Polynomials Definition and 740 Threads

Homework Statement Let A be the algebra \mathbb{Z}_5[x]/I where I is the principle ideal generated by x^2+4 and \mathbb{Z}_5[x] is the ring of polynomials modulo 5. Find all the ideals of A Let G be the group of invertible elements in A. Find the subgroups of the prime decomposition.Homework...

(I haven't encountered these before, also not in the book prior to this problem or in the near future ...) Show that the 1st derivatives of the legendre polynomials satisfy a self-adjoint ODE with eigenvalue $\lambda = n(n+1)-2 $ Wiki shows a table of poly's , I don't think this is what the...

Find all pairs of polynomials $$p(x)$$ and $$ q(x)$$ with real coefficients for which both equations are satisfied: $$p(x^2+1)=q(x)^2+2x$$ and $$q(x^2+1)=p(x)^2$$. These equations are set for all real $$x$$. I tried to substitute $$x$$ for $$-x$$ and others numbers like $$-1,1$$ etc. but...

Homework Statement Suppose a linear transformation T: [P][/2]→[R][/3] is defined by T(1+x)= (1,3,1), T(1-x)= (-1,1,1) and T(1-[x][/2])=(-1,2,0) a) use the given values of T and linearity properties to find T(1), T(x) and T([x][/2]) b) Find the matrix representation of T (relative to standard...

I have just finished a post entitled: http://mathhelpboards.com/linear-abstract-algebra-14/irreducible-polynomials-quotient-rings-rotman-proposition-3-116-a-16163.htmlon the Linear and Abstract Algebra Forum ... I want to have the following code recognised: k(z) \subseteq \text{ I am } \phi...

I am reading Joseph J. Rotman's book: A First Course in Abstract Algebra with Applications (Third Edition) ... I am currently focused on Section 3.8 Quotient Rings and Finite Fields ... [FONT=Times New Roman][FONT=Times New Roman]I need help with an aspect of the proof of Proposition 3.116...

Does anyone know any very simple to operate (intuitive) software for graphing polynomials in two variables. If the software allows you to plot two polynomials simultaneously then all the better ... Be a bonus if it allowed the finding of zeros also ... Peter

I am reading Joseph J. Rotman's book: A First Course in Abstract Algebra with Applications (Third Edition) ... I am currently focused on Section 3.6 Unique Factorization ... I need help with an aspect of the proof of Lemma 3.87 ... The relevant text from Rotman's book is as follows: In the...

I am reading Joseph J. Rotman's book: A First Course in Abstract Algebra with Applications (Third Edition) ... I am currently focused on Section 3.5 From Numbers to Polynomials ... I need help with an aspect of the proof of Lemma 3.70 ... The relevant text from Rotman's book is as follows:In...

I am reading Joseph J. Rotman's book: A First Course in Abstract Algebra with Applications (Third Edition) ... I am currently focused on Section 3.5 From Numbers to Polynomials ... I need help with an aspect of the proof of Lemma 3.67 ... The relevant text from Rotman's book is as follows:In...

I am reading W. Keith Nicholson's book: Introduction to Abstract Algebra (Third Edition) ... I am focused on Section 4.3:Factor Rings of Polynomials over a Field. I need some help with the proof of Lemma 2 on page 223-224. The relevant text from Nicholson's book is as...

I am reading W. Keith Nicholson's book: Introduction to Abstract Algebra (Third Edition) ... I am focused on Section 4.3:Factor Rings of Polynomials over a Field. I need some help with the proof of Part 1 of Lemma 1 on page 223-224. The relevant text from Nicholson's book is as follows...

I am reading W. Keith Nicholson's book: Introduction to Abstract Algebra (Third Edition) ... I am focused on Section 4.2:Factorization of Polynomials over a Field. I need some help with the proof of Part 1 of Theorem 12 on page 218 The relevant text from Nicholson's book is as...

I am reading W. Keith Nicholson's book: Introduction to Abstract Algebra (Third Edition) ... I am focused on Section 4.2:Factorization of Polynomials over a Field. I need some help (with an apparently very simple issue) with the proof Theorem 11 on pages 217 - 218 ... just seem to have a...

I am reading W. Keith Nicholson's book: Introduction to Abstract Algebra (Third Edition) ... I am focused on Section 4.2:Factorization of Polynomials over a Filed. I need some help with Example 10 on page 215 ... The relevant text from Nicholson's book is as follows:In the above text, we read...

Given a quadric equation (F(x,y) = 0), exist other quadric equation (G(x,y) = 0) such that the poinst of intersection between the graphics are ortogonals. So, how to find the coefficients of the new quadic equation? EDIT: I think that F and G needs to satisfy∇F • ∇G = 0. So, if F is known, how...

Homework Statement Solve for x. a) 2x^3-3x^2-5x+6=0 Homework Equations -Find possible values of x -Divide that factor by 2x^3-3x^2-5+6 using long division The Attempt at a Solution ƒ(2)=2(2)^3-3(2)^2-5(2)+6 ƒ(2)=16-12-10+6 ƒ(2)=0 Now using long division divide 2x^3-3x^2-5x+6 by (x-2)...

I am reading Joseph J.Rotman's book, A First Course in Abstract Algebra. I am currently focused on Section 3.5 From Polynomials to Numbers I need help with the statement and meaning of Corollary 3.58 The relevant section of Rotman's text reads as...

I am reading Joseph J.Rotman's book, A First Course in Abstract Algebra. I am currently focused on Section 3. Polynomials I need help with the a statement of Rotman's concerning the polynomial functions of a finite ring such as $$ \mathbb{I}_m = \mathbb{Z}/ m \mathbb{Z} $$ The relevant...

I'm currently reading a text which uses Hermite polynomials defined in the recursive manner. The form of the polynomials are such that C0 C1 are the 0th and 1st terms of a taylor series that generate the remaining coefficients. The author then says the standard value of C1 and C0 are used, but...

Homework Statement If ##P,Q## are polynomials of ##\mathbb{Z}[X]##, and ##p## is a prime number that divides all the coefficients of ##PQ##, show that ##p## divides the coefficients of ##P## or the coefficients of ##Q##. Homework Equations ##c_n = \sum_{k=0}^{n} a_k b_{n-k} ## is the n-th...

Homework Statement What is the greatest common divisor of ##X^a - 1 ## and ## X^b - 1 ##, ##(a,b) \in \mathbb{N}^\star## ? Homework EquationsThe Attempt at a Solution Assuming that ## a\le b ##, I find by euclidian division of ##b## by ##a## that ## b = an + r \Rightarrow X^b - 1 = (X^a-1)...

Homework Statement Find the polynomials P in R[X] such that ## X(X+1) P'' +(X+2) P' -P = 0 ## Homework Equations If P is a solution, I assume ## P = \sum_{n=0}^N a_nX^n## The Attempt at a Solution I get the answer ##\{\alpha (X+2),\alpha\in\mathbb{R}\} ##, but I've done many calculation...

Let F be a field extension of Q (the rationals) with [F:Q] = 24. Prove that the polynomial $$x^5+2x^4-16x^3+6x-10$$ has no roots in F. Proof: Let $$a$$ be a root of $$x^5+2x^4-16x^3+6x-10$$. Since the polynomial has degree 5 by theorem we know that $$[Q(a):Q]=5$$. If $$a \in F$$ and...

Dear Friends! I need to find roots of polynomials with variable coefficients, The command I used is w=0:50 A=w^2 B=w^3+2 C=w+2*w^2 D=w E=w./2 ss=[A B C D E] xi=roots(ss) by this I find all the roots of equation, I want to find velocities by setting v1=w/xi(1) v2=w/xi(2) v3=w/xi(3) v4=w/xi(4)...

In ring theory, a polynomial over a rings, say ## R[x] ## is presented as an abstract object of the form: ## p(x) = a_{n}x^{n} + ...+ a_{1}x + a_{0} ## where the coefficients ## a_{n}...a_{0} ## are from a ring R with unity and ##x## is a formal symbol. So what is the significance of ## p(x+1)...

I'm in desperate need of help with factoring. Basically, how do you do it? Below is an example of what I'm up against. 54c^2d^5e^3; 81d^3e^2 It wants me to find the greatest common factor. http://www.rempub.com/80-activities-to-make-basic-algebra-easier That is the book I'm working out of.

Find roots of: I need to set =0 then factorise. I know that this polynomial has coefficients of 1,2,4,8 and there is a rule to factorise this however, i don't know it. Also, i believe a high order polynomial will be included in my exams. Are there any other "special" polynomials such as this...

I really need help. f(x) is a fourth degree polynomial function f(x) has zeros of plus or minus 2 and plus or minus 3i f(0)=-108 Find an equation for f(x) in general form

I have just written a program to calculate Legendre Polynomials, finding for Pl+1 using the recursion (l+1)Pl+1 + lPl-1 - (2l+1).x.Pl=0 That is working fine. The next section of the problem is to investigate the recursive polynomial in the reverse direction. I would solve this for Pl-1 in...

Homework Statement If the polynomial P(x) = x^2+ax+1 is a factor of T(x)=2x^3-16x+b, find a, b Homework EquationsThe Attempt at a Solution Let (px+q) be a factor of P(x), p can possibly be 1 and so can q, according to factor theorem, Hence, factors (x+1) or (x-1) P(1) = 0, substituting I got...

Hi! I have a dataset that I fit to a 5th order and 4th order polynomial -- I was just trying to get the function that best fit the data. However, I realized that when I evaluate the integral for these 2 different functions (between 200 and 400), the answers are vastly different. I assumed...

Mod note: Moved from a technical math section, so missing the template. I have this question and the answer but my mathXL does not show me how it came to this conclusion. (4s3+4s2 + 72)/ s+3I got all the way to the answer 4s2 - 8s The correct answer is 4s2 - 8s + 24 I just don't know the...

1. The way we solved this problem was proposing that the wave function has to form of ##\Psi=\Theta\Phi R## where the three latter variables represent the anlge and radius function which are independent. The legendre polynomials were the solution to the ##\Theta## part. I am having some trouble...

Homework Statement Prove the following theorem: Let R be any ring and let f =! 0 and g =! 0 (they don't equal zero) be polynomials in R[x]. If the leading coefficient of either f or g is a unit in R, then: 1) fg =! 0 in R[x] 2) deg(fg) = deg(f) + deg(g) Homework EquationsThe Attempt at a...

Homework Statement A polynomial p(x) is such that p(0)=5, p(1)=4, p(2)=9 and p(3)=20. the minimum degree it can have a) 1 b) 2 c) 3 d) 4 Homework EquationsThe Attempt at a Solution a) Not Possible can't connect these points using straight line b) Not even possible to connect these points using...

I could really use some help on this problem as well! Thanks!

Homework Statement Could someone pls clarify if the value of x changes from just Laguerre polynomial to associated one? I am confused about the role of variable x. Homework Equations From what I have learned in the class, I understand that L1n(x) = d/dx Ln(x), n = 1, 2, 3... The Attempt at a...

I was tutoring a student and I came across the following question. I feel like I'm missing something obvious, but it seems like there are too many variables for an answer to be determined. The attached picture contains all of the question details.

Hi, I'm new to lisp and I've been set some coursework in it and I don't really know how to begin. I need to implement polynomial arithmetic so I can add, subtract and multiply polynomials. So like: $\left(x + y\right)\left(x + y\right) = \left({x}^{2} + 2xy + {y}^{2}\right)$ It also needs to...

(x2)3(2x3)3/(4x)2 I got 1/2x^13, is this correct?

I will be using /= to mean 'does not equal'. From my textbook: Division Algorithm: Let R be any ring and let f(x) and g(x) be polynomials in R[x]. Assume that f(x) /= 0 and that the leading coefficient of f(x) is a unit in R. then unique determined polynomials q(x) and r(x) exist such that 1)...

Hello! (Wave) I want to show that if $I,J$ ideals of $K[x_1, x_2, \dots , x_n]$, then $V(I \cap J)=V(I) \cup V(J)$. Do I have to show that $ V(IJ)=V(I\cap J)$ and then $V(IJ)=V(I)\cup V(J)$? (Thinking) If so, that's what I have tried: $x \in V(IJ) \leftrightarrow (f_i \cdot g_j)(x)=0$, where...

m=1 and l=1 x = cos(θ) What would be the solution to this? Thanks.

Hi, I am just curious, are Hermite and Legendre polynomials related to one another? From what I have learned so far, I understand that they are both set examples of orthogonal polynomials...so I am curious if Hermite and Legendre are related to one another, not simply as sets of orthogonal...

Hi, I need suggestions for picking up some standard textbooks for the following set of topics as given below: Ordinary and singular points of linear differential equations Series solutions of linear homogenous differential equations about ordinary and regular singular points...

Homework Statement Could someone explain how Legendre polynomials are derived, particularly first three ones? I was only given the table in the class, not steps to solving them...so I am curious. Homework Equations P0(x) = 1 P1(x) = x P2(x) = 1/2 (3x2 - 1) The Attempt at a Solution ...

I just had a few questions not directly addressed in my textbook, and they're a little odd so I thought I would ask, if you don't mind. :) -Firstly, I was just wondering, why is it that Legendre polynomials are only evaluated on a domain of {-1. 1]? In realistic applications, is this a limiting...

Homework Statement In the linear space of all real polynomials with inner product (x, y) = integral (0 to 1)(x(t)y(t))dt, let xn(t) = tn for n = 0, 1, 2,... Prove that the functions y0(t) = 1, y1(t) = sqrt(3)(2t-1), and y2 = sqrt(5)(6t2-6t+1) form an orthonormal set spanning the same subspace...

Hey! :o Let $K$ be a field. I want to show that the ring $K[x]$ has infinitely many irreducible polynomials.I have done the following: We suppose that there are finite many irreducible polynomials, $f_1(x), f_2(x), \dots, f_n(x)$ with $deg f_i(x)>0$. Let $g(x)=f_1(x) \cdot f_2(x) \cdots...