Polynomial Definition and 1000 Threads

Hey! :o I want to calculate the Taylor polynomial of order $n$ for the funktion $ f(x) = \frac{1}{ 1−x}$ for $x_0=0$ and $0 < x < 1$ and the remainder $R_n$. We have that \begin{equation*}f^{(k)}(x)=\frac{k!}{(1-x)^{k+1}}\end{equation*} I have calculated that...

Hi PF! In MATLAB I'm trying to use associated Legendre polynomials of the 1st and second kind, widely regarded as ##P_i^j## and ##Q_i^j##, where ##j=0## reduces these to simply the Legendre polynomials of the 1st and second kind (not associated). Does anyone here know if MATLAB has a built in...

Homework Statement Hello, I have a general question regarding to coefficient matching when spanning some function, say , f(x) as a linear combination of some other basis functions belonging to real Hilbert space. Homework Equations - Knowledge of power series, polynomials, Legenedre...

Hello, When choosing a polynomial stress function Φ to satisfy the biharmonic equation, how does once decide on which order of the polynomial to choose? For example, is it based upon the number of boundary conditions, like a 3rd order polynomial would satisfy 3 boundary conditions?

$f(x)$ is a degree 10 polynomial such that $f(p)=q$, $f(q)=r$, $f(r)=p$, where $p$, $q$, $r$ are integers with $p<q<r$. Show that not all the coefficients of $f(x)$ are integers.

Is this true? If the remainder of f(x) / g(x) is a (where a is constant), then the remainder of (f(x))n / g(x) is an I don't know how to be sure whether it is correct or wrong. I just did several examples and it works. Thanks

Hello, I followed an example in a book that compares polynomial regression with linear regression. We have one feature or explanatory variable. The code is the following: import numpy as np import matplotlib.pyplot as plt from sklearn.linear_model import LinearRegression from...

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...

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...

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...

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)?

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

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)\\...

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...

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...

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...

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...

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...

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...

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.

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...

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...

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...

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...

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...

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...

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...

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...

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...

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 ^...

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 !

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...

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 +...

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?

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?

Why is x^2 + 1 irreducible?

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?

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...

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...

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...

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.

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...

$\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...

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 ##\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...

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...

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...

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 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...

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...