Polynomials Definition and 784 Threads

I came across a problem in my homework to construct a MacLaurin polynomial of the nth degree for \sqrt{1+x}, and had some major problems. I gave up and looked up the answer on the internet, which was fairly complex: \sum \frac{(-1)^{n}(2n)!x^{n}}{(1-2n)(n!)^{2}(4^{n})} Well, I know I...

Is there a simple way to show that when we differentiate the following expression (call this equation 1): Y(x) = \frac{1}{n!} \int_0^x (x-t)^n f(t)dt that we will get the following expression (call this equation 2): Y'(x) = \frac{1}{(n-1)!}\int_0^x (x-t)^{n-1}f(t)dt It's simple...

Homework Statement Write P(x) = x^3+2x+3 as the product of Irreducible Polynomials over Z_5 Homework Equations Polynomial division The Attempt at a Solution I start out by taking out a factor of x+3 That is x+3 \div x^3+2x+3 I get P(x) = x^2-3x+1 which has zero...

Let R be a commutative semiring. That is a triple (R,+,.) such that (R,+) is a commutative monoid and (R,.) is a commutative semigroup. Let {\mathbf \alpha}_i = \alpha_1,\alpha_2,\ldots,\alpha_n . The n-variate indeterminate is just free monoid on n letters. However, it is common to...

Homework Statement X={4^n-3n-1/ n belongs to N} Y={9(n-1)/ n belongs to N } Homework Equations then, XUY is equals to X or Y or N or None of these

Homework Statement Find an approximated value for \sqrt[ ]{9.03} using a Taylors polynomial of third degree and estimate the error. Homework Equations The Attempt at a Solution I thought of solving it by using f(x)=\sqrt[]{x} centered at x_0=9 So...

Homework Statement Which of the subsets of P2 given in exercises 1 through 5 are subspaces of P2? Find a basis for those that are subspaces. (P(t)|p(0) = 2) Homework Equations The Attempt at a Solution The solution manual says that this subset is not a subspace because it...

Homework Statement Let x1, x2, x3 are the roots of the polynomial f(x)=x3+px+q, where f(x)\inQ[x], p\neq0. Find a polynomial g(y) of third degree with roots: y1=x1/(x2+x3-q) y2=x2/(x1+x3-q) y3=x3/(x1+x2-q) Homework Equations The Attempt at a Solution Any ideas? Thank you.

Homework Statement Let f(x)=x5-x2-1 \in C and x1,...,x5 are the roots of f over C. Find the value of the symmetric function: (2x1-x14).(2x2-x24)...(2x5-x54) Homework Equations I think, that I have to use the Viete's formulas and Newton's Binomial Theorem. The Attempt at a...

I'm trying to prove that a polynomial function of degree n has at most n roots. I was thinking that I could accomplish this by induction on the degree of the polynomial but I wanted to make sure that this would work first. If someone could let me know if this approach will work, I would...

Homework Statement Prove unique factorization for hte set of polynomials in x with integer coefficients Homework Equations The Euclidean algorithm may be of some use The Attempt at a Solution Let's say that the polynomial is of the form anx^n + a(n-1)x^(n-1) ... a1x + a0 There...

Basically, i am doing some cryptography, i need to show that a polynomial i have, which is not irreducibale, implies it is not primitive. I am having trouble factorising these rather large polynomials. I have checked to see whether the following polynomials are irreducible and found there...

Homework Statement Let L be the operator on P_3(x) defined by L(p(x)) = xp'(x)+p"(x) if p(x) = a_0(x)+a_1(x)+a_2(1+x^2) calculate L^n(p(x)) Homework Equations stuck between 2 possible solutions i) as powers of x decrease the derivatives of p(x) increase ii) as derivatives...

Homework Statement How do i solve this integral ? \int \big( \sqrt{x^{3}+1} + \sqrt[3] {x^{2}+2x} \big) \ dx Homework Equations The Attempt at a Solution what is the appropriate substitution to make here

Hi, I'm doing calc-2, and I have hard time understanding and visualizing the idea of Taylor approximation in my head. By the same time I have no problems solving homework on this topic. Can someone please explain how I should visualize and think about approximations using Taylor Polynomials...

I'm trying to understand the reminder of Maclaurin polynomials http://estro.uuuq.com/0.png http://estro.uuuq.com/1.png [PLAIN]http://estro.uuuq.com/2.png [PLAIN][PLAIN]http://estro.uuuq.com/3.png [PLAIN][PLAIN]http://estro.uuuq.com/4.png Here I show few attempts to use substitution...

Homework Statement Two spherical shells of radius ‘a’ and ‘b’ (b>a) are centered about the origin of the axes, and are grounded. A point charge ‘q’ is placed between them at distance R from the origin (a<R<b). Expand the electrostatic potential in Legendre polynomials and find the Green...

Homework Statement Using binomial expansion, prove that \frac{1}{\sqrt{1 - 2 x u + u^2}} = \sum_{k} P_k(x) u^k. Homework Equations \frac{1}{\sqrt{1 + v}} = \sum_{k} (-1)^k \frac{(2k)!}{2^{2k} (k!)^2} v^k The Attempt at a Solution I simply inserted v = u^2 - 2 x u, then...

Homework Statement The problem is to prove the identity B_k(1/4) = 2^{-k} B_k(1/2) for even k. Homework Equations The Bernoulli polynomials B_k(y) are defined by the generating function relation \frac{xe^{xy}}{e^x-1} = \sum_{k=0}^{\infty} \frac{B_k(y) x^k}{k!}. The Attempt at a Solution...

Homework Statement Find a Taylor or Maclaurin polynomial to apporximate ln(1.75) using 6 terms. Homework Equations The Attempt at a Solution I now that a MacLaurin polynomial is as follows.. c=0 and a Talyor polynomial is as follows.. so do I assume I'm working...

I am wondering how you determine how many polynomials of degree, let's say b, are in Zn[x]. From what I gather, it looks like it does not depend on what b is, but rather what n is. Namely, n^2. Is this correct?

Does anyone know if it possible to generate elementary symmetric polynomials in Maple (I am using version 12), and if so, how? I have scoured all the help files, and indeed the whole internet, but the only thing I have found is a reference to a command "symmpoly", which was apparently...

Homework Statement The first 3 parts of this 4 part problem were to derive the first 5 Hermite polynomials (thanks vela), The first 5 Legendre polynomials, and the first 5 Laguerre polynomials. Here is the last part: Write the polynomial 2x^4-x^3+3x^2+5x+2 in terms of each of the sets of...

Ok, I can plot a single polynomial easy enough such as 3*(x^2)-1 using fplot, but I want to graph multiple polynomials. When I try to use the plot it doesn't work even for one though. The graph is completely wrong. ie I make a new m-file. x = [-1:1]; y = 3*x.^2 - 1; Then call the...

Homework Statement I need to evaluate the following integral: \int_{-\infty}^{\infty}x^mx^ne^{-x^2}dx I need the result to construct the first 5 Hermite polynomials. Homework Equations The Attempt at a Solution First I tried arbitrary values for "m" and "n". I was not able to...

given the 'normalized' Chebyshev and Legendre Polynomials \frac{L_{2n}(x)}{L_{2n}(0)} and \frac{T_{2n}(x)}{T_{2n}(0)} for n even and BIG 2n--->oo then it would be true that (in this limit) \frac{L_{2n}(x)}{L_{2n}(0)}=\frac{sin(x)}{2x} and \frac{T_{2n}(x)}{T_{2n}(0)}=J_{0}(2x) here...

Hi. Thanks for the help. Homework Statement Find a basis for the set of polynomials in P3 with P'(1)=0 and P''(2)=0. Homework Equations P' is the first derivative, P'' is the second derivative. The Attempt at a Solution The general form of a polynomial in P3 is ax^3+bx^2+cx+d...

U and W are subspaces of V = P3(R) Given the subspace U{a(t+1)^2 + b | a,b in R} and W={a+bt+(a+b)t^2+(a-b)t^3 |a,b in R} 1) show that V = U direct sum with W 2) Find a basis for U perp for some inner product Attempt at the solution: 1) For the direct sum I need to show that it...

Homework Statement The problem is to find the inverse laplace of \frac{s^2-a^2}{(s^2+a^2)^2} I am supposed to use the residue definition of inverse laplace (given below) The poles of F(s) are at ai and at -ai and they are both double poles. Homework Equations f(t) =...

Homework Statement Compute the 6th derivative of f(x) = arctan((x^2)/4) at x = 0. Hint: Use the Maclaurin series for f(x). Homework Equations The maclaurin series of arctanx which is ((-1)^n)*x^(2n+1)/2n+1 The Attempt at a Solution I subbed in x^2/4 for x into the maclaurin...

Homework Statement M and N are positive integers with M>N. The division algorithm for integers tells us there exists integers Q and R such that M=QN+R with 0\leqR<N. The division algorithm for real polynomials tells us that there exist real polynomials q and r such that xM - 1 = q(xN - 1) +...

Homework Statement For each matrix A below, let T be the linear operator on R3 thathas matrix A relative to the basis A = {(1,0,0), (1,1,0), (1,1,1)}. Find the algebraic and geometric multiplicities of each eigenvalues, and a basis for each eigenspace. a) A =...

Homework Statement Let f be a polynomial of degree n >= 1 with all roots of multiplicity 1 and real on R. Prove that f has at most one more real root than f' f' has no more nonreal roots than f Homework Equations We are given the Gauss Lucas theorem: Every root of f' is contained in...

Hi everyone! Having spent many fruitless hours Googling this I stumbled upon this forum, and am hoping you may be able to help... I'm looking for a way to interpolate between two polynomials. These two lines are related and run along in a near-parallel fashion, and I want to divide the gap...

Alright, I'll be honest. I was extremely tired and slept all through the lesson in Algebra today lol. And now I need help with factoring polynomials. Example problems that I need help on: 7h3+448 Perfect square factoring - y4-81 Grouping - 3n3-10n2-48n+160 You don't have to answer...

Homework Statement find the rank and nullity of the linear transformation T:U -> V and find the basis of the kernel and the image of T Homework Equations U=R[x]<=5 V=R[x]<=5 (polynomials of degree at most 5 over R), T(f)=f'''' (4th derivative) The Attempt at a Solution Rank = 2...

Homework Statement Suppose A is a 2x2 real matrix with characteristic polynomial f(t) = t2 - 5t +4. Find a real polynomial g(t) of degree 1 such that (g(A))2 = A. Suppose A is a 2x2 complex matrix with A2 ≠ O. Show that there is a complex polynomial g(t) of degree 1 such that (g(A))2 = A...

I have problem understand in one step of deriving the Legendre polymonial formula. We start with: P_n (x)=\frac{1}{2^n } \sum ^M_{m=0} (-1)^m \frac{2n-2m)}{m!(n-m)(n-2m)}x^n-2m Where M=n/2 for n=even and M=(n-1)/2 for n=odd. For 0<=m<=M \Rightarrow \frac{d^n}{dx^n}x^2n-2m =...

Nevermind Solved it.

Hello, First of all, I am not trying to "spam" subforums. I found out that my thread shouldn't be posted under homework. Anyways, here it is. Integration Let say there's a polynomial, 5x+6 and you want to integrate from 0 to 3 respect to x, how do you input in MATLAB? (I guess you can't...

Homework Statement Prove for each square matrix B there is a real polynomial p(x) (not the zero polynomial) so p(B)=0 Homework Equations Rank-nullity? dimv = r(T) + n(T) The Attempt at a Solution I've found the dimension for nxn square matrices (n²) and a basis (1 in one place and...

Homework Statement I think I saw another thread answer this question, but I was a little lost whilst reading it. I have just recently learned of the rational root theorem and was using it quite happily; figuring out what possibly answers went with cubic and quartic polynomials gave new...

I have need to calculate the residues of some functions of the form \frac{f(x)}{p(x)} where p(x) is a polynomial. To be more specific I have already calculated the 2 residues of \frac{1}{x^2+a^2}. That one was quite easy. Now I'm asked to calculate the residues of...

POlynomials (or Taylor series ) of the form P(x)= \sum_{n}a_{2n}X^{2n} with a_{2n}\ge 0 strictly have ALWAYS pure imaginary roots ?? it happens with sinh(x)/x cos(x) could someone provide a counterexample ? is there an hypothesis with this name ??

Homework Statement I recently came across this integral while doing a problem in electromagnetism (I'm not sure if there exists a nice analytic answer): \int_{0}^{\pi}P_m(\cos(t))P_n(\cos(t)) \sin^2(t) = \int_{-1}^{1}P_m(x) P_n(x) \sqrt{1-x^2}, Homework Equations P_m(x) is the m^th...

hello everyone:smile: for i=1,2,...,(n+1) let P_{i}(X)=\frac{\prod_{1\leq j\leq n+1,j\neq i}(X-a_j)}{\prod_{1\leq j\leq n+1,j\neq i}(a_i-a_j)} prove that (P_1,P_2,...P_{n+1}) is basis of \mathbb{R}_{n}[X] . i already have an answer but i don't understand some of it. ... we have...

1. Let W be the linear subspace spanned by the polynomials 1 and x. Find an orthogonal projection of the polynomial p(x) = 1+x^2 to W. Find a basis in the space W(perp) My problem is that I don't know how to represent W as a matrix so that I could apply the orthogonal projection formula...

this is really perplexing. how can it be exact? simpsons rule uses quadratics to approximate the curve. how can it be exact if I am approximating a cubic with a quadratic?

trying to show that polynomials f(x), g(x) in Z[x] are relatively prime in Q[x] iff the ideal they generate in Z[x] contains an integer.Thanks .Not homework

given a set of orthogonal polynomials p_{n} (x) with respect to a certain positive measure \mu (x) > 0 on a certain interval (a,b) then i have notices for several cases that f(z) defined by the integral transform \int_{a}^{b}dx\mu(x)cos(xz)=f(z) has ALWAYS only real roots ¡¡ *...