Polynomials Definition and 740 Threads

Homework Statement Find T4(x), the Taylor polynomial of degree 4 of the function f(x)=arctan(11x) about x=0. (You need to enter a function.) Homework Equations The taylor polynomial equation Tn(x)= f(x)+(fn(x)(x-a)^n)/n!... The Attempt at a Solution When I take every...

Hello guise. I am familiar to a method of diagonalizing an nxn-matrix which fulfills the following condition: the sum of the dimensions of the eigenspaces is equal to n. As to the algorithm itself, it says: 1. Find the characteristic polynomial. 2. Find the roots of the characteristic...

I just had a discussion with someone who said he thought about quotient rings of polynomials as simply adjoining an element that is a root of the polynomial defining the ideal. For example, consider a field, F, and a polynomial, x-a, in F[x]. If we let (x-a) denote the ideal generated by x-a...

Now I'm trying to get my head around this question. I just know they're going to give us a large degree question like this in the exam... Let's say: I = ∫[e^(x^2)]dx with nodes being x=0 to x=0.5 The 5th degree polynomial is 1 + x^2 + (1/2)(x^4) So my queries are: How would I go about...

Let A \in M_n(F) and v \in F^n. Also...[g \in F[x] : g(A)(v)=0] = Ann_A (v) is an ideal in F[x], called the annihilator of v with respect to A. We know that g \in Ann_A(v) if and only if f|g in F[x]. Let V = Span(v, Av, A^2v, ... , A^{k-1}v).. V is the smallest A-invariant subspace containing...

z^4-4z^3+6z^2-4z-15 =0 How can i factor this polynomial in order to find the solutions?? I tried with the ruffini' rule. and i reached the following equation [(z+1)(-z^3-5z^2+11z-15)] =0 now how can i factor (-z^3-5z^2+11z-15) ? i tried it, but i can not solve it... :/

Homework Statement I'm working on problem 6.11 in Bransden and Joachain's QM. I have to prove 4 different recurrence relations for the associate legendre polynomials. I have managed to do the first two, but can't get anywhere for the last 2 Homework Equations Generating Function: T(\omega...

I'm having trouble with the following question: Construct a polynomial $q(x) \neq 0$ with integer coefficients which has no rational roots but is such that for any prime $p$ we can solve the congruence $q(x) \equiv 0 \mod p$ in the integers. Any hints on how to even start the problem will be...

Homework Statement Let A be an nxn matrix, and C be an mxm matrix, and suppose AB = BC. (a) Prove the following by induction: For every n∈ℕ, (A^n)B = B(C^n). What property of matrix multiplication do you need to prove this? Homework Equations The four basic properties of matrix...

Find a basis for and the dimension of the subspaces defined for each of the following sets of conditions: {p \in P3(R) | p(2) = p(-1) = 0 } { f\inSpan{ex, e2x, e3x} | f(0) = f'(0) = 0} Attempt: Having trouble getting started... So I think my issue is interpreting what those sets...

If i have to show a polynomial x^2+1 is irreduceable over the integers, is it enough to show that X^2 + 1 can only be factored into (x-i)(x+i), therefore has no roots in the integers, and is subsequently irreduceable?

Homework Statement We are given that a set of polynomials on [-1,1] have the following properties and have to show they are unique by induction. I have a way to show they are unique, but is not what he is looking for. I honestly have never seen it presented this way. P_n(x) = Ʃa_in*x^i All...

I am doing a Laplace's equation in spherical coordinates and have come to a part of the problem that has the integral... ∫ P(sub L)*(x) * P(sub L')*(x) dx (-1<x<1) The answer to this integral is given by a Kronecker delta function (δ)... = 0 if L...

In inner product spaces of polynomials, what is the point of finding the angle and distance between two polynomials? How does the distance and angle relate back to the polynomial?

In inner product spaces of polynomials, what is the point of finding the angle and distance between two polynomials? How does the distance and angle relate back to the polynomial?

This is a basic question but I don't think I've ever seen anything like it before. If Q(x) \ = \ b_0(x \ - \ a_1)(x \ - \ a_2)\cdot \ . \ . \ . \ \cdot (x \ - \ a_s) then \frac{P(x)}{Q(x)} \ = \ R(x) \ + \ \sum_{i=1}^s \frac{P(a_i)}{(x \ - \ a_i)Q'(a_i)}I just don't understand where the P(aᵢ)...

Find the number of polynomials of degree $5$ with distinct coefficients from the set $\left\{1, 2, 3, 4, 5, 6, 7, 8\right\}$ that are divisible by $x^2 - x + 1$

1. Find the Maclaurin polynomials of order n = 0, 1, 2, 3, and 4, and then find the nth MacLaurin polynomials for the function in sigma notation. cos(∏x) 2. Here is what I did: p0x = cos (0∏) = 1 p1x = cos(0∏) - ∏sin(0∏)x = 1 p2x = cos(0∏) - ∏sin(0∏)x -\frac{∏2(cos∏x)(x2)}{2!}(...

Homework Statement Factor the polynomial x^2 - 4x + 4 -4y^2 completely. Homework Equations The Attempt at a Solution Rearranging, I get x^2 - 4y^2 - 4x + 4 Then, I know that it is equal to (x -2y)(x+2y) - 4(x-1) and that is my final answer. but, my teacher only considered the...

Can someone help me to identify the type of power series for which the coefficients are palindromic polynomials of a parameter? More specifically, for a particular function f(x;a) with x, a, and f() in ℝ1, a > 0, and an exponential power series representation F0 + F1x/1! + F2x2/2! +...

Suppose that we've been given two polynomials and we want to find their Greatest Common Divisor. For integers, we have the Euclidean algorithm which gives us the G.C.D of the two given integers. Could we generalize the Euclidean algorithm to be used to find the G.C.D of any two given polynomials...

So I was working on eigenvalues of tridiagonal matrices, interestingly I get hermite polynomials as the solution. Is it possible to get an exact form for the zeros of hermite polynomials?

Is there such a thing as a "feral polynomial" ? Saw it mentioned on an Internet forum where someone claimed to be studying "feral polynomials". Closes I could find were "Farrel polynomials" .

x^2+2\\\\ \frac{2}{3} x^3 + \frac{13}{3} x\\\\ \frac{1}{3} x^4 + \frac{14}{3} x^2 + 2\\ \\ \frac{2}{15} x^5 + \frac{10}{3} x^3 + \frac{83}{15} x\\ \\ \frac{2}{45} x^6 + \frac{16}{9} x^4 + \frac{323}{45} x^2 + 2\\\\ \dots [SIZE="6"]?

Is it possible to have a polynomial with an uncountable number of turns? Like we could think of cos(x) as having a countable number of turns. could I have y=x^{\aleph_1} My question may not even make sense.

Ok, to start off I have been examining the structure of polynomials. For instance, consider the general polynomial P(x)=\sum^{n}_{k=0}a_{k}x^{k} (1) Given some polynomial, the coefficients are known. Without the loss of generality...

Homework Statement The two polynomial eqns have the same coefficients, if switched order: a_0 x_n+ a_1 x_n-1 + a_2 x_n-2 + … + a_n-2 x_2 + a_n-1 x + a_n = 0 …….(1) a_n x_n+ a_n-1 x_n-1 + a_n-2 x_n-2 + … + a_2 x_2 + a_1 x + a_0 = 0 …….(2) what is the connection between the roots of...

Homework Statement Find the value or values of k that make the equation true. Homework Equations Starting Equation: (2x + k)(x - 2k) = 2x2 + 9x - 18 The Attempt at a Solution (2x + k)(x - 2k) = 2x2 + 9x - 18 2x2 - 3xk - 2k2 = 2x2 + 9x - 18 <- I just foiled the left side...

Homework Statement Consider the set P4 of all real polynomials if degree <= 4. 1)Prove that P4 is a subspace of the vector space of all real polynomials 2)What is the dimension of the vector space P4. Prove answer by demonstrating a basis and verifying the proposed set is really a basis...

Homework Statement When 3x5 - ax + b is divided by x - 1 and x + 1 the remainders are equal. Given that a, b ε ℝ (a) the value of a; (b) the set of values of b. Homework Equations The Attempt at a Solution f(1) = f(-1) 3 - a + b = -3 + a + b 6 = 2a a = 3 ... [1] Substitute a -3 into 3 - a +...

Hello, I'm studing the hydrogen atom and I found an unified presentation of orhtogonal polynomials in the book by Fuller and Byron. I would like to learn more about it but in the same spirit(for physicits not for mathematicians). Can someone give some references where to find more?

The problem statement Write the following polynomials in x as polynomials of (x-3) Solution should be somewhat analytical in its approach. How would you do something like this? What does it mean? You can use any example to explain it, my specific homework question isn't necessary unless you...

Homework Statement I'm stuck in evaluating an integral in a problem. The problem can be found in Jackson's book page 135 problem 3.1 in the third edition. As I'm not sure I didn't make a mistake either, I'm asking help here. Two concentric spheres have radii a,b (b>a) and each is divided into...

Hi There, I posted this question over at MHF to no avail, I'm not really sure what the ruling is on this kind of thing, I know this site was setup when MHF was down for a long time but you seem to still be active and a lot of clever people are still here so hopefully you don't mind taking a...

Okay, so given a family of orthogonal polynomials under a weight w(x) is described by the differential equation Q(x) f'' + L(x) f' + \lambda f = 0, where Q(x) is a quadratic (at most) and L(x) is linear (at most). with the inner product \langle f | g \rangle \equiv \int_X f^*(x) g(x)...

Hi I know the dimension is 3, two polynomials has dimension 2 only so it cannot span P2. How would I go about showing it if I were to write it down mathematically? Thanks

I have two polynomials (1+x+x^2+x^3) and (1+x+x^2+x^3+x^4), I'm trying to figure out how to compute the product in Mathematica, but it's not working. Any help is appreciated, thanks.

Homework Statement Let K \subseteq L be fields. Let f, g \in K[x] and h a gcd of f and g in L[x]. To show: if h is monic then h \in K[x]. The Attempt at a Solution Assume h is monic. Know that: h = xf + yg for some x, y \in K[x]. So the ideal generated by h, (h) in L[x] equals...

Difficulty : College Homework Statement The Attempt at a Solution I am unsure how to approach this question. I think it involves a process where you add an expression & and subtract it (or multiply & divide) in order to manipulate the equation and rearrange it or reorder it. I've...

Our professor gave us an a problem to solve, she asked us to prove or verify the following identity: http://img818.imageshack.us/img818/5082/6254.png Where \Phi is the Generating function of Legendre polynomials given by: \Phi(x,h)= (1 - 2hx + h2)-1/2 2. This Identity is from...

Let \{ \phi_0,\phi_1,...,\phi_n\} othogonal set of polynomials on [a,b] n>0, with a weight function w(x) prove that \int_{a}^b w(x)\phi_n Q_k (x) \; dx = 0 for any polynomail Q_k(x) of degree k<n ? My work : I think there is a problem in the question since if we take x^2,x^3 on the...

If F is a field of characteristic p, with prime subfield K = GF(p) and u in F is a root of f(x) (over K), then u^p is a root of f(x). Now, I know that x^p \equiv x (\text{mod } p), so isn't it immediately true that f(x^p)=f(x) (over K)? So, 0=f(u)=f(u^p) . I only ask because this type...

Use the error estimate for Taylor polynomials to find an n such that | e - (1 + (1/1!) + (1/2!) + (1/3!) + ... + (1/n!) | < 0.000005 all i have right now is the individual components... f(x) = ex Tn (x) = 1/ (n-1)! k/(n-1)! |x-a|n+1 = 0.000005 a = 0 x = 1 I don't know where to go from here

Hi guys, I seem to still be having problems formatting polynomials in a standard way in Mathematica. I generate them randomly and would like them output using Print in a particular format. Say I have: theFunction=-2 + w^3 (-9 - 3 z) - 7 z + w^2 (4 + 5 z) + w (8 - 2 z^2) + w^4 (-5 z^2...

Polynomials help~~ Heh, so I posted this thread in the wrong category so I'm reposting it! =) Hello. So here was this problem I came across: If x^4-x^3+x^2-x^1+x^0=0, what is the numerical value of x^40-x^30+x^20-x^10+x^0? I did try doing many stuffs (symmetry) & factoring, but I think...

When explicitly given a set of polynomial equations, I am interested in describing its singular locus. I read this from several sources that a point is singular if the rank of a Jacobian at a singular point must be any number less than its maximal possible number. Or is it the locus where all...

Following relation seems to hold: \int^{1}_{-1}\left(\sum \frac{b_{j}}{\sqrt{1-μ^{2}}} \frac{∂P_{j}(μ)}{∂μ}\right)^{2} dμ = 2\sum \frac{j(j+1)}{2j+1} b^{2}_{j} the sums are for j=0 to N and P_{j}(μ) is a Legendre polynomial. I have tested this empirically and it seems correct. Anyway, I...

Homework Statement (see attachment) Homework Equations The Attempt at a Solution I have been attempting a proof by contradiction (for the last statement) for a while now, but I can't seem to reach a contradiction from these premises: 1 ≤ deg(f) ≤ deg(g) (without loss) N ≠...

Hey people, I need to calculate inner product of two Harmonic oscillator eigenstates with different mass. Does anybody know where I could find a formula for \int{ H_n(x) H_m(\alpha x) dx} where H_n, H_m are Hermite polynomials?

Homework Statement Let A_n be the algebraic numbers obtained as roots of polynomials with integer coeffiecients that have degree n. Using the fact that every polynomial has a finite number of roots. Show that A_n is countable. The Attempt at a Solution So an nth degree polynomial has n...