Polynomials Definition and 784 Threads

Four "proof that" exercises about taylor polynomials Homework Statement Definition: A function f is called C^n if f n times derivable and if the n-time derivable f^(n) is continuos. If is from class C^n then its called f ε C^n Exercise 1) Be f ε C^n in the interval [a,x]. Be P a polynomial...

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...

Homework Statement I encountered the following integral in my research, and I've yet to find an analytic solution: I(n_1,n_2,n_3) = \int_{-1}^{1} d(\cos\theta_1) \int_{-1}^{1} d(\cos\theta_2) P_{n_1}(\cos\theta_1) P_{n_2}[\cos(\theta_1-\theta_2)] P_{n_3}(\cos\theta_2) where P_n(x) is the nth...

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...

Homework Statement Let P denote the set of all polynomials whose degree is exactly 2. Is P a vector space? Justify your answer. Homework Equations (the numbers next to the a's are substripts P is defined as ---->A(0)+A(1)x+A(2)x^2 The Attempt at a Solution I really don't...

Hi! I have a sequence of difference polynomials (which I obtained by the method of finite differences) and I would like to find out if there is a recurrence relation between them. The generating function of general difference polynomials is given by: How would one write the...

Homework Statement I am trying to work out a solution to the following problem, where we are working in a field K complete with respect to a discrete valuation, with valuation ring \mathcal{O} and residue field k. Q: Let f(X) be a monic irreducible polynomial in K[X]. Show that if f(0) \in...

Homework Statement I'm supposed to show that the Hermite Polynomials are in Schwartz space h_n = \frac{1}{\sqrt{n!}}(A^{\dagger})^n h_0 where A^{\dagger} = \frac{1}{\sqrt{2}}(-\frac{d}{dx} + x) and h_0 = \pi^{-1/4}e^{-x^2/2} Homework Equations Seminorm...

Homework Statement Find x (17-x)^2(11-x)+256-32(17-x)-64(11-x)=0 Homework Equations The Attempt at a Solution This eq has 3 solutions. I solved this by multiplication. Is this some other easier way. Perhaps to group some of the factors 17-x and 11-x. Tnx for the answer.

Homework Statement Integrate the expression Pl and Pm are Legendre polynomials Homework Equations The Attempt at a Solution Suppose that solution is equal to zero.

I'm trying to prove the following, which is left unproven in something I'm reading on ruler-and-compass constructions: If ax^3+bx^2+cx+d is a polynomial over a subfield F of ℝ, and p+q\sqrt{r} is a root (with \sqrt{r}\notin F) then p-q\sqrt{r} is also a root. The theorem immediately before...

Given polynomials of degree n > 2, such that they have the form of p(x) \ = \ x^n \ + \ a_1x^{n - 1} \ + \ a_2x^{n - 2} \ + \ a_3x^{n - 3} \ + \ ... \ + \ a_{n - 2}x^2 \ + \ a_{n - 1}x \ + \ a_n. And \ \ all \ \ of \ \ the \ \ a_i \ \ are \ nonzero \ integers \ (which...

I know this may be a very stupid question, but I would really like to know. Is the determinant and the characteristic polynomial of an equation unique? I did several textbook questions and when I look at the solutions, they end up with completely different answers. Sometimes I am wrong and see...

http://bildr.no/view/1030479 The link above, it is my own and it is a bit disorderly, I think should explain taylor polynomials. In one assignent one had an assignment to derive taylor polynomials for cost^2 If one use the derivation rules with chain one get 2t for first derivative and...

we are given B = CAC^-1 Prove that A and B have the same characteristic polynomial given a hint: explain why ƛIn = CƛInC^-1 what I did was: B = CAC^-1 BC = CA Det(BC) = Det(CA) Det(B) Det(C) = Det(C) Det(A) Now they’re just numbers so I divide both sides by Det(C) Det(B) = Det(A)...

Homework Statement My professor wrote a paper about tennis a couple of years ago and he's asked us to recreate some of his results. So I have this vector: I'm supposed to multiply it with this matrix: to get (p1, 0, 0, 0, q1). p1 is: q1 is similar but the p's and q's...

Hello everyone, Sorry if this is in the wrong sub-forum, I wasn't sure exactly where to place it. I was wondering if there is an orthogonality relationship for the Legendre polynomials P^{0}_{n}(x) that have been converted to cylindrical coordinates from spherical coordinates, similar to...

Hello! This is my first post so forgive any errors of decorum. :o) I am a student working toward a degree in astrophysics but I'd like to jump a few years ahead when it comes to the study of exoplanets. While examining some data about the new discovery of Kepler-22b, I noticed a plotted data...

Homework Statement Nullity(B-5I)=2 and Nullity(B-5I)^2=5 Characteristic poly is: (λ-5)^12 Find the possible jordan forms of B and the minimal polynomials for each of these JFs. The Attempt at a Solution JFs: Jn1(5) or ... or Jni(5). Not sure how to find these jordan forms and minimal polynomials.

Does this actually work well? We won't learn isomorphisms in linear algebra, but a friend of mine showed me an example as I prefer to work with vectors and matrices rather than polynomials (All of my problem sets are with matrices and vectors). For example, if I wanted to find a basis for P3...

Homework Statement Let V be a finite dimensional complex vector space and T be the linear operator of V. Prove that the following are equivalent a V has a basis consisting of eigenvectors of T. b T can be represented by a diagonal matrix. c all the eigenvalues of T have multiplicity...

Homework Statement I'm given that the function f(x) is n times differentiable over an interval I and that there exists a polynomial Q(x) of degree less than or equal to n s.t. \left|f(x) - Q(x)\right| \leq K\left|x - a\right|^{n+1} for a constant K and for a \in I I am to show that Q(x)...

Is it true that polynomials of the form : f_n= x^n+x^{n-1}+\cdots+x^{k+1}+ax^k+ax^{k-1}+\cdots+a where \gcd(n+1,k+1)=1 , a\in \mathbb{Z^{+}} , a is odd number , a>1, and a_1\neq 1 are irreducible over the ring of integers \mathbb{Z}...

Homework Statement Do the polynomials t^{3} + 2t + 1,t^{2} - t + 2, t^{3} +2, -t^{3} + t^{2} - 5t + 2 span P_{3}? Homework Equations N/A The Attempt at a Solution My attempt: let at^{3} + bt^{2} + ct + d be an arbitrary vector in P_{3}, then: c_{1}(t^{3} + 2t + 1) + c_{2}(t^{2} -...

I have completed a difference table for 4 points, x0, x1, x2, x3 and found the third degree poly that goes through these four points. Now I need to know how to make the polynomial of second degree that interpolates x0, x2, and x3. Do I just need to remake the table for 3 points, now excluding...

Homework Statement Here is the entire problem set, but (obviously) you don't have to do it all, if you could just give me a few hints on where to even start, because I am completely lost. Recall that we found the solutions of the Schrodinger equations (x^2 - \partial_x ^2) V_n(x) =...

Hello all I had a simple question that I am intuitively sure I know the answer to but can't quite prove it. Suppose k is a polynomial in x and y, and k(x-1) = q for q some polynomial in y. Then is k = 0 ? How do I verify that k must be equal to 0? I can see that to just get a polynomial...

I am taking a first course in algebra and this is a problem in my textbook that has me stumped: Fix \ a \in \mathbb{Z}_{n} \ and \ f \in \mathbb{Z}_{n}[x] \ with \ \deg(f) = m. Show \ there \ is \ h_{m-1} \in \mathbb{Z}_{p}[x] \ with \ \deg(h_{m-1}) \leq (m-1) \ so \ f = a_{m}(x - a)^m +...

Let V= \mathbb{R}_3[x] be the vector space of polynomials with real coefficients with degree at most 3 and let D:V\to V be the linear operator of taking derivatives, D(f)=f'. I'm trying to check the Rank-nullity theorem for this example but it doesn't seem to hold: Since D is not injective...

Homework Statement Given the matrix: 0 1 0 0 0 1 12 8 -1 (sorry I don't know how to put proper matrix format) a) find polynomials a(λ)(λ+2)2+b(λ)(λ-3) = 1 (where a(λ) and b(λ) are the polynomials) The Attempt at a Solution Well the characteristic...

help withe this tow Question please ? Q1: A pair of dice, one red and the other green, is rolled six times. We know that the ordered pairs (1, 1), (1, 5), (2, 4), (3, 6), (4, 2), (4, 4), (5, 1), and (5, 5) did not come up. What is the probability that every value came up on both...