Polynomials Definition and 740 Threads

Hello everyone. I am working with generalized polynomial chaos. To represent a Normal random variable, the Hermite polynomials are used. However, as far as I understand, these represent N(0,1); if what I have read is correct, if I want to work with any other mean and variance, I shoud simply...

Hi all, In S. Weinberg's book "Cosmology", there is a derivation of the slightly modified temperature of the cosmic microwave background as seen from the Earth moving w.r.t. a frame at rest in the CMB. On Page 131 (1st printing), an approximation (Formula 2.4.7) is given in terms of Legendre...

Homework Statement _2F_1(a,b;c;x)=\sum^{\infty}_{n=0}\frac{(a)_n(b)_n}{(c)_nn!}x^n Show that Legendre polynomial of degree ##n## is defined by P_n(x)=\,_2F_1(-n,n+1;1;\frac{1-x}{2}) Homework Equations Definition of Pochamer symbol[/B] (a)_n=\frac{\Gamma(a+n)}{\Gamma(a)} The Attempt at a...

Homework Statement Given ## x^4+x^3+Ax^2+4x-2=0## and giben that the roots are ## 1/Φ, 1/Ψ, 1/ξ ,1/φ## find AHomework EquationsThe Attempt at a Solution ## (x-a)(x-b)(x-c)(x-d)=0## where a,b,c and d are the roots

Hi - does anyone know of a program library/subroutine/some other source, to find the zeros of a generalised Laguerre polynomial? ie. LαN(xi)=0

Say we have a polynomial ##f(x)=2x^3+3x^2-14x-21## and we want to find the upper and lower bounds of the real zeros of this polynomial. If no real zero of ##f## is greater than b, then b is considered to be the upper bound of ##f##. And if no real zero of ##f## is less than a, then a is...

using Rodrigues' formula show that $$\int_{-1}^{1} \,{P}_{n}(x){P}_{n}(x)dx = \frac{2}{2n+1}$$ $${P}_{n}(x) = \frac{1}{2^nn!}\frac{d^n}{dx^n}(x^2-1)^n$$ my thoughts $$\int_{-1}^{1} \,{P}_{n}(x){P}_{n}(x)dx = \frac{1}{2^{2n}(n!)^2}\int_{-1}^{1}...

Please see the attached image which is presented in my text. This is the end step after using the Rational Zeros Theorem to find possible rational zeros, testing by synthetic division, and then factoring. What I don't understand here is that we have the term...

hey i have doubt about Legendre polynomials and Associated Legendre polynomials what is Associated Legendre polynomials ? It different with Legendre polynomials ?

Can someone check if my answers are right and help me with the missing ones.

Hi all, I have solved part (a) which require verification if it is correct. However, for part b, I am not sure how to do. Appreciate your help. Thank you.

Hello everyone. The Legendre polynomials are defined between (-1 and 1) as 1, x, ½*(3x2-1), ½*(5x3-3x)... My question is how can I switch the domain to (1, 2) and how can I calculate the new polynomials. I need them to construct an estimation of a random uniform variable by chaos polynomials...

Assume that ##P## is a polynomial over a commutative ring ##R##. Then there exists a ring ##\tilde R## extending ##R## where ##P## splits into linear factor (not necessarily uniquely). This theorem, whose proof is given below, is difficult to find in the literature (if someone know a source, it...

Homework Statement Consider the function ##f\left(x\right)=\sqrt {x+2}##. Determine if the function is a one-to-one function, If so, find ##f^{-1}\left(x\right)## and state the domain and range of ##f\left(x\right)## and ##f^{-1}\left(x\right)## Homework Equations N/A The Attempt at a...

Homework Statement [/B] Polynomial P(x) when divided by (x-2) gives a remainder of 10. Same polynomial when divided by (x+3) gives a remainder of 5. Find the remainder the polynomial gives when divided by (x-2)(x+3). 2. Homework Equations Polynomial division, remainder theorem The Attempt...

Hi all. So to start I'll say I'm just dealing with functions of a real variable. In my linear algebra courses one thing was drilled into my head: "Algebraic invariants are geometric objects" So with that in mind, is there any geometric connection between two orthoganal functions on some...

Does there exist a polynomial P(x) with rational coefficients such that for every composite number x, P(x) takes an integer value and for every prime number x, P(x) does not take on an integer value? Can someone please guide me in the right direction? I've tried to consider the roots of the...

Homework Statement Solve for x: 3x^3 + 2x^2 + 75x - 50 = 0 The Attempt at a Solution I have tried substituting multiple values for "x" so that we get a factor, f(x)=0 I cannot seem to find an "x" value that will make this function=0. Is there a way to factor this function or did the book...

Homework Statement Find critical points and inflection points of: 1/[x(x-1)] Homework Equations 1/[x(x-1)] The Attempt at a Solution using quotient rule, we obtain (0-(2x-1)/(x^2-x)^2 set -2x+1=0 we get 1/2 for critical point. for second derivative, i get...

Homework Statement find maxima/minima of following equation. Homework Equations -(x+1)(x-1)^2 The Attempt at a Solution (-x-1)(x-1)^2 Using product rule, we obtain, -1(x-1)^2+(-x-1)*2(x-1) I don't know where to go from here. The software's factoring I had never seen before.

Homework Statement find derivative of (x-2)(x-3)^2 Homework Equations using product rule. The Attempt at a Solution 1(x-3)^2+2(x-3) x^2-6x-9 +2x-6 x^2-4x-15 doesn't factor.

Not too sure which forum this would be best suited to. Say I have lots of polynomials that have been obtained through conducting experiments, with the different coefficients in the polynomial representing different physical properties that have been changed in each case. How could I use this...

Homework Statement Prove that the roots of trigonometric polynomials with integer coefficients are denumerable. Homework EquationsThe Attempt at a Solution The book does not define what a trig polynomial is, but I am assuming it is something of the form ##\displaystyle a_0 + \sum^N_{n=1}a_n...

I am studying the linear oscillation of the spherical droplet of water with azimuthal symmetry. I have written the surface of the droplet as F=r-R-f(t,\theta)\equiv 0. I have boiled the problem down to a Laplace equation for the perturbed pressure, p_{1}(t,r,\theta). I have also reasoned that...

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...

Methods of factoring . Method of common factors Factorization by regrouping terms Factorization using identities Factors of the form ( x + a) ( x + b) Factor by Splitting Is this all the factoring methods out there ? Or are there more ? I am also looking for a book with lots of...

Hi everybody, I'm trying to calculate this: $$\sum_{l=0}^{\infty} \int_{\Omega} d\theta' d\phi' \cos{\theta'} \sin{\theta'} P_l (\cos{\gamma})$$ where ##P_{l}## are the Legendre polynomials, ##\Omega## is the surface of a sphere of radius ##R##, and $$ \cos{\gamma} = \cos{\theta'}...

I am reading David S. Dummit and Richard M. Foote : Abstract Algebra ... I am trying to understand the proof of Proposition 37 in Section 13.5 Separable and Inseparable Extensions ...The Proposition 37 and its proof (note that the proof comes before the statement of the Proposition) read as...

Dummit and Foote in Section 13.5 on separable extensions make some remarks about separable polynomials that I do not quite follow. The remarks follow Corollary 34 and its proof ... Corollary 34, its proof and the remarks read as follows: In the above text by D&F, in the remarks after the...

I am reading Paul E Bland's book: The Basics of Abstract Algebra and I am trying to understand his definition of "separable polynomial" and his second example ... Bland defines a separable polynomial as follows: ... and Bland's second example is as follows: I am uncomfortable with, and do...

I am reading both David S. Dummit and Richard M. Foote : Abstract Algebra and Paul E. Bland's book: The Basics of Abstract Algebra ... ... I am trying to understand separable polynomials ... ... but D&F and Bland seem to define them slightly differently and interpret the application of the...

Hi everyone. I started to look at different linearization techniques like: -linear interpolation - spline interpolation - curve fitting... Now Iam wondering (and I guess its very stupid) : As polynomials with a degree > 1 are not linear, why can I use them for linearization? With the...

I've always been curious about why we define polynomials the way we do. On the surface, it seems that they are expressions that naturally arise from combining the standard arithmetic operations on indeterminates. However, there are some points that I am generally confused about. Why are...

Homework Statement Evaluate the normalization integral in (22.15). Hint: Use (22.12) for one of the $H_n(x)$ factors, integrate by parts, and use (22.17a); then use your result repeatedly.Homework Equations (22.15) ##\int_{-\infty}^{\infty}e^{-x^2}H_n(x)H_m(x)dx = \sqrt{\pi}2^nn!## when ##n=m##...

Homework Statement When do two quadratic polynomials in ##\mathbb{Z}_3 [x]## generate the same ideal? Homework EquationsThe Attempt at a Solution I feel like they generate the same ideal only when they have the same coefficients, but am not sure how to show this.

I am reading "Introductory Algebraic Number Theory"by Saban Alaca and Kenneth S. Williams ... and am currently focused on Chapter 1: Integral Domains ... I need some help with the proof of Theorem 1.2.2 ... Theorem 1.2.2 reads as follows: https://www.physicsforums.com/attachments/6515 In the...

I am reading "Introductory Algebraic Number Theory"by Saban Alaca and Kenneth S. Williams ... and am currently focused on Chapter 1: Integral Domains ... I need some help with the proof of Theorem 1.2.2 ... Theorem 1.2.2 reads as follows: In the above text from Alaca and Williams, we read the...

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.

Hi, I have a question about taylor polynomials. https://wikimedia.org/api/rest_v1/media/math/render/svg/09523585d1633ee9c48750c11b60d82c82b315bfI was looking for proof that why every lagrange remainder is decreasing as the order of lagnrange remainder increases. so on wikipedia, it says, for a...

Hello all, I'm reading through Jackson's Classical Electrodynamics book and am working through the derivation of the Legendre polynomials. He uses this ##\alpha## term that seems to complicate the derivation more and is throwing me for a bit of a loop. Jackson assumes the solution is of the...

The sum of the $k$ th power of n variables $\sum_{i=1}^{i=n} x_i^k$ is a symmetric polynomial, so it can be written as a sum of the elementary symmetric polynomials. I do know about the Newton's identities, but just with the algorithm of proving the symmetric function theorem, what should we do...

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...

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...

Homework Statement Let S be a set of nonzero polynomials. Prove that if no two have the same degree, then S is linearly independent. Homework EquationsThe Attempt at a Solution We will proceed by contraposition. Assume that S is a linearly dependent set. Thus there exists a linear dependence...

I'm tutoring a student who is in a typical precalculus/trig course where they're teaching her about graphing various arbitrary polynomials. Among the rules of multiplicity and intercepts they seem to be phrasing the questions such that they expect the students to also find the maxima and minima...

Hello. I am currently studying at a specialized mechanical engineering high school. I'm in my first (or 10th) year, as I've stated before. I've done algebra I and algebra II, along with about one half of trigonometry (utilizing the trigonometric functions in practical problems, like splitting...

The question is : Is it true that two matrices with the same characteristic polynomials have the same trace? I know that similar matrices have the same trace because they share the same eigenvalues, and I know that if two matrices have the same eigenvalues, they have the same trace. But I am...

$\tiny{242.13.1}$ $\textsf{a. Find the $3^{rd}$ Taylor polynomial for $\sec{x}$ at $a=0$}\\$ \begin{align} \displaystyle f^0(x)&=f(x)=\sec{x}\therefore f^0(0)=1\\ &=\frac{1}{0!} x^0=1 \\ f^1(x)&=(\sec{x})'=\tan{x}\sec{x} \therefore f^1(x)=0 \\ &=\frac{1}{0!} x^0+\frac{0}{1!} x^1=1+0=1 \\...

$\tiny {11. 1.33-T} $ $\textsf{Find the nth order Taylor polynomials of the given function centered at a=100, for $n=0, 1, 2.$}\\$ $$\displaystyle f(x)=\sqrt{x}$$ $\textsf{using}\\$ $$P_n\left(x\right) \approx\sum\limits_{k=0}^{n} \frac{f^{(k)}\left(a\right)}{k!}(x-a)^k$$ $\textsf{n=0}\\$...

$\tiny{206.11.1.16-T}$ $\textsf{Find the nth-order Taylor polynomials centered at 0, for $n=0, 1, 2.$}\\$ $$\displaystyle f(x)=e^{-4x}$$ $\textsf{using}\\$ $$P_n\left(x\right) \approx\sum\limits_{k=0}^{n}\frac{f^{(k)}\left(a\right)}{k!}x^k$$ $\textsf{n=0}\\$ \begin{align} f^0(x)&\approx...