Polynomial Definition and 1000 Threads

Hi all, I would just like to get some clarity on units and zero-divisors in rings of polynomials. If I take a ring of Integers, Z4, (integers modulo 4) then I believe the units are 1 & 3. And the zero-divisor is 2. Units 1*1 = 1 3*3 = 9 = 1 Zero divisor 2*2 = 4 = 0 Now, If I...

Homework Statement Digital filter analysis - this is just one part of a multi-part question I can't move forward with. It's supposed to be an auxilliary question and isn't the "meat" of the problem. Find b, such that maximum of the magnitude of the frequency response function...

I apologize for the rather vague title. It's space-limited and I'm not sure how to concisely state what I want to know. Basically, I understand that the solutions to quadratic equations (and if I remember correctly cubic equations) often had surveying problems land surveying. However, quartic...

Homework Statement Factor x^16-x over the fields F4 and F8 Homework Equations factored over Z (or Q), x^16-x = (x*(x - 1)*(x^2 + x + 1)*(x^4 + x^3 + x^2 + x + 1)*(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1) The Attempt at a Solution I know the that quadratic and higher terms I have left...

Homework Statement 2n - 6m + 5n^2 - 15mn Homework Equations No particular equation since this is factoring The Attempt at a Solution Keep in mind that I struggle when it comes to grouping as I'm not sure where I'm supposed to start but... 2n - 6m + 5n^2 - 15mn Group first 2...

Homework Statement Let f be entire. Then if lim_{z\rightarrow \infty}|f(z)|=\infty then f must be a non-constant polynomial.Homework Equations The Attempt at a Solution So we know f is entire. Thus I suppose it makes sense to go ahead and expand it as a power series centered at zero. Thus...

Homework Statement Let P(n) be the set of all polynomial of degree n with integer coefficients. Prove that P(n) is countable, then show that all polynomials with integer coefficients is a countable set. 2. The attempt at a solution For this problem the book gives me a hint that using...

f(t) = t4 - 23 + 3t2 - 2t + 1 in Q[t] Am i right in thinking I just show by the rational root theorem that the only possible roots are +-1 f(+-1) =/= 0 so there are no repeated factors? Seems too easy..

Homework Statement I uploaded a picture of the question because I didn't want to confuse people from the way it looks because it is pretty long. http://tinypic.com/r/2itpm39/5 Homework Equations -None- The Attempt at a Solution I went from the original equation and took an 'x'...

I have some math people who say you can, and some who say you can't beyond quintic because of the Abel-Ruffini theorem. Which is it? Can I generalize all polynomials? Or at least can I manually make a formula for each individual degree?

Homework Statement \frac{x^3+x^2-5x+3}{x^3-3x+2} Homework Equations The Attempt at a Solution well I'm drawing that long division house with x^3-3x+2 on the outside and x^3+x^2-5x+3 on the inside. I'm seeing that x^3 goes into x^3 one time, so i put a 1 on top of the...

Homework Statement First find all rational zeros of f, then use the depressed equation to find all roots of the equation f(x) = 0. f(x) = x^3 + 5x^2 - 8x + 2Homework Equations The Attempt at a Solution Possible rational zeros: 2, -2, 1, -1 Synthetic division: 1 | 1 5 -8 2 _____1 6 -2...

I may have posted this back in the Old Country, but: let the polynomial: \[P(x)=x^n+a_1X^{n-1}+ ... + a_{n-1}x+1 \] have non-negative coeficients and \(n\) real roots. Prove that \(P(2)\ge 3^n \) CB

Polynomial functions... find "a" Homework Statement When ax3 - 4x2 + 5x - 3 is divided by (x+2) and (x-1), the remainders are equal. Find a. Don't know where to start. A little help? Hints?

Homework Statement Find eight elements r \in \mathbb{Q}[x]/(x^4-16) such that r^2=r. Homework Equations N/A The Attempt at a Solution The elements 0+(x^4-16) and 1+(x^4-16) clearly satisfy the desired properties, but I still need six more elements. Can anyone help me figure out a...

Homework Statement Find (f^{-1})'(a) of: f(x)=\sqrt{x^{3}+x^{2}+x+22} ; a=5. Homework Equations (f^{-1})'(a)=\frac{1}{f'((f^{-1})(a))} The Attempt at a Solution Well, I know to find an inverse: I need to set the equation equal to y, solve for x, then swap x and y. But I don't...

Homework Statement I have a polynimal equation as this - 0.00000000000049125*T^4 + 0.00000000021358333333333333333333333333333*T^3 + 0.00000290233125*T^2 - 0.032444109375*T + 19.891472013020833333333333333333 Homework Equations The Attempt at a Solution I insert those...

Is it possible to compute the Galois Group of a polynomial manually (without a computer)? If so, can someone please explain how? I can't seem to find any information (aside from computer algorithms) on how to find a Galois Group or how to factor a polynomial modulo a prime. If it helps to...

Homework Statement The function y=x^{2}+4x-6 has two inverses. What are they and which domains lead to these inverses? Homework Equations The Attempt at a Solution y=x^{2}+4x-6 x=y^{2}+4y-6 y(y+4)=x+6 Not really sure where to go from here.

Factoring a 4th order polynomial Homework Statement Example: (jw)^{3}+6(jw)^{2}+5jw+30=0 can be re-written into 6(5-w^{2})+jw(5-w^{2}). The fact that there are two identical (5-w^{2}) is a desirable outcome. Imaginary number j=\sqrt{-1} becomes -1 when raised to the power of 2. Homework...

Ok, so obviously, given some polynomial P(x) of degree r, it has r roots in the complex numbers by the FTOA, and if these roots are u_1, u_2,... it can be written as \begin{array}{l} P(x) = (x - {u_1})(x - {u_2})(x - {u_3}) \cdots \\ P(x) = {x^r} - ({u_1} + {u_2} + {u_3} + \ldots ){x^{r - 1}}...

Homework Statement If k is a natural number, prove that 2^{n} > n^{k} for all n \geq k^{2} + 1.Homework Equations We need to use a proof by induction.The Attempt at a Solution Let's do the case when k = 4. We check the base case directly: 2^{17} = 131072 > 83521 = 17^4 Suppose 2^{n} > n^{4} for...

Homework Statement The graph of the quadratic polynomial , y=ax2+bx+c is as shown below in the figure : http://postimage.org/image/nvkxv74yd/ Then : (A) b2-4ac<0 (B) c<0 (C) a<0 (D) b<0Homework Equations y=ax2+bx+c If y=0 , then ax2+bx+c=0 Then , x = {-b+-sqrt(b^2-4ac)}/2a The Attempt at...

Homework Statement I have a limit problem, however I do know how to work limits, I guess what I need is more of a refresher on how to work third degree polynomials. The polynomial(s) I am trying to work with are the following: x3-2x2+2x-15 -and- x3-5x2+10x-12 The limit is a limit where x...

Homework Statement Find one solution (or prove no solutions exist) to the equation x100 = 3 mod 83, where "=" means "congruent to" Homework Equations Possibly Fermat's theorem: If p is prime and p does not divide a, then ap-1 = 1 mod p. The Attempt at a Solution 83 is...

So I was just wondering why when you approximate using the Taylor Polynomials for something like e^x/x at x = 0 you can just find the approximation for e^x and make it all over x, could you do the same for like e^x/x^2 or e^x/x^3? I hope my question makes sense... thanks

Homework Statement Let Tn(x)=cos(narccosx) where x is real and belongs to [-1,1] and n E Z+ Find T1(x). Show that T2(x)=2x^2 - 1. Show that Tn+1 (x) + Tn-1 (x) = 2xTn(x) Hence, prove by induction that Tn(x) is a polynomial of degree n. The Attempt at a Solution Since cosθ=x and arccosx=θ we...

Homework Statement Homework Equations The Attempt at a Solution I keep getting the answer 1- 96/4!*x^4. Can't find where I'm going wrong.

I'm working on the following: "Prove that x^2 - 1=0" has infinitely many solutions in the division ring Q of quaternions." The Quaternions are presented in my book in the representation as two-by-two square matrices over ℂ. The book gives that for a quaternion (sorry for the terrible...

Homework Statement Consider the function sin(\pix) on [-1,1] and its approximations by interpolating polynomials. For integer n\geq1, let x_{n,j}=-1+\frac{2j}{n} for j=0,1,...,n, and let p_{n}(x) be the nth-degree polynomial interpolating sin(\pix) at the nodes x_{n,0},...,x_{n,n}. Prove that...

Homework Statement Let h: \mathbb{P_2} \rightarrow \mathbb{P_2} represent the transformation h(p(x)) = xp'(x) + p(1-x) for every polynomial p(x) \in \mathbb{P_2}. Find the matrix of h with respect to the standard basis \{1, x, x^2\} Homework Equations Matrix A of transformation: {\bf A}...

Hi all, I've learned that in unconstrained polynomial regression, the optimal order can be determined using two F tests : one to test for the significance of the overall regression, the other to test for the significance of the higher coefficients (assuming the first test passed of course)...

Hi. Can someone explain to me how to sort a Series so that the terms are in increasing powers of the exponent? For example the code: myseries = Normal[ Series[Sqrt[1 - w], {w, 0, 5}]] /. w -> 1/z produces 1-\frac{7}{256 z^5}-\frac{5}{128 z^4}-\frac{1}{16 z^3}-\frac{1}{8 z^2}-\frac{1}{2...

Homework Statement Suppose that A is a 6x6 matrix with real values and has a min. poly of p(s) = s^3. a) Find the Characteristic polynomial of A b) What are the possibilities for the Jordan form of A? c) What are the possibilities of the rank of A? Homework Equations See below...

Homework Statement Factor x^3+3x+6 over Z5, Z10, Z, Q and ℝ. 2. The attempt at a solution For Z5, I have the roots of x^3+3x+1 in Z5, x=2=-3 and x=3=-4, so x+3 and x+4 are factors. By long division of x^3+3x+1, it is found that x+3 is a repeated factor, so x^3+3x+6=(x+4)(x+3)^2. For Z and...

xy'^2 + yy' = 0 where y' = dy/dx The answer is C1 = y and C2 = xy but I get this: y'(xy' + y) = 0 where y' = 0 and thus y = C1 For the other solution: xy' + y = 0 y' = -y/x y = -y ln x + C2 C2 = y + y ln x Full question is here: http://www.cramster.com/solution/solution/640396

Homework Statement Given a random variable X with a known distribution (e.g. a beta distribution), find the distribution of f(X) = X^2 + X The Attempt at a Solution I've tried the normal approaches: the standard transformation theorem; conditioning on X; Laplace transformation, etc...

Homework Statement Prove that the polynomial $(x-1)(x-2)...(x-n) + 1$ is irreducible over Z for n\geq 1 and n \neq 4Homework Equations N/AThe Attempt at a Solution Let $f(x) = (x-1)(x-2) \cdots (x-n) + 1$ and suppose $f(x) = h(x)g(x)$ for some $h,g \in \mathbb{Z}[x]$ where $\deg(h), \deg(g) <...

I'm trying to find a general formula for an algebraic equation, I'm studying the behavior of ∏_{i=2}^n(1-\frac{1}{i^m}) for m=3 and so far I've seen that I can find a general formula if n^2 + n + 1 produces only prime numbers. if not, it would get way harder to find a general formula for it by...

Hi guys I'm getting into a little trouble when differentiating polynomial functions. How do you differentiate f(x)=ax+b/cx+d ? Is there other ways of calculating this apart from the chain rule ? Thanks for any help.

My quantum text, leading up to the connection formulas for WKB and the Bohr-Sommerfeld quantization condition states that for \begin{align}u'' + c x^n u = 0 \end{align} one finds that one solution is \begin{align}u &= A \sqrt{\eta k} J_{\pm m}(\eta) \\ m &= \frac{1}{{n + 2}} \\ k^2 &=...

For #4, I'm mostly confident I did it correctly. In determining the error, we're supposed to find the maximum absolute value on an interval I. I set I = (0,2pi). Is that right? http://i111.photobucket.com/albums/n149/camarolt4z28/4-1.png For #5...

Homework Statement I have list( Depth vs. Deflection) of data which requires plotting with the polynomial degree of 14. Homework Equations I need help to figure out how to plot with polynomial degree. The Attempt at a Solution

Homework Statement Let M be a subspace of the vector space \mathbb{R}_2[t] generated by p_1(T)=t^2+t+1 and p_2(T)=1-t^2, and N be a subspace generated by q_1(T)=t^2+2t+3 and q_2(T)=t^2-t+1. Show the dimension of the following subspaces: M+N, M \cap N, and give a basis for each...

The first hypothesis is that f is continuous on [a,b]... Is there a more concise mathematical way of saying... "because the function f is a polynomial it is continuous in its domain."? Because I rather not write that on my test it looks sloppy and non professional...

As part of solving a DE I need to make this friendlier to integrate: \frac{2u}{1+u^{-2}} I figured trying to divide it couldn't hurt. I got: u^{-1}+u^{-3} I can't type out all the steps easily, I'm on a mobile device at the moment. That answer looks suspect, did I do it correctly?

Homework Statement Prove, using Rodrigues form, that Hn+1=2xHn -2nHn-1 Homework Equations The Rodrigues form for Hermite polynomials is the following: Hn = (-1)nex2\frac{d^n}{dx^n}(e-x2)The Attempt at a Solution Hn+1 = (-1)n+1ex2\frac{d^(n+1)}{dx^(n+1)}(e-x2) where...

Homework Statement f:ℝ→ℝ x:→sqrt(x^2+1)-lxl Homework Equations calculate f^-1(]0,1]) The Attempt at a Solution well i chose a y from ]0,1] and tried to find an x that solves the problem like the following. 0<sqrt(x^2+1)-lxl≤1 at the end i got the following...

Homework Statement the question asks to find the quadratic maclaurin polynomial for f(x) Given f(x) = x sin(x)The Attempt at a Solution i know that a maclaurin series is when a=0 in a taylor series. i did the 1st-5th derivatives of f(x) and then used the formula for taylor polynomial and set...

I've heard there's a proof out there of this, basically that (I think) you can use the intermediate value theorem to prove that an Nth-degree polynomial has no more than N roots. I'm not in school anymore, just an interested engineer. Does anyone know where I can find this proof or any...