Polynomials Definition and 784 Threads

Are there any general methods to solve the following complex exponential polynomial without relying on numerical methods? I want to find all possible solutions, not just a single solution. e^(j*m*\theta1) + e^(j*m*\theta2)+e^(j*m*\theta3) + e^(j*m*\theta4) + e^(j*m*\theta5) = 0 where...

Homework Statement Let V=C^4 and consider the linear map V->V given by the matrix: {{12,-6,6,-6},{2,21,21,51},{-3,12,12,30},{1,-9,-9,-21}} (Each {...} denotes a row, tried to use Latex but got extremely confused!) Given that chA(X)=(X-6)^4, calculate: (i) The power such that...

In my third year math class we were asked a question to prove that Ho(X) and H1(x) are orthogonal to H2(x), with respect to the weight function e^(-x^2) over the interval negative to positive infinity where Ho(x) = 1 H1(x) = 2x H2(x) = (4x^2) - 2 i know that i have to multiply Ho(x) by...

Homework Statement Define the inner product of two polynomials, f(x) and g(x) to be < f | g > = ∫-11 dx f(x) g(x) Let f(x) = 3 - x +4 x2. Determine the inner products, < f | f1 >, < f | f2 > and < f | f3 >, where f1(x) = 1/2 , f2(x) = 3x/2 and f3(x) = 5(1 - 3 x2)/4 Expressed as a...

I'm trying to show that a set W of polynomials in P2 such that p(1)=0 is a subspace of P2. Then find a basis for W and dim(W). I have already found that the set W is a subspace of P2 because it is closed under addition and scalar multiplication and have showed that. The thing I'm stuck on...

Do Orthogonal Polynomials have always real zeros ?? the idea is , do orthogonal polynomials p_{n} (x) have always REAl zeros ? for example n=2 there is a second order polynomial with 2 real zeros if we consider that there is a self-adjoint operator L so L[p_{n} (x)]= \mu _{n} p_{n} (x)...

this seems more like a physics word problem, I am not even sure how to set this problem up to use it as a polynomial. "A person holds a pistol straight upward and fires. The initial velocity of most bullets is around 1200 ft/sec. The hieght of the bullet is a function of time and is...

Divide the polynomials by using long division. (-9x^6+7x^4-2x^3+5)/(3x^4-2x+1) When I attempted it I started by pulling using 3x^2 . multiplied that by the (3x^4-2x+1) and from there I had to use a fraction of 7/3 or something and then couldn't divide into x cubed. If anyone can...

Homework Statement Find the Taylor Polynomial T2(x) (degree 2) for f(x) expanded about X0. f(x)=3x + cos(3x) X0= 0 Find the error formula and then find the actual (absolute) error using T2(0.6) to approx. f(0.6). The Attempt at a Solution As I've said on this forum before...

Trying to solve a question in linear algebra. P2 is a polynomial space with degree 2. Is P(t): P'(1)=P(2) (P' is the derivative) a subspace of P2. What is the basis ? It seems that it is a subspace with basis 1-t,2-t2. Can anybody explain how this can be found?

Homework Statement If P is the set of all 4-degree polynomials, and W is the subset of all 4-degree polynomials such that p(-2) = p(2), find a set S such that W = span(S). Homework Equations The Attempt at a Solution My guess is that one set that works is x^4, x^2, and 1. My...

Homework Statement A chare +Q is distributed uniformly along the z axis from z=-a to z=+a. Find the multipole expansion. Homework Equations Here rho has been changed to lambda, which is just Q/2a and d^3r to dz. The Attempt at a Solution I have solved the problem correctly...

Homework Statement I need to simplify this: ((84/13)x4y - 4) / (-x + (21/13)x5)y)Homework Equations The Attempt at a Solution I don't know if it can be simplified further. I can't factor anything out that will cancel. I multiplied both the top and the bottom by 13 to get rid of those fractions...

Polynomials divisible by... Homework Statement 1) Explain why n^3 - n is always divisible by 3 for any n that is an element of the natural numbers. 2) Give 2 other polynomials that are always divisble by 3. 3) Give 2 polynomials that are divisible by 2 but not 4, and 2 other polynomials...

Homework Statement A polynomial p(x) leaves the rest 3 when divided by (x+2) and the rest 8 when divided by (x-6). What's the rest r(x) when p(x) is divided by (x+2)(x-6)? Homework Equations The Attempt at a Solution I wrote the three equations: p(x)=q1(x+2) + 3 p(x)=q2(x-6) +...

Homework Statement Consider the polynomial p(x)=x^6-1. (Apply over any field F). (a) Find two elements a,b \in F so that p(a)=p(b)=0. Then use your answer to find two linear factors of p(x). (b) Show that the other factor of p(x) is x^4+x^2+1 (c) Verify the identity...

I see in MATLAB that you can call legendre(n,X) and it returns the associated legendre polynomials. All I need is is the simple Legendre polynomial of degrees 0-299, which corresponds to the first element in the array that this function returns. I don't want to call this function and get this...

Hi I have a set of nonlinear equations f_i(x_1,x_2,x_3...) and I want to find their solutions. After doing some reading I have come across commutative algebra. So to simplify my problem I have converted my nonlinear equations into a set of polynomials p_i(x_1,x_2,x_3...,y_1,y_2...) by...

Homework Statement Let P(n,m) be the space of all polynomials z with complex coefficients, in two variables s and t, such that either z = 0 or else the degree of z(s, t) is <= m - 1 for each fixed s and <= n - 1 for each fixed t. Prove that there exists an isomorphism between Pn (x)...

Homework Statement (x_1 - x_2)(x_3 - x_4) Find permutations of subscripts that leave value unchanged Homework Equations The Attempt at a Solution Okay so I know that it's asking how I should rearrange things and still not change the value. Switching 1 with 2 or 3 with 4 would work but I know...

There's a question in Calculus by Spivak about polynomials and I was wondering about how to construct them to have specific roots or values at certain points. For example it says if x_{1}, ..., x_{n} are distinct numbers, find a polynomial f_{i} such that it's of degree n-1 which is 1 at x_{i}...

I've been trying to work out a bunch of problems that have to do with finding irreducible polynomials, and this one really seemed to stump me... What is the probability that a random monic polynomial over F_3 of degree exactly 10 factors into a product of polynomials of degree less than or...

HI there, I have a tiny question concerning the gcd of polynomials. Assume, \chi is the greatest common divisor of the polynomails p_{ij}, i,j=1,2. I then form q_{11}=p_{11}^2+p_{12}^2,\quadd q_{12}=p_{11}p_{21}+p_{12}p_{22},\quadd q_{21}=p_{21}p_{11}+p_{22}p_{12},\quadd...

Very Interesting Question on Division of Polynomials! [b]1. Question: 'When a polynomial f(x)= x^4 - 6x^3 + 16x^2 - 25x + 10 is divided by another polynomiall g(x)= x^2 - 2x + k, the remainder is x+a. Find the value of a k and a'. Homework Equations [b]3. I tried solving it by...

Here is something that always bugged me, and I think I have an explanation for it now, but I am wondering if it is correct. Alright, the problem to me was that back when I was in Diff-eq, to use undetermined coefficients with polynomials, we would always group together the terms on one side, and...

Homework Statement A is a square matrix of size n, B is of size m, C is an m*n(typo,should be n*m) matrix and n>m ,Rank(C)=m. if AC=CB, prove characteristic polynomial of B divides that of A. Homework Equations nothingThe Attempt at a Solution I think I need to prove any eigenvalue of B is an...

Homework Statement Let S denote the collection of all polynomials of the form p(t) = (2a - b)t^2 + 3(c - b)t + (a - c), where a,b,c are real numbers. Determine whether or not S is a subspace of P2. The Attempt at a Solution Okay, so I know that in order for S to be a subspace, it must...

Question: Can someone explain the following to me: "... the entries of the matrix A - tI_n are not scalars in the field F. They are, however, scalars in another field F(t), the field of quotients of polynomials in t with coefficients from F." I asked someone earlier today, and I got an...

Homework Statement Let P3 be the space of all polynomials (with real coefficients) of degree at most 3. Let D : P3 -> P3 be the linear transformation given by taking the derivative of a polynomial. That is D(a + bx + cx2 + dx3) = b + 2cx + 3dx2: Let B be the standard basis {1; x; x2; x3}...

Homework Statement Show the the equation f(z) = z^4 + iz^2 + 2 = 0 has two roots with |z|=1 and two roots with |z|=sqrt(2), without actually solving the equation.Homework Equations Rouche's theorem, the argument principle?The Attempt at a Solution This is what I have done so far: First show...

Hello, I'm working through Polynomials by Barbeau. I think I may have provided a decent proof for one of the exercises, but I'd like a second opinion. Here's the exercise: Show that every quadratic equation with complex coefficients has at least one complex root, and therefore can be written...

Please somebody help me with this it is very urgent. I have that f(x) = x^5-5x+1 has S_5 as galois group over rationals. ANd M is the splitting field of f(x) over rationals. Then how can I show that : determine f(x) is irreducible over Q({-51}^{1/2})[x] or not? Determine if there is...

I am having trouble evaluating the Legendre Polynomials (LPs). I can do it by Rodrigues' formula but I am trying to understand how they come about. Basically I have been reading Mary L. Boas' Mathematical Methods in the Physical Sciences, 3E. Ch.12 §2 Legendre Polynomials pg566. In the...

Hi, there are a few questions and concepts I am struggling with. The first question comes in 3 parts. The second question is a proof. Question 1: Please Click on the link below :smile: Question 2: Please Click on the link below :smile: For Q2, could you please show me how to...

Homework Statement Find a polynomial with integer coefficient for which \sqrt(2) + i is a zero. Homework Equations The Attempt at a Solution I'm not sure where to really start with this one. It is on my review sheet, and I can't remember how to do it. Could someone give me a hand?

My book reads as follows: "This is an entirely elementary algebraic formula concerning a polynomial in x or order n, say f(x) = a_0 + a_1x + a_2x^2 + ... + a_nx^n . If we replace x by a + h = b and expand each term in powers of h, there results immediately a representation of the form...

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 Show that (x^p^n) - x = product (product c(x)) where the product is taken over irreducible polynomials c(x) in F[x] (order of F[x]=p). (the inside product is taken over polynomials of degree d and the outside product is taken for all d such that d divides n)...

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...