{x} \end{align*} \[ \begin{array}{llll}\displaystyle f^0(x)&=\sin{x}&\therefore f^0(\frac{5x}{6})&=\frac{1}{2}\\ \\ f^1(x)&=\cos{x}&\therefore...
  40. K

    I Can Taylor series be used to get the roots of a polynomial?

    I'm using this method: First, write the polynomial in this form: $$a_nx^n+a_{n-1}x^{n-1}+...a_2x^2+a_1x=c$$ Let the LHS of this expression be the function ##f(x)##. I'm going to write the Taylor series of ##f^{-1}(x)## around ##x=0## and then put ##x=c## in it to get ##f^{-1}(c)## which will be...
  41. L

    Finding the Constant for Cancellation in Polynomial Factoring

    Homework Statement ##\frac{3x^2+x+C}{x+5}## find value of constant C such that the clause can be canceled in some manner. What will be the canceled form of the clause. Homework Equations -presumably C is a constant, and also an integer. -polynomial factorization will be attempted -cancelling...
  42. B

    MHB How can I factorize this polynomial?

    Decompose $$6a^2-3ab-11ac+12ad-18b^2+36bc-45bd-10c^2+27cd-18d^2$$ I noticed that the factorized form would be $$(Aa+Bb+Cc+Dd)(Wa + Xb + Yc + Zd)$$ Which is similar to the factorized form $$(Aa+Bb+Cc)(Wa+Xb+Yc)$$ $$Yc(Aa+Bb)+Cc(Wa+Xb) = c(CX+BY)$$ Is there a way that I can somehow use...
  43. J

    I Newton Divided Difference Interpolation Polynomial

    f(x)= a(0) + a1(x-x(0)) + a2(x-x(1))(x-x(0)) I am having a hard time understanding the intuition of (x-x(1))(x-x(0)) being multiplied by the coefficient a(2). For example, if I added a(3) to the equation, I would have had to multiply a(3) by (x-x(0))(x-x(1))(x-x(2)). I've researched the Mean...
  44. Quadrat

    Factoring a four term polynomial

    Homework Statement I just want to know how get from ##4x^3+3x^2-6x-5=0 ## to ##(x+1)^2(4x-5)=0##. What's the trick when dealing with these nasty polynomials? I got the answer by taking another approach (solving a root equation) but I noted this is also a way to go, but I can't figure out the...
  45. CynicusRex

    Prove: polynomial is uniquely defined by three of its values

    Homework Statement Algebra - I.M. Gelfand, Problem 164. Prove that a polynomial of degree not exceeding 2 is defined uniquely by three of its values. This means that if P(x) and Q(x) are polynomials of degree not exceeding 2 and P(x1) = Q(x1), P(x2) = Q(x2), P(x3) = Q(x3) for three different...
  46. F

    Solving a 3rd Degree Polynomial: What are the Options?

    Homework Statement Solve for the roots of the following. (What do you notice about the complex roots?) b) x3 + x2 + 2x + 1 = 0 Homework Equations To find roots of a polynomial of degree n > 3, look at the constant and take all its factors. Those are possible roots. Then plug them into see...
  47. S

    Verifying Subspace of P3: Closure of Addition & Scalar Multiplication

    Homework Statement Determine if the following is a subspace of ##P_3##. All polynomials ##a_0+a_1x+a_2x^2+a_3x^3## for which ##a_0+a_1+a_2+a_3=0## Homework Equations use closure of addition and scalar multiplication The Attempt at a Solution Let ##P=a_0+a_1x+a_2x^2+a_3x^3## and...
  48. Vitani11

    Finding the roots of a polynomial with complex coefficients?

    Homework Statement z2-(3+i)z+(2+i) = 0 Homework EquationsThe Attempt at a Solution [/B] Does the quadratic formula work in this case? Should you deal with the real and complex parts separately?
  49. M

    MHB Finding Taylor Polynomial for tan(x) - Wondering

    Hey! :o Let $f :\rightarrow \mathbb{R}$, $f(x) := tan(x)$. I want to find a $N\in \mathbb{N}$ such that for the $N$-th Taylor polynomial $P_N$ at $0$, that is defined as follows $P_N(x)=\sum_{n=0}^N\frac{f^{(n)}(0)}{n!}x^n$, it holds that $$\left |f(x)-P_N(x)\right |\leq 10^{-5}, \ \ x\in...
  50. doktorwho

    How to Solve a Polynomial Function with Complex Zeros?

    Homework Statement Given the polynomial function ##x^4+x^3+2x^2+4=0## solve it if you know that it has at least one complex zero whose real part equals the complex part. Homework Equations 3. The Attempt at a Solution [/B] My guess is that if this function has one complex zero it must have a...
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