Polynomials Definition and 740 Threads

Homework Statement Show that if f^{(n)}(x_0) and g^{(n)}(x_0) exist and \lim_{x \rightarrow x_0} \frac{f(x)-g(x)}{(x-x_0)^n} = 0 then f^{(r)}(x_0) = g^{(r)}(x_0), 0 \leq r \leq n . Homework Equations If f is differentiable then \lim_{x \rightarrow x_0}\frac{f(x)-T_n(x)}{(x-x_0)^n}=0 ...

Homework Statement Prove that the set of all trigonometric polynomials with integer coefficients is countable. Homework Equations t(x)= a+\sum a_ncos(nx)+ \sum b_n sin(nx) the sum is over n and is from 1 to some natural number. The Attempt at a Solution So basically we have to look at all...

Hello , i need to calculate the following integral \int_{-\infty}^{\infty} x^4 H(x)^2 e^{-x^2} dx i tried using the recurrence relation, but i don't go the answer

Two polynomials are considered orthogonal if the integral of their inner product over a defined interval is equal to zero... is that a correct and complete definition? From what I understand, orthogonal polynomials form a basis in a vector space. Is that the desirable quality of orthogonal...

I was wondering if anyone could provide some examples of when/where the following orthogonal polynomials are used in physics? I'm starting a research project in the math department, and my professor is trying to steer the project back to physics, asking for specific applications of the...

Hello, I'm here because I lack experience and education in the subject of systems of polynomial equations (univariate and multivariate). I've also recently found a need to work with them. If this is the wrong subforum, then I apologize for my ignorance and would be grateful if a moderator...

Prove that product of sum of roots and sum of reciprocal of roots of a polynomial with degree n is always greater than or equal to n2. I tried the same on a polynomial of degree 4: ax4+bx3+cx2+dx+e = 0 Let the roots be p, q, r, and s The following equations show the relation of roots to...

Hello, I am trying to understand how to solve a system of two variables (let's say s and p representing two physical quantities), where there is a third-order polynomial representing each. I'm not sure I am describing this correctly in words, but here is the system I need to solve. P = a0 +...

How does one show that the set of polynomials is infinite-dimensional? Does one begin by assuming that a finite basis for it exists, and then reaching a contradiction? Could someone check the following proof for me, which I just wrote up ? We prove that V, the set of all polynomials over a...

Let n belongs to N, let p be a prime number and let $$Z/p^n Z$$denote the ring of integers modulo $$p^n$$ under addition and multiplication modulo $$p^n$$ .Consider two polynomials $$f(x) = a_0 + a_1 x + a_2 x^2 +...a_n x^n$$ and $$g(x)=b_0 + b_1 x + b_2 x^2 +...b_m x^m$$,given the coefficients...

The question was: How many real number solutions are there for 2^x=-x^2-2x. I tired for an hour to isolate x but i couldn't do it. Then i used wolfram alpha and it gave me two solutions and graph. I realized that question was, how many not what are the solutions, and i could do that by graphing...

On page 222 of Nicholson: Introduction to Abstract Algebra in his section of Factor Rings of Polynomials Over a Field we find Theorem 1 stated as follows: (see attached) Theorem 1. Let F be a field and let A \ne 0 be an ideal of F[x]. Then a uniquely determined monic polynomial h exists...

I am reading Dummit and Foote on Polynomial Rings. In particular I am seeking to understand Section 9.4 on Irreducibility Criteria. Proposition 9 in Section 9.4 reads as follows: Proposition 9. Let F be a field and let p(x) \in F[x] . Then p(x) has a factor of degree one if and only if...

Not sure if this is the right place to post (but its related to a complex analysis questions) I'm doing a past paper for my revision and am stuck at the first hurdle. I simply cannot factor this polynomial in z for the life of me. I've tried completing the square and the usual quadratic...

Homework Statement Prove that \sum_{n=0}^{\infty}{\frac{r^n}{n!}P_{n}(\cos{\theta})}=e^{r\cos{\theta}}J_{0}(r\sin{\theta}) where P_{n}(x) is the n-th legendre polynomial and J_{0}(x) is the first kind Bessel function of order zero. Homework Equations...

I am struggling to understand interpolating polynomials and their errors. I have a problem off of a study guide here: http://terminus.sdsu.edu/SDSU/Math541_f2012/Resources/studyguide-mt01.pdf I understand that the composite simpsons rule is only exact for polynomials up to order 3, with error...

Homework Statement If we define \xi=\mu+\sqrt{\mu^2-1}, show that P_{n}(\mu)=\frac{\Gamma(n+\frac{1}{2})}{n!\Gamma(\frac{1}{2})}\xi^{n}\: _2F_1(\frac{1}{2},-n;\frac{1}{2}-n;\xi^{-2}) where P_n is the n-th Legendre polynomial, and _2F_1(a,b;c;x) is the ordinary hypergeometric function...

Why polynomials are used in numerical analysis?

For 2 polynomials f,g, resultant Res(f,g) vanish if and only if f and g has at least a common root. However, is there any way to construct a coefficients polynomial of 3 polynomials f,g,h [Res(f,g,h)] that vanish if and only if f,g,h has at least a common root?

I recently discovered that for a 3rd degree polynomial I was studying, f(5) - 4f(4) + 6f(3) - 4f(2) + f(1) = 0. At first I just though it was coincidental that the coefficients were the 5th row of Pascal's Triangle, but then I tried a 2nd degree polynomial and found that f(4) - 3f(3) + 3f(2) -...

So I've been reading about minimax polynomial approximations and have found them to be pretty impressive. However, i am confused on exactly how to determine the constants. The first step is supposed be solving for the Chebyshev polynomials as an initial guess. I'm reading wikipedia but I'm a...

Orthogonal polynomials are perpendicular?? hi.. So as the title suggests, i have a query regarding orthogonal polynomials. What is the problem in defining orthogonality of polynomials as the tangent at a particular x of two polynomials are perpendicular to each other, for each x? This...

The Laguerre polynomials, L_n^{(\alpha)} = \frac{x^{-\alpha}e^x}{n!}\frac{d^n}{dx^n}\left(e^{-x}x^{n+\alpha} \right) have n real, strictly positive roots in the interval \left( 0, n+\alpha+(n-1)\sqrt{n+\alpha} \right] I am interested in a closed form expression of these roots...

Homework Statement Is the set of all polynomials with positive coefficients a vector space? It's not. But after going through the vector space conditions I don't see how it can't be.

Dummit and Foote on page 284 give the following definitions of irreducible and prime for integral domains. (I have some issues/problems with the definitions - see below)...

Homework Statement Find all integers x such that 7x \equiv 11 mod 30 and 9x \equiv 17 mod 25 Homework Equations I guess the Chinese Remainder theorem and Bezout's theorem would be used here. The Attempt at a Solution I can do this if the x-terms didn't have a...

I am working on Exercise 8 of Dummit and Foote Section 9.2 Exercise 8 ==================================================================================== Determine the greatest common divisor of a(x) = x^3 - 2 and b(x) = x + 1 in \mathbb{Q} [x] and write it as a linear...

Homework Statement Expand f(x) = 1 - x2 on -1 < x < +1 in terms of Legendre polynomials. Homework Equations The Attempt at a Solution Unfortunately, I missed the class where this was explained and I have other classes during my professor's office hours. I have no idea how to begin this...

So I needed to factor -4x5-8x4+8x3+4x. I factored out a -4x and I am left with x4+2x3-2x2-4. The problem is I am unsure how to factor x4+2x3-2x2-4. I know how to long divide polynomials but I have not done synthetic division in over 4 years. From what I have seen on the internet it seems...

(HUN,1979) Prove the following statement: If a polynomial $f(x)$ with real coefficients takes only nonnegative values, then there exists a positive integer $n$ and polynomials $g_1(x),g_2(x),...,g_n(x)$ such that $f(x)=g_1(x)^2+g_2(x)^2+\cdots+g_n(x)^2$. A related question of my own, but I...

Homework Statement I'm trying to prove, for part of a homework problem, that if the ratio of two polynomials ##p## and ##q## with real coefficients is a polynomial, then all of its coefficients are real. Homework Equations N/A The Attempt at a Solution Well, we can first note...

Homework Statement Find the minimum possible value for a^{2}+b^{2} where a and b are real such that the following equation has at least one real root. Homework Equations x^{4}+ax^{3}+bx^{2}+ax+1 The Attempt at a Solution I tried to find the roots of the equation and then find a...

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...

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)...