Polynomial Definition and 1000 Threads

Homework Statement Given p(x) = x4+2x2+1 and A = [[1 1 -2 0] [0 1 0 2] [1 1 -1 1] [0 0 -2 -1]] p(A) = 0 Find a polynomial q(x) so that q(A) = A-1 a) What is q(x)? b) Compute q(A) = A-1 Homework Equations I found the Cayley-Hamilton theorem, which states: p(x) = det(A-xIn)...

As it turns out, this first part was an extremely pervasive user error that I have not seen for days. Still, though, it is interesting that equations can be factored in many different ways, so I will post that here instead: 3x^3 = -5x^2 +2x -3x^3 -5x^2 + 2x = 0 Factored: x(-3x+1)(x+2) and...

Homework Statement Examining the answers of the previous two questions, write the quadratic polynomial f(x1,x2,x3)=x1x2−6x22+3x2x3−3x23 in the form f(x1,x2,x3)=[x1x2x3]A[x1x2x3]<-this last group is a column matrix where A is a symmetric matrix. Homework Equations Matrix multiplication...

Homework Statement (There is a comparison question I have, so i'll post my question with these) 8) Each of the following polonomials has x-intercepts of -6, 5 , and 0. Determien the appropriate equation for each. Then, sketch a graph of the function. b) A quartic function that extends...

To compute the checksum for some frame with m bits, corresponding to the polynomial M(x), the frame must be longer than the generator polynomial. Why...?

Homework Statement Suppose R is an integral domain and I is a principal ideal in R[x], and I \neq {0} a) Show I = <g(x)> for some g(x)\inR[x] that has minimal degree among all non-zero polynomials in I. b) Is it necessarily true that I = <g(x)> for every g(x)\inR[x] that has minimal...

Homework Statement Prove that 0.493948<\int_0^{1/2} \frac{1}{1+x^4} dx<0.493958Homework Equations This chapter is about Taylor Polynomials, and specifically this section deals with Taylor's formula with remainder: f(x)=\sum_{k=0}^n \frac{f^{(k)}(a)}{k!} (x-a)^k + E_n(x) The general formula for...

Two particles move along the x-axis. Particle one has the position x = 8t^2 + 7t + 2 Particle two has the acceleration a = -8t, and when t=0 v=23. When the velocity of the particles match what is their velocity? I thought of approaching this problem by changing both equations into the...

Homework Statement Give a formula for the extended function that is continuous at the indicated point. (x3-4x2-11x+30)/(x2-4), x=2 Homework Equations The Attempt at a Solution I know that I have to factor the top and bottom so that I can cancel terms that cause the function to...

Hi, I'm trying to learn Fourier transforms by myself. I'm a bit confused about how the trignometric polynomial complex form was derived. I'm referring to this: http://en.wikipedia.org/wiki/Trigonometric_polynomial Now, I haven't taken complex analysis so I only know the basics. I used Euler's...

I want to write an algorithm that gives as output the numbers a_n,\ldots, a_1,a_0, when a matrix A\in\mathbb{R}^{n\times n} is given as input, such that \det (A - \lambda) = a_n\lambda^n + \cdots + a_1\lambda + a_0,\quad\quad\forall\lambda\in\mathbb{C} If n=2, a_2 = 1,\quad a_1 =...

\frac{r^3+3r^2+4r-8}{r-1} how do i solve that? and is there a general formula? thanks! [SIZE="1"]Edit by bored Borek: LaTeX corrected.

Homework Statement A n × n-matrix A satisfies the equation A2 = A. (a) List all possible characteristic polynomials of A. (b) Show that A is similar to a diagonal matrix Homework Equations The Attempt at a Solution A2 = A so, A2 - A = 0 A(A-I) = 0 Our minimal polynomial is x2 - x = m(x)...

Hi guys, Thanks for taking the time to read the post. I have a question related to curve fitting and polynomials that i was hoping someone might be able to help me with. I have a set of x and y data points, all on a graph. I have then calculated the 4th order least squares polynomial...

Today I read about the AKS - Primality test in which the simple theorem For gcd(a, n) = 1, we have (X - a)^n\equiv X^n - a\ (mod\ n)if and only if n is prime. was proven. The if direction is quite trivial from the fact that \binom{p}{k}\equiv 0\ (mod\ p) for 1\leq k < p. The other direction...

Homework Statement f(x,y) = ln(3y-8x) Derive the first and second order Taylor polynomial approx, L(x,t) and Q(x,t), for T(x,T) about the point (1,1) Homework Equations -None- The Attempt at a Solution I do not understand what the question wants, nor do i want a solution. I...

[b]1. Homework Statement The swinging period T(θ) for a small amplitude simple pendulum, is given by T, a constant for a constant length pendulum. If the initial angle θ is large, then the amplitude becomes large and the period needs to be corrected. The correction to the large amplitude...

Homework Statement Suppose that you are given a set of observations (tk,yk), k = 1,...,M. You plot these points on a sheet & it seems that the relationship between (t,y) could be approximated with a second order polynomial. a) Write down the model in the form y = Ax + c. Specify the vectors...

Homework Statement Divide A(x)= [x3+2x2+3 -4x3-x2-5] [3x2-2 x3-2x2+x+4] by B(x) = [x+4 -3] [-x+6 x+2] on both the right side and the left side. Homework Equations The Attempt at a Solution I am...

Why must a polynomial be finite in length? Is there such thing as a recursive polynomial? Thanks guys!

Please Help me out with this one. Can all patterns in 2D or 3D, have an equivalent equation in polynomial?

Hello, I'm trying to figure out connection between the characteristic polynomials for real matrices [3x3] and their powers. Suppose A is a real matrix [3x3] which's c.p is t^3+t^2+t-3, how can i find the c.p. of A^2. Now suppose p(t)=a_1t^3+a_2t^2+a_3t+a_4 Right away I can know that...

Hi all, I came across a linear recurrence with polynomial coefficients and realized that I don't have a clue as to how to solve it. The usual methods like generating functions or guessing seem not to work in that case. Here is the equation: i (i - 1) (i - 2) b[i] = 1/3 (i + 1) i (1 -...

Homework Statement I'm working out a differential equation problem that I am supposed to solve with the formula \mathcal{L}\{t^\alpha\} = \frac{\Gamma{(\alpha + 1)}}{s^{\alpha+1}}. The problem is \mathcal{L}\{t^{\frac{1}{2}}\} (finding the Laplace transform of the given function)...

Consider the space of all polynomials in n variables of degree at most d. The dimension of that space is C(n+d,d). How do I calculate the dimension of that same space when I restrict the domain of the polynomials to the unit ball? In that case all the polynomials (sum(i=1..n) x_i^2)^p with p a...

Homework Statement If an even degree polynomial of order 2n intersects the x-axis twice, how do I show that the (2n-1) th derivative has a zero in that interval? Homework Equations example: let g(x)=x^3(1-x). Show without computation that g'''(c) =0 for some c in (0, 1). The...

Homework Statement No problem exactly I am just reading a book that refrences symmetric polynomials but i don't know what a symmetric polynomial is. I looked at the wiki page but i didn't really get what it was saying. Any help on clearing up the meaning would be greatly appreciated...

I have this function (1-6*x^2)^-1 and i want to polynomial interpolation (lagrange and spline) in 21 equidistant points [-1,1] I made this function x =linspace(-1,1,21); y = (1-6*x^2)^-1; z=[-1:0.01:1] c=polyfit(x,y,20) p=polyval(c,z) s=spline(x,y,z) plot(z,(1-6*x^2)^-1, z, p, z, s)...

For two matrices A and B, when p(A)=0 iff p(B)=0 for any polynomial, what will happen? i read that A and B have the same minimal polynomial, why?

Graph each function given below on a graphing calculator to find a general rule for determining when a graph crosses the x-axis at an x intercept or when the graph just touches and turns away from the x axis. State the rule that you find. y = (x + 1)^2(x - 2) y = (x - 4)^3(x - 1)^2 y =...

I want to interpolate a set of points (x(i),y(i)) for i=0, n; using Newton's polynomial of degree n. I've Kincaid's book of numerical analysis. He gives an algorithm that doesn't make entirely sense to me. He says for i=0, n do di <- f(xi) end for j=1, n do etc. Basically I'm stuck...

Hey I'm studying for an exam and one of the things i need to know is this: 4. Given the eigenvalues of a matrix: a) Determine the characteristic polynomial. b) Find vectors than can act as bases for the associated eigenspaces. Part a seems relatively straight forward but for part b I...

Can someone tell me if the following statement is true? Say you have P(x)e^{Q(x)} where P(x) is any finite-degree polynomial and Q(x) is a polynomial of integer order k. Is it true that for any positive epsilon, I can find some real numbers A and B such that |P(x)e^{Q(x)}| \leq...

Let K be a field of characteristic p. Suppose f(x)=(xk+ck-1xk-1+...+c0)(xp-k+...) in K[x] with 1≤k≤p-1. My question is: 1. since f(x) in K[x], can I conclude g(x)=xk+ck-1xk+...+c0 in K[x] as well? 2. We see that in general if g(x)=xk+ck-1xk-1+...+c0 then ck-1=-(α1+α2+...+αk) where...

Polynomial Inequalities - Finding the solution set?? Homework Statement Solve the Inequality 2x^3 >-8x^2 Homework Equations The Attempt at a Solution Ok I am able to solve this by first figuring out the zeroes, and then testing with regions, So my answer is x=0 and x =...

Input a polynomial into FORTRAN? Hello all, I took a Fortran programming class last semester and can write Fortran well. However, I never recall anyone ever asking if there would be a way to input a polynomial into the command prompt. I'm assuming there is not. It would be great though. I have...

Homework Statement Suppose an/n+1 +...+a0/1=0. Prove f(x) =anxn +...+a0 has a root between zero and one. Homework Equations I'm pretty sure this is induction, but I'm not completely sure. Mean Value Theorem probably The Attempt at a Solution Well f(0)=a0 and f(1)=an + ... +...

Homework Statement Given a n-sided polygon. Divide it into multiple small triangles that have same radius r. Compute the following limits: Homework Equations The Attempt at a Solution When I plug in a random number to guess the answer, somehow it is always near 0 (very small) so...

Homework Statement Interpolate the function ln(x+1) using a Newton's polynomial of degree 2. The points to be interpolated are x_0=0, x_1=0.6 and x_2=0.9. Homework Equations p_2(x)=c_0+c_1(x-x_0)+c_2(x-x_0)(x-x_1). The Attempt at a Solution So I used divided differences in order...

If \mathbf{p(x) = 2009-2008x^{100}+2007x^{99}-2006x^{98}+.....+1909x} Then Calculate \mathbf{p(2008)}

Homework Statement Divide 2x5+x-1 by 3x2+1 in Q[x], Z5[x], and R[x] The Attempt at a Solution I believe the answer should be the same in Q[x] and R[x] and after division I got 2/3x3-2/9x with remainder 1 2/9x-1. I had trouble in Z5 and would like someone to help me. So far I...

Homework Statement This question was on a test in a math contest I was recently in, and I cannot seem to figure out how to get the answer: Let f(x)=x7+x6+x5+x4+x3+x2+x+1. If k is a positive integer such that 53 < k < 115, find the sum of all distinct k such that the numerical remainder...

Homework Statement Let n\,\in\,\mathbb{N}. Let F be a field, and suppose that p(x)\,\in\,F[x] is a polynomial of degree (n + 1). Let S be the set: S\,=\,\left\{\left(a_0,\,\ldots,\,a_n\right)\,:\,a_i\,\in\,F\right\} Define \phi: S\,\rightarrow\,F[x]/\left(p(x)\right) via...

Hello my friends, I have this problem and would appreciate someones help: Determine all intervals where the following function is increasing or decreasing. F(x) = -x^(5)+(5/2)x^(4)+(40/3)x^(3)+5 Solution To determine if the function is increasing or decreasing we will need the...

First of all this is from a book I own. It is not for any class or assignment. I am just curios on how you would go about solving such a problem. The radii of four different-sized ball bearings differ by 1.0mm in radius from one size to the next. If the volume of the largest equals of the...

Homework Statement Find the remainder of x^4 + 3x +2 after division by x+3 in Z5[x] Homework Equations my quotient after dividing was: x^3 + 2X^2 + 4x +1 The Attempt at a Solution I found the remainder to be 4. If anyone has time, I believe I made a mistake somewhere and would...

Hi, I'm looking for a numerical method to solve simultaneous polynomial equations that can be implemented in a computer program. I have included an example of a typical pair of equations that I may need to solve. In this case the two variables that I need to solve for are x and y, all other...

Homework Statement 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...

Hi, given a polynomial ring R=\mathbb{C}[x_1,\ldots,x_n] and an ideal I=\langle f_1, f_2 \rangle, \quad f_1, f_2 \in R, is it always true that R/I \cong (R/\langle f_1 \rangle)/\phi(\langle f_2 \rangle), with \phi: R \rightarrow R/I being the quotient map? That is, is quotienting by I always...

Homework Statement I need to find an expression for: y^{2}H(y) I know how to find: yH(y) with: yH(y)=\frac{1}{2}H_{n+1}(y)+nH_{n-1}(y) I looked through the miscellaneous relations but nothing stuck out to me. Can someone give me some guidance on how to go about finding a relation...