Polynomials Definition and 740 Threads

Homework Statement For each n\,\in\,\mathbb{N}, let p_n(x)\,\in\,\mathbb{Z}_2[x] be the polynomial 1\,+\,x\,+\,\cdots\,x^{n\,-\,1}\,+\,x^n Use mathematical induction to prove that p_n(x)\,\cdot\,p_n(x)\,=\,1\,+\,x^2\,+\,\cdots\,+x^{2n\,-\,2}\,+\,x^{2n}Homework Equations Induction steps...

Homework Statement Find two polynomials, each of degree 2, in Z6[x] whose product has degree 0. Can you repeat the same in Z7[x]? Explain. Homework Equations In Z6[x] and Z7[x] can the only variable be x? The Attempt at a Solution I know Z6 consists of {0,1,2,3,4,5} and Z7...

Homework Statement I'm doing work finding the centroids of 2d graphs. I'm working these problems using double integrals of regions that are horizontally or vertically simple. To do this I have to be able to convert line equations from one variable to the other. Some are simple but others...

In a past exam paper at my uni I am asked to show that the hermite polynomials are solutions of the hermite diff. equation but first there is the following \Phi(x,t)=\exp (2xh-h^2)=\sum_{n=0}^{\infty} \frac{h^n}{n!}H_n(x) So I need to find the form of H_n first, and I'm stuck. I tried...

Homework Statement Factor x^{16}-x in F_8[x] Homework Equations The Attempt at a Solution I know how to do it in F_2[x] . I also feel the factorizations are the same in the two fields..but not sure how to prove it.

Find a polynomial that is orthogonal to f(x)=x2-1/2 using L2[0,1]. I have looked all in the textbook and all over the internet and have found some hints if the interval is [-1,1], but still do not even know where to start here. I think I was gone the day our professor taught this because I do...

1.Let F be an extension field of K and let u be in F. Show that K(a^2)contained in K(a) and [K(u):K(a^2)]=1 or 2. 2.Let F be an extension field of K and let a be in F be algebraic over K with minimal polynomial m(x). Show that if degm(x) is odd then K(u)=K(a^2). 1. I was thinking of...

Homework Statement Show that x^2\,+\,x can be factored in two ways in \mathbb{Z}_6[x] as the product of nonconstant polynomials that are not units.Homework Equations Theorem 4.8 Let R be an integral domain. then f(x) is a unit in R[x] if and only if f(x) is a constant polynomial that is a...

Homework Statement Hi, I need to show that \alpha+1=[x] is a primitive element of GF(9)= \mathbb{Z}_3[x]/<x^{2}+x+2> I have already worked out that the function in the < > is irreducible but I do not know where to go from this. Homework Equations there are 8 elements in the...

Hello, I'm currently doing an undergrad project on this topic and I was wondering if any of you guys know what is the fastest algorithm (asymptotically) that has been discovered so far, for such purpose. Here is paper by Shoup (1993) which gave the fastest algorithm up to then...

Hello, I'm currently doing an undergrad project on this topic and I was wondering if any of you guys know what is the fastest algorithm (asymptotically) that has been discovered so far, for such purpose. Here is paper by Shoup (1993) which gave the fastest algorithm up to then...

I've recently been working with Legendre polynomials, particularly in the context of Spherical Harmonics. For the moment, it's enough to consider the regular L. polynomials which solve the differential equation [(1-x^2) P_n']'+\lambda P=0 However, I've run into a problem. Why in the...

Homework Statement If the characteristic polynomial of an operator T is (-1)^n*t^n, is T nilpotent? Homework Equations The Attempt at a Solution My first instinct for this question is that the answer is yes, because the matrix form of T must have 0's on the diagonal and must...

If the characteristic polynomial of an operator T is (-1)^n*t^n, is T nilpotent? My first instinct for this question is that the answer is yes, because the matrix form of T must have 0's on the diagonal and must be either upper triangular or lower triangular. This is what I found when I tried...

I would like to know if the following correct. Suppose I start with a connection on a real vector bundle and extend it to the complexification of the bundle. The curvature forms of the complexification seem to be the same as curvature 2 forms of the real bundle. From this it seems that the...

Homework Statement Prove that a polynomial f of degree n with coefficients in a field F has at most n roots in F. Homework Equations The Attempt at a Solution So we could prove this by induction by using a is a root of f if and only if x-a divides f. My question is: why do...

Why should the exponents of polynomials just be whole numbers ?

Hi everybody! I have this problem: Either P = (X+2)m+(X+3)n and Q = x2+5x+7; Determine m, n such that Q | P;( m, n = ? (Q divide P)); May you help me please? Thank You!

Homework Statement attached Homework Equations The Attempt at a Solution what is x_i? is it the coefficient of x or simply add up 1-5? i found the notation different from http://mathworld.wolfram.com/PolynomialNorm.html so i am confused. Thx!

Homework Statement 1. Use the definition of "f(x) is O(g(x))" to show that 2^x + 17 is O(3^x)'). 2. Determine whether the function x^4/2 is O(x^2) 2. The attempt at a solution 1. From my understanding, I would say of course 2^x + 17 is O(3^x) because the constant is of such low...

Homework Statement How to prove that the only irreducible polys over the reals are the linear ones and the quadratic ones no real roots? What about the ones with higher degree? I feel that I'm missing something that's really obvious. Homework Equations The Attempt at a Solution

Anyone who knows the limits of orthogonality for Bessel polynomials? Been searching the Internet for a while now and I can't find a single source which explicitly states these limits (wiki, wolfram, articles, etc). One thought: since the Bessel polynomials can be expressed as a generalized...

Homework Statement Let K be a field, and f,g are relatively prime in K[x]. Show that f-yg is irreducible in K(y)[x]. Homework Equations There exist polynomials a,b\in K[x] such that af+bg=u where u\in K. We also have the Euclidean algorithm for polynomials. The Attempt at a...

Vector Spaces, Polynomials "Over Fields" What does it mean when a vectors space is "over the field of complex numbers"? Does that mean that scalars used to multiply vectors within that vector space come from the set of complex numbers? If so, what does it mean when a polynomial, p(x) is...

Homework Statement [PLAIN]http://img443.imageshack.us/img443/3096/questiond.jpg The Attempt at a Solution If P(z)Q(z)=0 then \displaystyle a_0b_0 + (a_0b_1 + a_1 b_0)z + ... + \left( \sum_{i=0}^k a_i b_{k-i} \right) z^k + ... + a_n b_m z^{n+m} =0 Now what? Equate coefficients...

Homework Statement The Hermite polynomials H_n(x) may be defined by the generating function e^{2hx-h^2} = \sum_{n=0}^{\infty}H_n(x)\frac{h^n}{n!} Evaluate \int^{\infty}_{-\infty} e^{-x^2/2}H_n(x) dx (this should be from -infinity to infinity, but for some reason the latex won't work!)...

Let K be a subfield of C, the field of complex numbers, and f an irreducible polynomial in K[X]. Then f and Df are coprime so there exist a,b in K[X] such that af + bDf = 1 (D is the formal derivative operator). Now what I don't understand is why this equation implies f and Df are coprime when...

Homework Statement Determine which of the following are subspaces of P3: a) all polynomials a0+a1x+a2x^2+a3x^3 where a0=0 b) all polynomials a0+a1x+a2x^2+a3x^3 where a0+a1+a2+a3=0 c) all polynomials a0+a1x+a2x^2+a3x^3 for which a0, a1, a2, a3 are integers d) all polynomials of the form...

I am having some difficulty with a homework problem I was recently assigned. The problem says to "Replace each trigonometric function with its third Maclaurin polynomial and then evaluate the function at f(0.1)" This is what I have done so far: f(x)=(x cosx- sinx)/(x-sin⁡x) 1st trig...

So long story short, I have a friend who wants me to help her learn how to factor cubic polynomials. Normally I would just fess up and say I don't remember but it's something I'd like to review myself and lessons online aren't the clearest. Here's one of the questions: 2x3+3x2-8x+3 I...

Homework Statement I am following a derivation of Legendre Polynomials normalization constant. Homework Equations I_l = \int_{-1}^{1}(1-x^2)^l dx = \int_{-1}^{1}(1-x^2)(1-x^2)^{l-1}dx = I_{l-1} - \int_{-1}^{1}x^2(1-x^2)^{l-1}dx The author then gives that we get the following...

Hey, I'm a high school student (11th grade) and I'm working on a computer algebra system for a research project. Most things are are going well (sums, products, derivatives, integrals, series, expansion, complex analysis, factoring basic expressions, etc.). However, I am having difficulty...

(a) Find all irreducible polynomials of degree less than or equal to 3 in Z2[x]. (b) Show that f(x) = x4 + x + 1 is irreducible over Z2. (c) Factor g(x) = x5 + x + 1 into a product of irreducible polynomials in Z2[x]. We have an irreducible polynomial if it cannot be factored into a...

Homework Statement Solve for x, 225*sin(x)/x^6-225*cos(x)/x^5-90*sin(x)/x^4+15*cos(x)/x^3-5/(2*x^3)=0 Homework Equations Finding this very complicated to solve, are there any useful hints or techniques we should know about? The Attempt at a Solution Have used numerical...

[FONT="Palatino Linotype"]Given the Legendre polynomials P0(x) = 1, P1(x) = x and P2(x) = (3x2 − 1)/2, expand the polynomial 6(x squared) in terms of P l (x). does anyone know what this question is asking me? what is P l (x)? thanks in advance

i find that chebyshev polynomials are quite useful in numerical computations is there any good references?

Diff Eq's -- orthogonal polynomials [PLAIN]http://img27.imageshack.us/img27/566/39985815.jpg I managed to do the first part, stuck in the part circled. Any help will be appreciated, thanks.

So I'm studying for a final, and it just so happens my professor threw taylor polynomials at us in the last week.. I understand the concept of a taylor polynomial but i need some help fully understand the LaGrange remainder theorem if we have a function that has n derivatives on the interval...

Im given two polynomials: f(x) =(2x^6) + (x^5) - (3x^4) + (4x^3) + (x^2) -1 and g(x)=(x^3)-(x^2)+2x+3 find polynomials Q(x),R(x) in the set of R[x] s.t f(x) =g(x)Q(X) +R(X) and deg(R)<deg(g) Am i even in the right area? and something to do with manipulating numbers in C[x]...

Homework Statement Expand (a+b)n Homework Equations The Attempt at a Solution Substituting n=2, (a+b)n = a2 + 2ab + b2 Substituting n=3, (a+b)n = a3 + 3a2b + 3ab2 + b3 It's easy to see the powers of a decrease at the same time as the powers of b increase by order 1 each...

I just wanted to know what kind of math is needed to solve questions like 1, 2 and 3 of http://www.math.toronto.edu/deljunco/354/ps4.fall10.pdf and number 5 of http://www.math.toronto.edu/deljunco/354/354final08.pdf . I don't need solutions, I just need to know what book or online source can...

Homework Statement We know that the gcd of two polynomials can be written as gcd(p(x),q(x))=p(x)m(x) + q(x)n(x) for some n(x) and m(x) in F[x] F a fieldI want to show gcd(n(x),m(x))=1 for a fixed gcd(p(x),q(x)) The Attempt at a Solution Well, what I tried was to let D(x)=gcd(p(x),q(x))...

Hey everyone Im doing control engineering and was wondering what methods i could use to find the roots of a 4th order polynomial? For example: (x^4) + (8x^3) + (7x^2) + 6x = 5 Could I separate that into two brackets of quadratics or will i need to use a really long winded method...

Homework Statement Given a set of polynomials in x: x^{r_1}, x^{r_2},...,x^{r_n} where r_i \neq r_j for all i \neq j (in other words, the powers are distinct), where the functions are defined on an interval (a,b) where 0 < a < x < b (specifically, x \neq 0), I'd like to show that this...

Homework Statement (a) Find a primitive root β of F3[x]/(x^2 + 1). (b) Find the minimal polynomial p(x) of β in F3[x]. (c) Show that F3[x]/(x^2 + 1) is isomorphic to F3[x]/(p(x)). The Attempt at a Solution I am completely lost on this one :confused:

Homework Statement Let f(x)=x2 +3x -5, and let the summation (from n=0 to infinity) an (x-4)n be the Taylor series of f about 4. Find the values of a0, a1, a2, a3, and a4. Homework Equations The Attempt at a Solution What am I supposed to do with the summation? And what does it mean...

Homework Statement use the third degree Taylor polynomial of cos at 0 to show that the solutions of x2=cos x are approx. \pm\sqrt{2/3}, and find bounds on the error. Homework Equations P2n,0(x) = 1-x2/2!+x4/4!+...+(-1)nx2n/(2n)! The Attempt at a Solution when it says "third...

Can somebody prove the equivalence statement of two real polynomials in one variable x for me? My Math teacher just told us to remember it as a definition and so I didn't get any proof for it; I attempted to prove it myself and ended up confusing myself with a lot of symbols. So, can somebody...

How to prove that a polynomial of degree 2 or 3 over a filed F is a prime polynomial if and only if the polynomial does not have a root in F? and i can't think of an example of polynomial of degree 4 over a field F that has no root in F but is not a prime polynomial. it says each...

Homework Statement Which of these subsets of P2 are subspaces of P2? Find a basis for those that are subspaces. (Only one part) {p(t): p(0) = 2} Homework Equations The Attempt at a Solution So, I know the answer is that it's not a subspace via back of the book, but I don't...