Polynomials Definition and 784 Threads

Homework Statement Find the ##gcd(x^3+x^2-x, x^5+x^4+2x^2-x-1) ##and write it as a linear combination. Homework Equations The Attempt at a Solution I know the ##gcd(x^3+x^2-x, x^5+x^4+2x^2-x-1)=1## What I have so far is ##1. x^5+x^4+2x^2-x-1=(x^3+x^2-x)(x^2+1)+(x^2-1)## ##2...

Is there anyway around this problem? syms m,n; x1 = [0, 1, 4, m]; x2 = [3, n, 9, 27]; conv(x1,x2) Undefined function 'conv2' for input arguments of type 'sym'

Recently, I've developed a habit of trying to separate the idea of a function from the idea of the image of the function. This has mostly just confused me, but I am adamant about sticking to it. I think the two terms, "ring of polynomials" and "ring of polynomial functions," are not...

Given that $k^2 + k + n$ is always prime for all positive integer $k$ in the interval $\left (0, (n/3)^{1/2} \right )$. Find the largest interval for which the same can be stated. This easily follows from Heegner-Stark theorem, but can you show the same bypassing it, without going through the...

The generalized Rodrigues formula is of the form K_n\frac{1}{w}(\frac{d}{dx})^n(wp^n) The constant K_n is seemingly chosen completely arbitrarily, & I really need to be able to figure out a quick way to derive whether it should be K_n = \tfrac{(-1)^n}{2^nn!} in the case of Jacobi...

My professor wants to convert propositional statements such as X ^ Y into polynomeals such as P[(X^Y)] = xy Now, we may have multiple propositional formulas and wish to determine if they are consistent or inconsistent using Boolean polynomials. I'm having a tough time finding material...

Hi, Homework Statement A quadratic piecewise interpolation is carried out for the function f(x)=cos(πx) for evenly distributed nodes in [0,1] (h=xi+1-xi, xi=ih, i=0,1,...,πh). I am asked to bound the error. Homework Equations The Attempt at a Solution I believe the error in this case is...

how do you write a polynomial in three variables say x,y,z in standard form?

Homework Statement A conducting spherical shell of radius R is cut in half and the two halves are infinitesimally separated (you can ignore the separation in the calculation). If the upper hemisphere is held at potential V0 and the lower half is grounded find the approximate potential for...

Hi all, Suppose I have a system which can be described using something like: y(t) = a_1 x(t) + a_2 x^2(t) + \dots + a_p x^p(t) I want to find the coefficients using samples from x(t) and y(t) (pairwise taken at same times) using as fewest samples as possible. Clearly this is linear in...

Hi, I am trying to make progress on the following integral I = \int_0^{2\pi} \sqrt{(1+\sum_{n=1}^N \alpha_n e^{-inx})(1+\sum_{n=1}^N \alpha_n^* e^{inx})} \ dx where * denotes complex conjugate and the Fourier coefficients \alpha_n are constant complex coefficients, and unspecified...

Consider \[ f(x) = \begin{cases} 1, & 0\leq x\leq 1\\ -1, & -1\leq x\leq 0 \end{cases} \] Then \[ c_n = \frac{2n + 1}{2}\int_{0}^1\mathcal{P}_n(x)dx - \frac{2n + 1}{2}\int_{-1}^0\mathcal{P}_n(x)dx \] where \(\mathcal{P}_n(x)\) is the Legendre Polynomial of...

Homework Statement Can you find a basis {p1, p2, p3, p4} for the vector space ℝ[x]<4 s.t. there does NOT exist any polynomials pi of degree 2? Justify fully.Homework Equations The Attempt at a Solution We know a basis must be linearly independant and must span ℝ[x]<4. So intuitively if there...

i was trying to solve this problem when i got confused on how to arrange the terms in descending powers of the literal factors because some term contain two variables and the polynomial has 3 variables. how can i properly arrange the dividend here? $\displaystyle...

Let $p(x,y)$ and $q(x,y)$ be two polynomials with coefficients in $\mathbb R$. Define $P=\{(a,b)\in\mathbb R^2 : p(a,b)=0\}$ and $Q=\{(a,b)\in \mathbb R^2:q(a,b)=0\}$. Now assume that there is a sequence of points $(x_n,y_n)$ in $\mathbb R^2$ such that: 1. $(x_n,y_n)\to (0,0)$. 2. $(x_n,y_n)\in...

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...

Let R be a UFD and K be its field of fractions. Let f(x) be in R[x] and f(x) = a(x)b(x) where a(x), b(x) are in K[x]. Show that there exists a c in K such that c*a(x) and c-1*b(x) are both in R[x] and such that f(x) = (c*a(x))(c-1*b(x)) in R[x]. I have been stuck on this for a while now as I...

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 Zdenote 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 are in Z/p^nZ...

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 Irreducibility in Polynomial Rings. In particular I am studying Proposition 12 in Section 9.4 Irreducibility Criteria. Proposition 12 reads as follows: Let $I$ be a proper ideal in the integral domain $R$ and let $p(x)$ be a non-constant monic polynomial in...

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...

Homework Statement I just need to deduce the expression for the associated Legendre polynomial P_{n}^{-m}(x) using the Rodrigues' formula Homework Equations Rodrigues formula reads P_{n}(x)=\frac{1}{2^{n}n!}\frac{d^n}{dx^n}(x^2-1)^n and knowing that...

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...