Polynomial Definition and 1000 Threads

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...

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...

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...

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...

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) =...

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...