Polynomials Definition and 740 Threads

Homework Statement f(\lambda) = \frac{8\pi hc\lambda ^{-5}}{e^{hc/\lambda kT}-1} Is Planck's Law where h\ =\ Planck's\ constant\ =\ 6.62606876(52)\ \times\ 10^{-34}\ J\ s; c\ =\ speed\ of\ light\ =\ 2.99792458\ \times\ 10^{8}\ m\ s^{-1}; and\ Boltzmann's\ constant\ =\ k\ =\ 1.3806503(24)\...

[SOLVED] Algebta (factoring polynomials) Homework Statement x^2 * [ (1/2) * (1-x^2)^(-1/2) * (-2x)] + (1 - x^2)^(1/2) * (2x) factor this down. Homework Equations n/a The Attempt at a Solution -x^3*(1- x^2)^(-1/2) + 2x(1 - x^2)^(1/2) I understand how you get to this part...

Homework Statement Okay, basically why does (a_{}n + b_{}n) ignore the Fundamental Theory of Polynomials? Homework Equations ... I could post them here, but basically when n is odd (a_{n} + b_{n}) = a series that looks like this: (a+b) (a_{n-1} b_{0} - a_{n-2}b +a_{n-3}b_{2} + ... +...

(1): Find all irreducible polynomials of the form x^2 + ax +b , where a,b belong to the field \mathbb{F}_3 with 3 elements. Show explicitly that \mathbb{F}_3(x)/(x^2 + x + 2) is a field by computing its multiplicative monoid. Identify [\mathbb{F}_3(x)/(x^2 + x + 2)]* as an abstract group...

Homework Statement Show that the set of polynomials with integer coefficients with at least 1 real root is decidable. The Attempt at a Solution The question did not ask for specific language, just an intuitive finite algorithm will do.

Hello! First post here. My question is, is it possible to use a matrix to solve a system where you have the same variable, but a different degree. i.e. 2x^2 + 2x + y = 2 -3x^2 - 6x + 2y = 4 4x^2 + 6x - 3y = 6 Now, I know this is possible other ways, but seeing as each of those would...

Homework Statement Show that x^{p^n}-x is the product of all the monic irreducible polynomials in \mathbb{Z}_p[x] of a degree d dividing n.Homework Equations The Attempt at a Solution So, I want to prove that the zeros of all such monic polynomials are also zeros of x^{p^n}-x and vice versa. I...

Help, I am infinitly confused :) When solving the limit for this type and factoring the largest power of a variable in the polynomial in order to make its coefficient become a limit multiplied by another limit of Infinity I get lost. I just do not understand how (Infinity)(5 + Infinity +...

Hey! Could someone please help me find out how to express the product of two Legendre polynomials in terms of a sum of Legendre polynomials. I believe I have to use the recursion formula (l+1)P_{l+1}(x)-(2l+1) x P_l(x) + l P_{l-1}(x)=0 but I am not sure how to do this. What is basically...

Hi , is there a method to obtain the roots of Polynomials: P(x)=a_{0}+a_{1}x+a{2}x^{2}+...+a_{n}x^{n} i know there are , but my problem is this if we knew that are complex roots of the form z=a+ib , would be a method to obtain the complex root with BIGGER and SMALLER real part ?? , i mean...

I need to show that the coefficients of a complex polynomial P(z) are real iff P(x) is real for all real x. Thanks!

Homework Statement Is it possible to describe some infinite degree polynomials by their roots in a way analagous to finite degree polynomials? Homework Equations The Attempt at a Solution I know that, since not all infinite degree polynomials have roots (e.g. the power series...

Homework Statement The roots of the equation x^3-x-1=0 are \alpha,\beta,\gamma S_n=\alpha^n +\beta^n +\gamma^n (i)Use the relation y=x^2 to show that \alpha^2,\beta^2,\gamma^2 are roots of the equation y^3-2y^2+y-1=0 (ii)Hence, or otherwise find the value of S_4 (iii)Find...

http://img100.imageshack.us/img100/9016/linalggp1.jpg for (a): does that mean i must compute l0(t), l1(t) and l2(t), and i wasn't sure how to do this with the lagrange polynomial formula given, so i found one online and did it, I'm not sure if this is correct, but my l0(t) looks like this: =...

[SIZE="2"]\frac{1}{1^4+1^2+1}+\frac{1}{2^4+2^2+1}+...+\frac{1}{2007^4+2007^2+1}

[SOLVED] polynomials/ galois field question Im reading through a section that deals with polynomials Galois fields and ran into something that I am not quite understanding. Say we have an irreducible polynomial, f(x), which has coefficients from GF(2) and roots \beta, \beta^{2}...

I remember learning an iterative method that gives the answer to trigonometric polynomials such as sin(x)-0.7-0.611cosx = 0 where x is the angle in degrees. The person who I learned this method from called it the method for solving transcendentals. Now I can't seem to find any...

Hi, Can someone explain why the following is true? It seems to be an "accepted fact" everywhere I search, and I can't tell why. Let F be a field. Let E be the function from F[x] to F^F, where F[x] is the set of all polynomials over F, and F^F is the set of all functions from F to F. Then...

Homework Statement I need to find the critical points of f(x) = (x^3 - 2x)e^x I found the derivative, and set it equal to zero ended up with e^x (x^3 +3x^2 -2x -2) = 0 I am having trouble factoring the second factor, any suggestions?

any ideas on how to go about conducting these please. i will attempt them once i have a clear idea on how to do this. thanks :) let V be the vector space of polynomials over C of degree <= 10 and let "D: V -----> V" be the linear map defined by D(f) = df/dx show (1) D^11=0 (2)...

I am attempting to construct a field containing 625 elements and should be in the form Zn[x] mod f(x). Factoring 625 leads to 5^4. So I'm guessing my field will be GF(5^4). So in order for me to construct a field with all elements in it, I need f(x) to be some irreducible polynomial mod 5...

okay, so i know how to factor( and the expand( but i have a question for dividing polynomials such as... 9x^5-6x^3+x-63 / x-8 how do i enter this into my calculatoor?! AHH its driving me INSANEEEEEEEEEe i get 9x^5-6x^3+x-63/x-8 but my teacher gets a different everytime for every...

[b]1. Consider the vector space of polynomials 1+x^3 , 1-x+x^2, 2x, 1+x^2 Are they linearly dependent or independent? dimension of vecotr space spanned by these vectors? [b]3. I have tried to solve this by letting a1 = 1+x^3 a2 = 1-x+x^2 a3 = 2x a4 = 1+x^2 Then I let (alpha)a1 +...

There is a question where you should find a formula for P-n(0) using the Legendre polynomials: P-n(x)=1/(2^n*n!) d^n/dx^n(x^2-1)^n , n=0,1,2,3... I tried to derive seven times by only substituting the n until n=7,I did that because i wanted to find something that i can build my formula but i...

Homework Statement A cubic polynomial gives remainders (5x+4) and (12x-1) when divided by x^{2} - x + 2 and x^{2} + x - 1 respectively. Find the polynomial Homework Equations :S Well, I am using the root theorem, the factor theorem, and possibly just basics on long division.. We know...

Homework Statement (6x^4-3x^2+x-4) / (2x^2+1) Homework Equations Relevant equations? The Attempt at a Solution Here is my attempt, but I want to make sure that I didn't break any laws by changing the number to be divided by switching the last two terms around by using the commutative law...

Let s1 be the set spanned by the polynomials: x^3+x+1, x^3-3x^2+x-2, 2x^3-1. Let s2 be the set spanned by the polynomials: x^3-1, x^2+x+1. What is the intersection of s1 and s2? I really don't know where to begin, I don't know how to define these sets, s1 and s2. since i don't know what...

Homework Statement factorise the following as far as possible 1) x^3 + y^3 2) x^4 + x^3 - 3x^2 - 4x - 4 Homework Equations The Attempt at a Solution 1) Not quite sure really what to do, lol, only just been taught how to divide polynominals, and the factor and remainder...

15. Determine wheter the set is a vector space. The set of all fifth-degree polynomials with the standard operations. AXIOMS 1.u+v is in V 2.u+v=v+u 3.u+(v+w)=(u+v)+w 4.u+0=u 5.u+(-u)=0 6. cu is in V 7.c(u+v)=cu+cv 8.(c+d)u=cu+cd 9.c(du)=(cd)u 10.1(u)=u the axioms that fail are...

Homework Statement Course - Control systems engineering, chapter: design using root locus I'm familiar with dividing a polynomial when given a factor using the remainder theorem however is there another way when only the third or fourth order equation is given and nothing else? We aren't...

I have two... Homework Statement The the limit Homework Equations \lim_{x \rightarrow 1} \frac{1-cosx}{x^2} The Attempt at a Solution I figured to just plug in 1, but I wanted to make sure... Homework Statement Find the limit Homework Equations \lim_{x \rightarrow 3}...

Hello, I face this problem: X^3 + X - 71 = (X^2 + 4X + 16)Q(X) + R(X), where Q and R are polynomials. Decide which they are. I got that Q(X) = (X + 1/4) and that R(X) = - 75, but apparently it is wrong. I am stuck and don't know what to do. Thanks in advance.

Hello everyone, I need some help with this one: I need to write a routine that tests the irreducibility of a polynomial over Fp, (where Fp is the finite field with p elements and p is a prime). It should take as input: p,the polynomial and its degree. It should return TRUE if the...

Homework Statement Is it there a method to find out if a polynomial has no integer roots? The Attempt at a Solution I tried the division of polynomials, as well as the Horner's Method, but no luck.

Trying to ask my method of doing the question is correct or not? Try check is there any mistake please? I am beginner. The Question Find the real solutions to each of the following questions. (a) X³+X²-10X+8=0My attempt at a solution X³+X²-10X+8=0 f(1)=1³+1²-10(1)+8 f(1)=0 X=1 X-1=0 (one of...

How do i know under which circumstances to use synthetic and when to just do regular polynomial division? do they not both give the same results?

1. Given the history of polynomials and there application why are they important? 3. When I researched the history all I found on the internet all I found was who was the first to solved certain types of poynomial. It didn't help me figure out why they might be important. I know...

help me its so hard working in the finite field Z_p show that the all the factors of polynomial x^{p^n}-x have degree "d" such that d|n. thanx

Homework Statement Show x^2 + 2 in Z_5[x] is irreducible. This is before the section on the factor theorem (j is a root -> (x-j) is a factor). So I'm not so sure I want to start checking for zero's since "its not available" per se. Homework Equations The Attempt at a Solution...

Homework Statement Prove that x = 2^(1/2)+2^(1/3) is irrational. Homework Equations The Attempt at a Solution The hint to the exercise says that I should first show that x satisfies an equation of the form x^6+a_1*X^5+...+a_0 = 0, where the coefficients are integers. I am...

Hello! I have just a small question - does the theory of polynomials say something about their coefficients? I mean: is the polynomial with all the coefficients being imaginary still considered as a "normal' polynomial?

I got a problem in quantum physics that i have come to a differential equation but I don't see how to solve it, its on the form F''(x)+(Cx^2+D)F(x)=0. How should I solve it? Thanks

So I have an assignment due in a few hours and I am pretty happy with it, aside from the fact that I am completely lost on the following section: - The polynomials of degree 3, denoted P3, form a vector space. 1. Show that when added, two general polynomials of degree 3 will always produce...

Hey everyone, I was looking at another thread about factoring polynomials, and now I'm wondering if anyone knows any unusual or cool ways to factor them (i.e. not using numerical techniques such as Newton's Method). Unfortunately the only one I do know is the rational root theorem...

Prove: if P is a polynomial function and P(a) and P(b) have opposite signs, then there exists at least one value c between a and b for which P(c) = 0

Why do you guys think that given two 3x3 matrices, they are similar if and only if their characteristic polynomial and minimal polynomial are equal (this reasonably fails for 4v4 matrices though)?

Homework Statement find closest function a+bx3 to x2 on the iterval [-1,1] (we consider standard inner product (f,g) = integral(-1 to 1):fgdx So, here is my attempt, but I got a suspicious result: [(1,1) (1,x3)] [a] [(1,x3) (x3,x3)][b] = [(1,x2)]...

Homework Statement We have this theorem: Let f(x)\in F[x] Then f(x) has multiple roots if and only if gcd(f(x),f'(x))=d(x) and d(x)\geq 1 We went BRIEFLY over the proof and we are supposed to be able to apply it on an upcoming exam. I'm not exactly sure how it works or what I'm...

Hello, I'm trying to prove the fundamental thereom of algebra using the minimal modulous principal. There is one step that I cannot make sense out of logically and I was wondering if one of you could explain it to me. If the limit as |z| approaches infinity implies |P(z)| approaches infinity...

Homework Equations The Attempt at a Solution i am confused as to how one determines a given polynomial does not satisfy the eisenstein crtiterion, for example... ...i am told x^4 + 4x^3 + 6x^2 + 13x + 13 fails to satisfy eisentein