Polynomials Definition and 784 Threads

Homework Statement show that: \sqrt{3+2\sqrt{2}}-\sqrt{3-2\sqrt{2}}=2 Homework Equations I remember over-hearing someone talking about the modulus? I don't know how that's suppose to help me The Attempt at a Solution I'm still at the conceptual stage :cry: I don't know which...

Let $F$ be any field. Let $p_1,\ldots, p_n\in F[x]$. Assume that $\gcd(p_1,\ldots,p_n)=1$. Show that there is an $n\times n$ matrix over $F[x]$ of determinant $1$ whose first row is $p_1,\ldots,p_n$. When $n=2$ this is easy since then there exist $a_1,a_2\in F[x]$ such that $p_1a_1+p_2a_2=1$...

Ok for the longest while I've been at war with polynomials and isomorphisms in linear algebra, for the death of me I always have a brain freeze when dealing with them. With that said here is my question: Is this pair of vector spaces isomorphic? If so, find an isomorphism T: V-->W. V= R4 ...

Homework Statement Let {p, q} be linearly independent polynomials. Show that {p, q, pq} is linearly independent if and only if deg p ≥ 1 and deg q ≥ 1. Homework Equations λ1p + λ2q = 0 ⇔ λ1 = λ2 = 0 The Attempt at a Solution λ1p + λ2q + λ3pq = 0 I know if λ3 = 0, then the coefficients of...

My teacher said that, No one knows of any quadratic polynomial that produces an infinite amount of primes. I was thinking could we use a polynomial like x^2+1 and then do a trick similar to Euclids proof of the infinite amount of primes and assume their are only finitely many of them...

I quote an unsolved problem posted in another forum on December 5th, 2012.

A function f : \mathbf{R}^n\rightarrow\mathbf{R} is multilinear if it's linear in every variable. Is there a multilinear function that's not a multilinear polynomial? Given a function defined on the n dimensional hypercube, values of which are 0 or 1, there is a unique multilinear extension...

Anyone how to evaluate this integral? \int_{-1}^{1} (1-x^2) P_{n}^{'} P_m^{'} dx , where the primes represent derivative with respect to x ? I tried using different recurrence relations for derivatives of the Legendre polynomial, but it didn't get me anywhere...

Problem: Let f be monic irreducible polynomial over a field F, k be monic irreducible polynomial over a field K, deg f = deg k. Let u be common root of f and k. Prove (or disprove by counterexample), that f=k over field (F intersection K), i.e. polynomials f and k are identical. Proof would be...

You have: Ʃ(n+1)/n! x^n = (1+x)e^x Is it in general true with a polynomium that: ƩP(n)/n! x^n = P(x)e^x ?

Homework Statement Where P_n(x) is the nth legendre polynomial, find f(n) such that \int_{0}^{1} P_n(x)dx = f(n) {1/2 \choose k} + g(n)Homework Equations Legendre generating function: (1 - 2xh - h^2)^{-1/2} = \sum_{n = 0}^{\infty} P_n(x)h^n The Attempt at a Solution I'm not sure if that...

For what complex numbers, x, is Gn = fn-1(x) - 2fn(x) + fn+1(x) = 0 where the terms are consecutive Fibonacci polynomials? Here's what I know: 1) Each individual polynomial, fm, has roots x=2icos(kπ/m), k=1,...,m-1. 2) The problem can be rewritten recursively as Gn+2 = xGn+1 +...

When I use the factor or expand functions on my TI-89 it outputs a matrix with values that are seemingly coming from no where. For example, if I ask my calculator to expand (3-x)^2 it gives the matrix [45, 12; 12,13]. Why is it doing this? How do I fix it?

Are these assertions true? I am referring to polynomials with real coefficients. 1. There exists of polynomial of any even degree such that it has no real roots. 2. Polynomials of odd degree have atleast one real root which implies that polynomial of even degree has atleast one critical...

Hi all. I am trying to evaluate high-degree Chebyshev polynomials of the first kind. It is well known that for each Chebyshev polynomial T_n, if -1\le x\le1 then -1\le T_n(x)\le 1 However, when I try to evaluate a Chebyshev polynomial of a high degree, such as T_{60}, MATLAB gives...

Homework Statement Determine the values of x for which the function can be replaced by the Taylor polynomial if the error cannot exceed 0.001. e^x ≈ 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} For x < 0 Homework Equations Taylor's Theorem to approximate a remainder: |R(x)| =...

Homework Statement I was wondering how people intuitively see how to decompose functions? For example: x^2 + 5x - 14, how do you solve that to be (x+7)(x-2) without a calculator? Do you use a specific method or do you just sit for a while trying and failing? The question is a...

I am reading Anderson and Feil - A First Course in Abstract Algebra. On page 56 (see attached) ANderson and Feil show that the polynomial f = x^2 + 2 is irreducible in \mathbb{Q} [x] After this they challenge the reader with the following exercise: Show that x^4 + 2 is irreducible in...

Homework Statement Find the degree 3 Taylor polynomial approximation to the function f(x)=5ln(sec(x)) about x=0. Homework Equations the taylor polynomial equation The Attempt at a Solution Here are my derivatives f(x)=5ln(secx) f'(x)=5tanx f''(x)5sec^2(x)...

Homework Statement Find T4(x), the Taylor polynomial of degree 4 of the function f(x)=arctan(11x) about x=0. (You need to enter a function.) Homework Equations The taylor polynomial equation Tn(x)= f(x)+(fn(x)(x-a)^n)/n!... The Attempt at a Solution When I take every...

Hello guise. I am familiar to a method of diagonalizing an nxn-matrix which fulfills the following condition: the sum of the dimensions of the eigenspaces is equal to n. As to the algorithm itself, it says: 1. Find the characteristic polynomial. 2. Find the roots of the characteristic...

I just had a discussion with someone who said he thought about quotient rings of polynomials as simply adjoining an element that is a root of the polynomial defining the ideal. For example, consider a field, F, and a polynomial, x-a, in F[x]. If we let (x-a) denote the ideal generated by x-a...

Now I'm trying to get my head around this question. I just know they're going to give us a large degree question like this in the exam... Let's say: I = ∫[e^(x^2)]dx with nodes being x=0 to x=0.5 The 5th degree polynomial is 1 + x^2 + (1/2)(x^4) So my queries are: How would I go about...

Let A \in M_n(F) and v \in F^n. Also...[g \in F[x] : g(A)(v)=0] = Ann_A (v) is an ideal in F[x], called the annihilator of v with respect to A. We know that g \in Ann_A(v) if and only if f|g in F[x]. Let V = Span(v, Av, A^2v, ... , A^{k-1}v).. V is the smallest A-invariant subspace containing...

z^4-4z^3+6z^2-4z-15 =0 How can i factor this polynomial in order to find the solutions?? I tried with the ruffini' rule. and i reached the following equation [(z+1)(-z^3-5z^2+11z-15)] =0 now how can i factor (-z^3-5z^2+11z-15) ? i tried it, but i can not solve it... :/

Homework Statement I'm working on problem 6.11 in Bransden and Joachain's QM. I have to prove 4 different recurrence relations for the associate legendre polynomials. I have managed to do the first two, but can't get anywhere for the last 2 Homework Equations Generating Function: T(\omega...

I'm having trouble with the following question: Construct a polynomial $q(x) \neq 0$ with integer coefficients which has no rational roots but is such that for any prime $p$ we can solve the congruence $q(x) \equiv 0 \mod p$ in the integers. Any hints on how to even start the problem will be...

Homework Statement Let A be an nxn matrix, and C be an mxm matrix, and suppose AB = BC. (a) Prove the following by induction: For every n∈ℕ, (A^n)B = B(C^n). What property of matrix multiplication do you need to prove this? Homework Equations The four basic properties of matrix...

Two part question... Homework Statement Question 1: Let v = (a, b, c)T be a column vector which represents a coordinate vector of a polynomial in P2 with respect to the Bernstein basis. Find the 4 × 3 matrix which transforms v to the standard basis of P3. (Hint: First transform v to...

Find a basis for and the dimension of the subspaces defined for each of the following sets of conditions: {p \in P3(R) | p(2) = p(-1) = 0 } { f\inSpan{ex, e2x, e3x} | f(0) = f'(0) = 0} Attempt: Having trouble getting started... So I think my issue is interpreting what those sets...

If i have to show a polynomial x^2+1 is irreduceable over the integers, is it enough to show that X^2 + 1 can only be factored into (x-i)(x+i), therefore has no roots in the integers, and is subsequently irreduceable?

Homework Statement We are given that a set of polynomials on [-1,1] have the following properties and have to show they are unique by induction. I have a way to show they are unique, but is not what he is looking for. I honestly have never seen it presented this way. P_n(x) = Ʃa_in*x^i All...

I am doing a Laplace's equation in spherical coordinates and have come to a part of the problem that has the integral... ∫ P(sub L)*(x) * P(sub L')*(x) dx (-1<x<1) The answer to this integral is given by a Kronecker delta function (δ)... = 0 if L...

In inner product spaces of polynomials, what is the point of finding the angle and distance between two polynomials? How does the distance and angle relate back to the polynomial?

In inner product spaces of polynomials, what is the point of finding the angle and distance between two polynomials? How does the distance and angle relate back to the polynomial?

This is a basic question but I don't think I've ever seen anything like it before. If Q(x) \ = \ b_0(x \ - \ a_1)(x \ - \ a_2)\cdot \ . \ . \ . \ \cdot (x \ - \ a_s) then \frac{P(x)}{Q(x)} \ = \ R(x) \ + \ \sum_{i=1}^s \frac{P(a_i)}{(x \ - \ a_i)Q'(a_i)}I just don't understand where the P(aᵢ)...

Find the number of polynomials of degree $5$ with distinct coefficients from the set $\left\{1, 2, 3, 4, 5, 6, 7, 8\right\}$ that are divisible by $x^2 - x + 1$

Let A be an integral domain. If c ε A, let h: A[x] → A[x] be defined by h(a(x))=a(cx). Prove that h is an automorphism iff c is invertible. This one really had me stumped. I have a general idea of what the function is doing. Now, assuming that h is an automorphism, we want to show that...

1. Find the Maclaurin polynomials of order n = 0, 1, 2, 3, and 4, and then find the nth MacLaurin polynomials for the function in sigma notation. cos(∏x) 2. Here is what I did: p0x = cos (0∏) = 1 p1x = cos(0∏) - ∏sin(0∏)x = 1 p2x = cos(0∏) - ∏sin(0∏)x -\frac{∏2(cos∏x)(x2)}{2!}(...

Homework Statement Factor the polynomial x^2 - 4x + 4 -4y^2 completely. Homework Equations The Attempt at a Solution Rearranging, I get x^2 - 4y^2 - 4x + 4 Then, I know that it is equal to (x -2y)(x+2y) - 4(x-1) and that is my final answer. but, my teacher only considered the...

Can someone help me to identify the type of power series for which the coefficients are palindromic polynomials of a parameter? More specifically, for a particular function f(x;a) with x, a, and f() in ℝ1, a > 0, and an exponential power series representation F0 + F1x/1! + F2x2/2! +...

Suppose that we've been given two polynomials and we want to find their Greatest Common Divisor. For integers, we have the Euclidean algorithm which gives us the G.C.D of the two given integers. Could we generalize the Euclidean algorithm to be used to find the G.C.D of any two given polynomials...

So I was working on eigenvalues of tridiagonal matrices, interestingly I get hermite polynomials as the solution. Is it possible to get an exact form for the zeros of hermite polynomials?

Is there such a thing as a "feral polynomial" ? Saw it mentioned on an Internet forum where someone claimed to be studying "feral polynomials". Closes I could find were "Farrel polynomials" .

x^2+2\\\\ \frac{2}{3} x^3 + \frac{13}{3} x\\\\ \frac{1}{3} x^4 + \frac{14}{3} x^2 + 2\\ \\ \frac{2}{15} x^5 + \frac{10}{3} x^3 + \frac{83}{15} x\\ \\ \frac{2}{45} x^6 + \frac{16}{9} x^4 + \frac{323}{45} x^2 + 2\\\\ \dots ?

Is it possible to have a polynomial with an uncountable number of turns? Like we could think of cos(x) as having a countable number of turns. could I have y=x^{\aleph_1} My question may not even make sense.

Ok, to start off I have been examining the structure of polynomials. For instance, consider the general polynomial P(x)=\sum^{n}_{k=0}a_{k}x^{k} (1) Given some polynomial, the coefficients are known. Without the loss of generality...

Homework Statement The two polynomial eqns have the same coefficients, if switched order: a_0 x_n+ a_1 x_n-1 + a_2 x_n-2 + … + a_n-2 x_2 + a_n-1 x + a_n = 0 …….(1) a_n x_n+ a_n-1 x_n-1 + a_n-2 x_n-2 + … + a_2 x_2 + a_1 x + a_0 = 0 …….(2) what is the connection between the roots of...

Homework Statement Find the value or values of k that make the equation true. Homework Equations Starting Equation: (2x + k)(x - 2k) = 2x2 + 9x - 18 The Attempt at a Solution (2x + k)(x - 2k) = 2x2 + 9x - 18 2x2 - 3xk - 2k2 = 2x2 + 9x - 18 <- I just foiled the left side...

Homework Statement Consider the set P4 of all real polynomials if degree <= 4. 1)Prove that P4 is a subspace of the vector space of all real polynomials 2)What is the dimension of the vector space P4. Prove answer by demonstrating a basis and verifying the proposed set is really a basis...