Polynomials Definition and 740 Threads

Homework Statement (3y-2/y+3) - (3y+1)/(y2+6y-9) Homework EquationsThe Attempt at a Solution Ok, I have attempted to solve in 2 steps, step 1: solve 3y-2/y+3 step 2: solve 3y+1/y2+6y-9 and then subtract the answers. This doesn't seem to work, as I get: 3y-2/y+3 = 3 remainder 11 and...

When you have a polynomial say ax^4+bx^3+cx^2+dx+e where a,b,c,d and e are constants and divide this by a polynomial say ax+b it follows that the quotient will be a cubic polynomial. Assuming that a remainder exists, then the remainder will be a constant because in my reasoning, the remainder...

Hi. I'm off to solve this integral and I'm not seeing how \int dx Hm(x)Hm(x)e^{-2x^2} Where Hm(x) is the hermite polynomial of m-th order. I know the hermite polynomials are a orthogonal set under the distribution exp(-x^2) but this is not the case here. Using Hm(x)=(-1)^m...

Hey! In my calculus book they claim that a second degree polynomial always can be rewritten as x^2 - a^2 or as x^2 + a^2, if you use an appropriate change of variable. I was thinking about how this works. Let's say we have a second degree polynomial (on the general form?) ax^2 +bx + c = 0...

Whats up guys ! currently studying for calculas exam and could use someone going over my answers ! Homework Statement Q1. Calculate the taylor polynomial of degree 5 centred 0 for f(x) = e-x. Simply coeffcients and use the error formula to estimate the error when p5(0.1) Q.2 Q1...

I am reading Chapter 2: Commutative Rings in Joseph Rotman's book, Advanced Modern Algebra (Second Edition). I need help with Exercise 2.47 on page 114. Problem 2.47 reads as follows: I need help with showing that $$f(x)$$ has a root $$ \alpha \in \mathbb{F}_4 $$. My work on this part of the...

I came across the Legendre differential equation today and I'm curious about how to solve it. The equation has the form: $$(1 - x^2)y'' - 2xy' + \nu(\nu +1)y = 0, (1)$$ Where ##\nu## is a constant. The equation has singularities at ##x_1 = \pm 1## where both ##p## and ##q## are not analytic...

Homework Statement Trying to make sense of my notes... "A polynomial in n variables on an n-dimensional F-vector space V is a formal sum of the form: p(x)= ∑(C_i)x^β" so basically can somebody help me understand how polynomials represent vector spaces? Whatever degree the polynomial is...

[/itex]Homework Statement Find the first three coeficents c_n of the expansion of Cos(x) in Hermite Polynomials. The first three Hermite Polinomials are: H_0(x) = 1 H_1(x) = 2x H_0(x) = 4x^2-2The Attempt at a Solution I know how to solve a similar problem where the function is a polynomial of...

I was studying a article that solves the cube and quartic equation in the inverse sense: ##x = \sqrt[3]{A} + \sqrt[3]{B}## ##x = \sqrt[4]{A} + \sqrt[4]{B} + \sqrt[4]{C}## https://www.physicsforums.com/attachment.php?attachmentid=70239&stc=1&d=1401676309I found this relationship too...

Homework Statement Reason or prove whether there exist polynomials A, B and C such that the following is satisfied where y=e^{k\cdot arcsinx}: A\cdot y''+B\cdot y'+C\cdot y=0 Note that this is high school level calculus so it shouldn't be something too complicated. While I said...

I'm in the first of 3 courses in quantum mechanics, and we just started chapter 4 of Griffiths. He goes into great detail in most of the solution of the radial equation, except for one part: translating the recursion relation into a form that matches the definition of the Laguerre polynomials...

(1) P_{l}(u) is normalised such that P_{l}(1) = 1. Find P_{0}(u) and P_{2}(u) We have the recursion relation: a_{n+2} = \frac{n(n+1) - l(l+1)}{(n+2)(n+1)}a_{n} I'm going to include a second similar question, which I'm hoping is solved in a similar way, so I can relate it to the above...

Homework Statement Write ##sin(ax)## for ##a \in \mathbb{R}##. (Use generating function for appropriate ##z##) Homework Equations ##e^{2xz-z^2}=\sum _{n=0}^{\infty }\frac{H_n(x)}{n!}z^n## The Attempt at a Solution No idea what to do. My idea was that since...

(a) Use Taylor's Theorem to estimate the error in using the Taylor Polynomial of f(x)=sqrt{x} of degree 2 to approximate sqrt{8}. (The answer should be something like 1/2 * 8^{-7/2}. (b) Find a bound on the difference of sin(x) and x- x^{3}/6 + x^{5}/120 for x in [0,1]This is a problem on a...

Pl(u) is normalized such that Pl(1) = 1. Find P0(u) and P2(u) note: l, 0 and 2 are subscript recursion relation an+2 = [n(n+1) - l (l+1) / (n+2)(n+1)] an n is subscript substituted λ = l(l+1) and put n=0 for P0(u) and n=2 for P2(u), didnt get very far please could someone...

If $P(0)=3$ and $P(1)=11$ where $P$ is a polynomial of degree 3 with integer coefficients and $P$ has only 2 integer roots, find how many such polynomials $P$ exist?

i.e., does the set of functions of the form, \{ x^{\frac{n}{m}}\}_{n=0}^{\infty} for some fixed m produce a linearly independent set? Either way, can you give a brief argument why or why not? Just curious :)

What is the maximum number of roots of a multivariate polynomial over a field? Is there a multivariate version of the fundamental theorem of algebra?

Hello! :cool: I want to find the greatest common divisor of $x^4+1$ and $x^2+x+1$. I applied the Euclidean division and found that $x^4+1=(x^2+x+1) \cdot (x^2-x)+(x+1)$. So,isn't it like that: $gcd(x^4+1,x^2+x+1)=x+1$ ? But.. in my textbook,the result is $1$! Which of the both results is...

Homework Statement Suppose that u and v are real numbers for which u + iv has modulus 3. Express the imaginary part of (u + iv)^−3 in terms of a polynomial in v.Homework Equations The Attempt at a Solution |u+iv|=3 then sort(u^2+i^2) = 3 then u = 3 and v=0 or u=0 and v=3(0+3i)^-3 i swear i am...

Hello. I open this 'thread', in number theory, but he also wears "calculation". I've done a little research, I share with you. Let \ r_1, r_2, \cdots, r_n, roots of the polynomial. P(x)=p_0x^n+p_1x^{n-1}+ \cdots+p_{n-1}x+p_n Let \ Q(x)=q_0 x^n+q_1x^{n-1}+ \cdots +q_n, such that its roots...

I've been taught that with the basic form of a function's maclaurin series, complex forms of the same series can be found. For example, the first three terms for arctan(x) are x-x^3/3 + x^5/5, meaning the first three terms for arctan(x^2+1) at a=0 should be (x^2+1) - ((x^2+1)^3)/3 +...

Homework Statement I want to transform a polynomial of kind p(x)=ax³+bx²+cx+d in another like p(t)=At³+B. Is possible? Homework Equations Is possible to transform a polynomial of kind ax³+bx²+cx+d in another like t³+pt+q...

Hello. Homework Statement Basically I want to evaluate the integral as shown in this document: Homework Equations The Attempt at a Solution The integral with the complex exponentials yields a Kronecker Delta. My question is whether this Delta can be taken inside the integral...

Greetings! :biggrin: Homework Statement Starting from the Rodrigues formula, derive the orthonormality condition for the Legendre polynomials: \int^{+1}_{-1} P_l(x)P_{l'}(x)dx=(\frac{2}{2l + 1}) δ_{ll'} Hint: Use integration by parts Homework Equations P_l=...

Hey :o ! Could you help me at the following exercise? $k, n \in \mathbb{N}$ $f(x)=cos(k \pi x), x \in [0,1]$ $x_i=ih, i=0,1,2,...,n, h=\frac{1}{n}$ Let $p \in \mathbb{P}_n$ the Lagrange interpolating polynomials of $f$ at the points $x_i$. Calculate an upper bound of the maximum error...

Contemporary Abstract Algebra by Gallian This is Exercise 14 Chapter 3 Page 69 Question Let $G$ be the group of polynomials under the addition with coefficients from $Z_{10}$. Find the order of $f=7x^2+5x+4$ . Note: this is not the full question, I removed the remaining parts. Attempt...

Homework Statement Use a Taylor Polynomial about pi/4 to approximate cos(42){degrees} to an accuracy of 10^-6. *To get an accuracy of 10^-6, use the error term to determine an nth Taylor Polynomial to use. Homework Equations x = 45 or pi/4, x0 = 42 or 7pi/30 cos(x) = Pn(x) + Rn(x)...

Homework Statement I want to varify that the components of a homogenous electric field in spherical coordinates \vec{E} = E_r \vec{e}_r + E_{\theta} \vec{e}_{\theta} + E_{\varphi} \vec{e}_{\varphi} are given via: E_r = - \sum\limits_{l=0}^\infty (l+1) [a_{l+1}r^l P_{l+1}(cos \theta) - b_l...

show that the first derivative of the legendre polynomials satisfy a self-adjoint differential equation with eigenvalue λ=n(n+1)-2 The attempt at a solution: (1-x^2 ) P_n^''-2xP_n^'=λP_n λ = n(n + 1) - 2 and (1-x^2 ) P_n^''-2xP_n^'=nP_(n-1)^'-nP_n-nxP_n^' ∴nP_(n-1)^'-nP_n-nxP_n^'=(...

Hi everyone, :) Here's a question I encountered and I need your help to solve it. Question: Let \(V\) be the space of real polynomials of degree \(\leq n\). a) Check that setting \(\left(f(x),\,g(x)\right)=\int_{0}^{1}f(x)g(x)\,dx\) turns \(V\) to a Euclidean space. b) If \(n=1\), find...

I am reading R Y Sharp: Steps in Commutative Algebra. In Chapter 3 (Prime Ideals and Maximal Ideals) on page 44 we find Exercise 3.24 which reads as follows: ----------------------------------------------------------------------------- Show that the residue class ring $$ S $$ of the ring of...

Does anyone understand this project? I desperately need your help! Please let me know. Appreciate a lot!

Homework Statement Let T: R^6 -> R^6 be the linear operator defined by the following matrix(with respect to the standard basis of R^6): (0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 ) a) Find the T-cyclic subspace generated by each standard basis vector...

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...