Polynomials Definition and 740 Threads

Let V=C[x]10 be the fector space of polynomials over C of degree less than 10 and let D:V\rightarrowV be the linear map defined by D(f)=f' where f' denotes the derivatige. Show that D11=0 and deduce that 0 is the only eigenvalue of D. find a basis for the generalized eigenspaces V1(0), V2(0)...

What's the difference between polynomials (as elements of a ring of polynomials) and polynomial functions??

Homework Statement Take two polynomials f(x) and g(x) over a field, and suppose that gcd(f,g)=1. Show that any rational function, \frac{h(x)}{f(x)g(x)} can be written in the form (\frac{p(x)}{f(x)}) + (\frac{q(x)}{g(x)}) for some polynomials, p and q. The Attempt at a Solution I...

Homework Statement I have a question which asks me to find the dimensions of the subspace of even polynomials (i.e. polynomials with even exponents) and odd polynomials. I know that dim of Pn (polynomials with n degrees) is n+1. But how do I find the dimensions of even n odd polynomials...

Hi guys. I am new, and i need help badly. I have been asked this question and I have no idea how to do it. Any help would be appreciated! Show that the Hermite polynomials H2(x) and H3(x) are orthogonal on x € [-L, L], where L > 0 is a constant, H2(x) = 4x² - 2 and H3(x) = 8x³ - 12x...

Homework Statement Working in (Z/7Z)[x], compute the greatest common divisor of the polynomials (X^2 - 3X + 2) and (X+6). The Attempt at a Solution I need help understanding how to compute this. It would be greatly appreciated if someone gave me an example with different modulus and...

Homework Statement Find all monic polynomials of degrees 2 and 3 in (Z/2Z)[x]. Determine which ones are irreducible, and write the others as products of irreducible factors. The Attempt at a Solution I know that factors of degree 1 correspond to roots in Z/2Z and that monic...

Problem: We want to calculate a polynomial of degree N-1 that crosses N known points in the plane. Solution A: solving a NxN system of linear equation (Gauss elimination) Solution B: construction from Lagrange basis polynomials. One of my professors said that the first solution is...

Hello. I don't know what to do with one integral. I am sure it is something very simple, but I just don't see it... For some reason I am not able to post the equations, so I am attaching them as a separatre file. Many thanks for help.

Hey there, physics forums! A question occurred to me the other day: Is it true that if f \in \mathbb{Z}[x] is monic and irreducible over \mathbb{Q} , then for at least one a \in \mathbb{Z} , f(a) is prime? I can't prove it, but I suspect it's true. Does anyone know if this problem...

Suppose I have a REALLY big polynomial: a_0 + a_1 x + a_2 x^2 + a_3 x^3+a_4 x^4+ \cdots + a_n x^n I can rewrite the polynomial as a combination of multiplication and addition operators (instead of exponents) that a computer tends to like as such: a_0 + x \left( a_1 + x \left( a_2 + x \left(...

I have troubles arriving at the solution to this question: Consider the transformation T: P3-->P3 given by: T(f)=(1-x^2)f '' - 2xf ' Determine the bases for its range and kernel and nullity and rank Can anyone explain how should i go about finding the bases for its kernel and range?? i get 0...

Homework Statement I want to prove that a polynomial f(x) and a polynomial g(x) with degrees of k,n where k,n are positive even integer, n>k that limit x-> - infinity of f(x)-g(x)=-infinity Homework Equations a polynomial can be written as a1x^n+a2x^(n-1)...+a(n-1)x+an The...

Homework Statement For spherical coordinates, we will need to use Legendre Polynomials, a.Sketch graphs of the first 3 – P0(x), P1(x), and P2(x). b.Evaluate the orthogonality relationship (eq 3.68) to show these 3 functions are orthogonal to each other. (3 integrals). c.Show that the...

Homework Statement Does a general formula exist? \sum \limits_{k=0}^{m_1} a_kx^k\cdot\sum \limits_{k=0}^{m_2} b_kx^k=\sum \limits_{k=0}^{m_1+m_2} c_kx^k Homework Equations The Attempt at a Solution I am having trouble understanding the relation between the c coefficients in the product and...

Show that the sum of the cubes of the root of the equation x3 + (lambda)x + 1 = 0 is -3 Show also that there is no real value of (lambda) for which the sum of the fourth powers of the roots is negative. This question is in one of the past papers of further maths and I don't know how...

Homework Statement Solve for X. Homework Equations (3 / x+2) - (1 / x) = 1 / 5x The Attempt at a Solution (NOTE: I always had difficulties with fractions) (3x - x - 2 / x² + 2) - (1 / 5x) = 0 (2x - 2 - x² - 2) / 5x³ + 10x (-x² + 2x - 4) / 5x³ + 10x ^ I attempted a few...

Homework Statement Find the values of Σ(a^2), Σ(1/a), Σ(a^2)(B^2) and ΣaB(a + B) for: x^4 - x^3 + 2x + 3 = 0 Homework Equations Σa = 1, ΣaB = 0, ΣaBC = -2, aBCD = 3 The Attempt at a Solution I found the Σ(a^2) and Σ(1/a) successfully correct bt could neither find Σ(a^2)(B^2)...

i'm pretty new to polynomials and I have qns, which I do not know how to solve. x5 + ax3 + bx2 - 3 = (x2 - 1)Q(x)- x - 2 Q(x) is a polynomial. Solve a and b. I know the degree of Q(x) is 3, so I subst Q(x) into x3, but I got stuck. Help. Thanks in advance.

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...

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 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...

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...