Polynomials Definition and 740 Threads

So I am going through Serge Lang's Algebra and he left a proof as an exercise, and I simply can't figure it out... I was wondering if someone could point me in the right direction: If f is a polynomial in n-variables over a commutative ring A, then f is homogeneous of degree d if and only if...

A question in my textbook asks me to show that the first four Hermite polynomials form a basis of P3. I know how to do the problem, but I don't really understand what is behind the scenes. Why can we use the coefficients of a polynomial as a column vector? I don't really know how to ask...

I was wondering if it were possible to efficiently solve the common root of 4 polynomials in 4 variables algebraically. I am currently using a gradient descent method, which can find these roots in a couple seconds; however, I am concerned about local minima. So far I have attempted to use...

an idea i had: factorizing taylor polynomials Can any taylor polynomial be factorized into an infinite product representation? I think so. I was able to do this(kinda) with sin(x), i did it this way. because sin(0)=0, there must be an x in the factorization. because every x of...

Homework Statement Apologies in advance as I can't use any formatting yet... In linear algebra class, we're finding the length of vectors (polynomials) by computing its inner product with itself and finding the square root of the resultant value. The inner product is defined in this case (not...

I was hoping someone could point me in the direction of a suitable extension of Chebyshev polynomials to mutple dimensions? I find Chebyshev polynomials useful in situations when I need to fit some function in a general way, imposing as little pre-concieved ideas about the form as possible...

Homework Statement Let f, g be nonzero polynomials with deg (f) \geq deg (g). Show that there is a unique monomial bx^{k} where deg(f(x) - bx^{k}g(x)) < deg (f). Homework Equations see above The Attempt at a Solution I define polynomials f and g, with deg(f) = n and deg (g) = m...

I know this isn't in the right format, but I figured I'd get a better answer here than anywhere else. In my last exam, there was a question asking to prove (a + bi - except there were values for a and b, but i forgot them) was a solution to a polynomial of the 3rd degree. Said polynomial was...

Hey guys, I've been working on a quantum related problem in my math physics class and I've run into a snag. When dealing with Legendre Polynomials ( specifically : P_{lm} (x) ), I can find the general expression that can be used to derive the polynomial for any sets of l and m (wolfram math...

Homework Statement Show that the one-dimensional Schr¨odinger equation ˆ (p^2/2m) ψ+ 1/2(mw^2)(x)ψ = En ψ can be transformed into (d^2/d ξ ^2)ψ+ (λn- ξ ^2) ψ= 0 where λn = 2n + 1. using hermite polynomials Homework Equations know that dHn(X)/dX= 2nHn(x) The Attempt at a...

Hi: I am trying to show: If f is analytic in C (i.e., f is entire) and : |z|>1 implies |f(z)|>1. Prove that f(z) is a polynomial. I have tried using the fact that f(z)=Suma_nz^n (Taylor series) valid in the whole of C, and derive a contradiction assuming |f(z)|>1 for |z|>1 . I...

This is problem 13.3 from Rudin's Real and Complex analysis. It is not homework. Is there a sequence of polynomials {Pn} such that Pn(0) = 1 for n = 1,2,3,... but Pn(z) -> 0 for all z != 0 as n -> infinity? My guess here is no. Sketch of proof: Assume such a sequence existed. Then we...

Homework Statement f(x) = x^3 + x^2 - 11x^2 -9x +18 when f(x) = 6 Homework Equations Using Mathcad Polyroots, solve function, The Attempt at a Solution I don't know how to paste mathscad, but i can get solution using polyroots for f(x) = 0 I don't know how...

Homework Statement Let W equal the set of all polynomials in F[x] with degree less than or equal to n-1 such that the sum of the coefficients of the terms is 0. Find a basis of W over F. The Attempt at a Solution I don't know where to begin to find the basis. Do I use the fact that the sum...

"Let f is a polynomial from k[x,y], where k is a field. Suppose that x appears in f with positive degree. We view f as an element of k(y)[x], that is polynomial in x whose coefficients are rational functions of y." I think I am missing something...why do we need rational functions here? can't...

I don't understand how to factor a polynomial over Z3 [x], Z7 [x], and Z11 [x] I need to factor the polynomail x3 - 23x2 - 97x + 291 PLEASE HELP!

Homework Statement lim x-> -1 x3-6x2+11x - 6 x3-4x2+19x + 14 Homework Equations The Attempt at a Solution

Lagrange Polynomals are defined by: lj(t)= (t-a0) ...(t-aj-1)(t-aj+1)...(t-an) / (aj-a0)...(aj-aj-1)(aj-aj+1)...(aj-an) A) compute the lagrange polynomials associated with a0=1, a1=2, a2=3. Evaluate lj(ai). B) prove that (l0, l1, ... ln) form a basis for R[t] less than or equal to n...

Hello! Is there any way of calculating the integral of H_n(x) * H_m(x) * exp(-c^2 x^2) with x going from -infinity to +infinity and c differs from unity. I'm aware that c=1 is trivial case of orthogonality but I'm really having a problem with the general case. (I should say that this isn't a...

Hi everyone, I have to demonsrate that for every real polynomial, P Q and R, I have : P²=X(Q²+R²) ==imply==> P=Q=R=0 Using degrees, we can easily demonsrate the above. However, I'm looking for another way, without using that.

It is basic knowledge that if a polynomial P(x) of nth degree has a root or zero at P(a), then (x-a) is a factor of the polynomial. However, can this be proved? or is this more of a definition of roots of polynomials?

Would the GCD of x^2+x+c and (x-a)^2+(x-a)+c always be 1?

Whats a good book to learn about these topics?

I'm not reading a text in English, so I should clarify some notation first: P[X], is the set of all polynomials. P[[X]], is the set of formal power series, which may infinitely many nonzero coefficients. I'm asked to prove that X is the only prime(?correct term?) element in P[[X]]. By saying...

I'm looking right now at what purports to be the normalisation condition for the associated Laguerre polynomials: \int_0^\infty e^{-x}x^k L_n^k(x)L_m^k(x)dx=\frac{(n+k)!}{n!}\delta_{mn} However, in the context of Schroedinger's equation in spherical coordinates, I find that my...

Homework Statement Find all the monic irreducible polynomials of degree \leq 3 in \mathbb{F}_2[x], and the same in \mathbb{F}_3[x].Homework Equations n/aThe Attempt at a Solution n/a

Hello again my favourite helpers (or should I say, saviours!) :-p Long time no speak, but I am in more need of an explanation than an answer. In this mathematics textbook I have, it gives an explanation under the heading Polynomials in general. It goes as follows: "If f(x)=...

Homework Statement 3/x^2+2x - 2/x^2+x-2 + 4/x^2(x-1) Find the lowest common denominator and solve. Homework Equations The Attempt at a Solution I factored x(x+2) - (x-1)(x+2) + x^2(x-1) It looks like (x+2), (x-1) are common but what to do with the x & x^2 left over? Thank...

Homework Statement Let V=Pn(C) (polynomials of nth degree with complex coefficients), where n >=1. Find a basis for W=V s.t. every f(x) belonging to the basis satisfies f(0)f(1)= -1. (Demonstrate that the set you find is linearly independent and spans W.) Homework Equations For...

Hi! Brief question: I wonder which conditions should a polynomial function of odd degree fulfill in order to be symmetric to some point in the plane. Are there such conditions?

1) As far as i think i understand, stability of a polynomial means; the polynomial correspond to a (its inverse laplace transform) differential equation, and the differential equations solution is dependant on the coefficiants of this polynomial? if the polynomial is unstable the solution of...

how would i write the depressed polynomial for the obvious root of f(x) = x5 - .5x4 - 5.5x3 - 3.5x2 - 6.5x - 3

Hi, In Wikipedia it's stated that "... Legendre polynomials are useful in expanding functions like \frac{1}{\sqrt{1 + \eta^{2} - 2\eta x}} = \sum_{k=0}^{\infty} \eta^{k} P_{k}(x) ..." Unfortunately, I am failing to see how this can be true. Is there a way of showing this...

Okay, I'm doing my yr 11 Mathematics assignemnt and I need help. How do you find a Logistic or Polynomial function/equation using data? (in steps please!) I could just give some example data as... an example, but I think I'll just throw 1 small section of my assignment, use the data or...

Can anyone PROOVE how to find out the normalisation of hermite polynomial?

Homework Statement How to use eigenfunction expansion in Legendre polynomials to find the bounded solution of (1-x^2)f'' - 2xf' + f = 6 - x - 15x^2 on -1<= x <= 1 Homework Equations eigenfunction expansion The Attempt at a Solution [r(x)y']' + [ q(x) + λ p(x) ]...

Can anyone tell me how to find the exact number of primitive polynomials of degree n over a finite field F_q? I believe the answer is φ(q^n-1)/n, but I cannot find a proof of this. Thanx in advance.

Since the Hermite Polynomials are orthogonal, could one state that they span all polynomials f where f : R \rightarrow R? This would be EXTREMELY useful for the harmonic oscillator potential in quantum mechanics...

I'm trying to figure out how to prove that every polynomial in \mathbb{Z}_9 can be written as the product of two polynomials of positive degree (except for the constant polynomials [3] and [6]). This basically is just showing that the only possible irreducible polynomials in \mathbb{Z}_9 are the...

Are there any ploynomials p(x) such that p(x)^2 -1 = p(x^2+1) for all x? To cut it short: With CAS software I have verified that there are no solutions except p(x) = 1.618... and p(x) = -0.618... (constants) up to order 53 or so, but I have to prove this (or find the other solutions)...

Homework Statement Recall that two polynomials f(x) and g(x) from F[x] are said to be relatively prime if there is no polynomial of positive degree in F[x] that divides both f(x) and g(x). Show that if f(x) and g(x) are relatively prime in F[x], they are relatively prime in K[x], where K is an...

Hey Everyone, I'm reading a paper by Claude LeBrun about exotic smoothness on manifolds and he is talking about a connection between polynomials and groups that I am not familiar with (or at least I think that's what he's talking about). He's creating a line bundle (which happens to be...

I defined x as a syms, then work with polynomials involving x. But then I can't find an evaluation command for these polynomials at some x value. Anyone knows how I can get it to work? Thank you in advance. There's another system where you can define polynomials just like matrices, e.g. x^2+1...

I need to show that: \sum_{n=0}^{\infty}\frac{H_n(x)}{n!}y^n=e^{-y^2+2xy} where H_n(x) is hermite polynomial. Now I tried the next expansion: e^{-y^2}e^{2xy}=\sum_{n=0}^{\infty}\frac{(-y)^{2n}}{n!}\cdot \sum_{k=0}^{\infty}\frac{(2xy)^k}{k!} after some simple algebraic rearrangemnets i...

I thought this was rather odd, and wanted to just show it to see what you all thought of it. Well, also, if anyone knows what I should read to exactly understand what I did. 1: Let's define each answer of a polynomial such as (ax + b)(cx + d) as x1 = (-b/a), x2 = (-d/c). 2: The...

Homework Statement Prove that f(x)=x^3-7x+11 is irreducible over Q Homework Equations The Attempt at a Solution I've tried using the eisenstein criterion for the polynomial. It doesn't work as it is written so I created a new polynomial...

Homework Statement Suppose there are two polynomials over a field, f and g, and that gcd(f,g)=1. Consider the rational functions a(x)/f(x) and b(x)/g(x), where deg(a)<deg(f) and deg(b)<deg(g). Show that if a(x)/f(x)=b(x)/g(x) is only true if a(x)=b(x)=0. Homework Equations None The Attempt at...

Homework Statement For polynomials over a field F, prove that every non constant polynomial can be expressed as a product of irreducible polynomial. Homework Equations No relevant equations. The Attempt at a Solution Well a hint the teacher gave me was that the degree of the...

are hermite interpolationg polynomials necessarily cubic even when used to interpolate between two points? this page would have me believe so in calling it a "clamped cubic" : http://math.fullerton.edu/mathews/n2003/HermitePolyMod.html

Hello, I'm having trouble with this question and was wondering if someone could give me hints or suggestions on how to solve it. Any help would be greatly appreciated thankyou! :) Find the Taylor polynomial of degree 3 of f (x) = e^x about x = 0 and hence find an approximate value for...