Polynomials Definition and 740 Threads

the directions are: determine whether each expression is a polynomial.:rolleyes: i got the easyones like: 12x3-2x2+0.5 is a polynomial but i can't figure these out: (x3/11)+(x2/8) I don't know if the variables can be fractions but i think this could be this: 1/11x3+1/8 x2 so i think...

Hi I have the following problem: To calculate the fine structure energy corrections for the hydrogen atom, one has to calculate the expectation value for (R,R/r^m), where R is the solution of the radial part of the schroedinger equation (i.e. essentially associated laguerre polynomial) and...

Question that I came across and that has stumped me for about a week hehe. Let p(z)=z^n +i z^{n-1} - 10 if \omega_j are the roots for j=1,2,...,ncompute: \sum_{j=1}^n \omega_j} and \prod_{j=1}^n \omega_j}

PF Member Careful pointed to the website of Gerardus 't Hooft, Dutch physicist and winner of 1999 Nobel Prize in Physics with Martinus J.G. Veltman. 't Hooft has a very interesting and useful website, which includes the following useful pdf file about 'Special Functions and Polynomials'...

find a formula that depends on p that determines the number of reducible monic degree 2 polynomials over Zp. so the polynomials look like x^2+ax+b with a,b in Zp. I examined the case for Z3 and Z5 to try and see what was going on. in Z3 we had 9 monic degree 2 polynomials, 6 of them...

Let U and V denote, respectively, the spaces of even and odd polynomials in Pn. Show that dimU + dimV = n+1 [Hint: Consider T: Pn ---> Pn where T[p(x) - P(-x) ] So where to begin? I thought that i should let p(x) = a + a0x + a2x^2 + ... + anX^n So if U is the space of even polynomials...

Okay, so i have this problem in my text, and I've almost figured it out (i think) but i need a little help "Let V be the set of all polynomials of degree 3. Define addition and scalar multiplication pointwise. Prove that V with respect to these operations of addiont and scalar multiplication...

Can someone explain to me why the following is true (ie, show me the proof, or at least give me a link to one): Over the field Zq the following polynomial: x^q^n-x is the product of all irreducible polynomials whose degree divides n Thanks.

"Conjecture a classifucation rule for all irreducible polynomials of the form ax^2 + bx + c over the reals. Prove it." I'm stuck cold at the start. classification rule ? "Let R be an integral domain. A nonzero f in R[x] is irreducible provided f is not a unit and in every factorization f...

Hi all Jut had a question. How do I go about finding the general formula for roots of the complex poly {z}^{n}-a where a is another complex number. Do I just go {z}^{n}=a? :S so complicated this things! Thanks in advance!

Hey there, does anyone know where I could find a list of Legendre Polynomials? I need them of the order 15 and above, and I haven't been able to find them on the net. Thanks!

Does anyone know if this is true and if so where they know it from? Given a polynomial over the integers there exists a finite field K of prime order p, such that p does not divide the first or last coefficient, and the polynomial splits over K. I realize this could be considered an...

Hi This is the character equation for a polynomial of degree where n \geq 0 p(x) = a_0 x^{n} + a_{1} x^{n-1} + a_2 x ^{n-2} + \cdots + a_{n-1}x + a_{n} I'm presented with the following assignment: Two polynomials \mathrm{p, q} where n = 3. These polynomials can derived using the...

1/sin(phi) * d/d\phi(sin(phi) * du/d\phi) - d^2u/dt^2 = -sin 2t for 0<\phi < pi, 0<t<\inf Init. conditions: u(\phi,0) = 0 du(\phi,0)/dt = 0 for 0<\phi<pi How do I solve this problem and show if it exhibits resonance? the natural frequencies are w = w_n = sqrt(/\_n) =2...

taylor differentition polynomials? hi got a question here that involves this extremely difficult question anyone that can point me in the right direction on what to do will be most appreciated :) Find Exactly the tayor polynomial of degree 4 f(x) = cos ( pi*x / 6 ) about x=-1 i know...

I was just wondering if it was possible to prove anything about the normality of the number: \sum_{x=0}^{\infty} \left((P(x) \mod b)\left(b^{-x}\right)\right) Where P(x) is a Polynomial with integer coefficients and b is the base of decimal representation. Is anything even known for...

I just want to check my answer. The question asks for the Taylor polynomial of degree 6 for ln(1-x^2) for -1<x<1 with c=0. I got tired after differentiating 6 times so I'm worried I made some mistakes along the way. The question also said: hint: evaluate the derivatives using the formula...

Show that Pn(x^2) is the 4n+2-nd Taylor polynomial of sin(x^2) by showing that \lim_{n\rightarrow infinity} R2n+1(x^2) = 0. note that Rn(x) represents the remainder I'm stuck on this question, can anyone help me please?

Im having difficulty computing large Hermitian polynomials in C++. I fear I may have to steer away from a recursive formula. Any help would be greatly appreciated. John

Hi everyone, in this problem we are asked to get a coord vectors of polynomials with B as standard basis for P3 and then express one of the coord vectors as lin. combination of the others. So the set is this: {1-4x+4x^2+4x^3, 2-x+2x^2+x^3, -17 -2x-8x^2 + 2x^3} The way I was thinking is...

What is the fastest and easiest way to factor these? ex. 3x^2+8x-3

I just found this really old book. In it, I found a way of solving quadratic equations using calculus. I've never seen this method in any other book. Ok, here's the method : The discriminant of the quadratic formula i.e sqrt(b^2 - 4ac) is equal to the first derivative of the original...

I'm having a problem with a proof I came across in one of my calculus books but it's not the calculus part of the proof that I'm having trouble with. Here's the actual proof: "Prove: The number of distinct derivatives of order n is the the same as the number of terms in a homogeneous...

Ok, I have a final in Pre-Calc comming up, and I am still a bit confused on finding asymtotes (vertical, horizontal, and slant) could someone help me with equations i can use to find the asymtotes or how i do? I am just really confused. Heres the problem. Find the vertical asymtote(s): F(x)...

Ok, I have a final in Pre-Calc comming up, and I am still a bit confused on finding asymtotes (vertical, horizontal, and slant) could someone help me with equations i can use to find the asymtotes or how i do? I am just really confused. Also, I am having trouble with finding a fourth degree...

I'm having problems finding all integer solutions to some of the higher degree polynomials. for p(x)= x^3− 3x^2+ 27 ≡ 0 (mod 1125), i get that 1125 = (3^2)(5^3). p(x) ≡ 0 (mod 3^2), p(x) ≡ 0 (mod 5^3). x ≡ 0, 3, 6 (mod 3^2) for 3^2 for 5^3, x ≡ 51 (mod 5^3) then i get x=801, 51, 426 (mod...

Simplify (x+1)/(x-1) multiplied by (x+3)/(1-x^2) divided by (x+3)^2/(1-x) Im not sure how to factor the 1-x^2 and what to do with 1-x I don't know how to simplify this please help someone. The answer to this question is 1/(x-1)(x+3) x cannot = 1,-1, and -3

For function f(x)=1/(1+x^2), calculate Taylor polynomials for the 2nd and 4th degree about the point a=0. The answer was: P2 = 1-x^2; P4 = 1-x^2+x.^4 for 2nd degree I got -2x/[(1+x^2)^2] for 4th degree I got 12x/[(1+x^2)^4]

Compute the minimal polynomials for each of the following operators. Determine which of the following operators is diagonalizable. a) T : P_2(\mathbb{C}) \to P_2(\mathbb{C}), where: (Tf)(x) = -xf''(x) + (i + 1)f'(x) - 2if(x). b) Let V = M_{k \times k}(\mathbb{R}). T : V \to V[/itex]...

What is the theorem that states if \Omega is a polynom with degree > 1 with real coefficients. If there exists a complex number z = a + bi such that \Omega(a+bi)=0 then \overline{z} = a - bi is also a root of \Omega ? For \Omega(x) = x^2 + px + q with p and q real then if a+bi is a...

Hi, I fail finding a proof (even in MathWorld, in my Mathematic dictionary or on the Web) for the following property of Chebyshev polynomials: (T_i o T_j)(x) = (T_j o T_i)(x) = T_ij(x) when x is in ] -inf ; + inf [ Example : T_2(x) = 2x^2-1 T_3(x) = 4x^3-3x T_3(T_2(x)) = T_2(T_3(x)) =...

hi folks! I have been trying to figure out some plausible geometric intrepretation to legendre polynomials and what are they meant to do. I have come across the concept of orthogonal polynomials while working with some boundary value problems in solid mechanics and wasn't able to come to...

well i could not get anything really mibd boggling, so u will have to put up with this one what is the product of: (x-a)(x-b)(x-c)..... = ?

Hi I got a Linear Algebra question. I'm suppose to find two polynomials p1 and p2 both of highest degree 3, and which satisfies the following: p1(-1) = 1 p1'(-1) = 0 p2(1) = 3 p2'(1) = 0 p1(0) = p2(0) p1'(0) = p2'(0) I hope that there is somebody out there...

Please go to the bottom of this page for the problem that I am having trouble with.

Find a nonzero polynomial f(w, x, y, z) in the four indeterminates w, x, y, and z of minimum degree such that switching any two indeterminates in the polynomial gives the same polynomial except that its sign is reversed. For example, f(z, x, y,w) = -f(w, x, y, z). Prove that the degree of the...

I don't get it. I use it to approximate f for some x, but the formula for Taylor Polynomials already has f in it?

I know that legendre polynomials are solutions of the differential equation is (1-x^2)d^2y/dx^2 - 2x dy/dx+l(l+1)y=0 where l is an integer. The first five solutions are P0(x)=1, P1(x)=x, P2(x)=3/2x^2-1/2, P3(x)=5/2x^3-3/2x, P4(x)=35/8x^4-15/4x^2+3/8 The problem is that I don't understand what...

Let f be a function that has derivatives of all orders for all real numbers. Assume f(1)=3, f'(1)=-2, f"(1)=2, and f'''(1)=4 a. Write the second-degree Taylor polynomial for f about x=1 and use it to approximate f(0.7). b. Write the third-degree Taylor polynomial for f about x=1 and use it...

Ok, I have been trying to divide this polynomial. (x^3-15x-7)/(x^2-3x-3) After I factor the first part I get stuck. This is last problem on my homework and is due in less than an hour. Please some one help me out. Thanks