Polynomial Definition and 1000 Threads

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...

Homework Statement Let A be an nxn matrix with real number entries, in which all entries are 1. Find the characteristic polynomial of A. Homework Equations characteristic polynomial: f(t)=det(A-tI), I is identity matrix The Attempt at a Solution I've tried to do this by various...

How do you convert binary into polynomial form? I understand hexadecial conversion so {4e} = 01001000 now how do I go about changing that into a polynomial?

1.Let F=K(u) where u is transcedental over the field K. If E is a field such that K contained in E contained in F, then Show that u is algebraic over E. Let a be any element of E that is not in K. Then a = f(u)/g(u) for some polynomials f(x), g(x) inK[x] 2.Let K contained in E...

Homework Statement Consider the polynomial p(x) = x^5+45x^3+324x-3x^4 - 135x^2 - 972 . Given that p has some integer roots on the real and imaginary axes, factorise p into linear and quadratic factors with real coefficients. Enter your answer a set of factors in the form { x-1, x+4...

What would be the best way to show that if F is an infinite field and f(x) is a polynomial in F[x] and f(a)=0 for an infinite number of elements a of F, that f(x) must be the zero polynomial? It kind of just makes logical sense to me, so I can't think of a way to actually show this. please help

I started to answer this question, and I have quite a bit an answer, but still not complete, let's say that we write a Lagrangian in QFT, which an unknown function of the scalar field \phi and its derivative \partial \phi. We can always Taylor-expand it and get: L(\phi,\partial\phi) = a + b \phi...

Homework Statement Show the polynomial f(x) = x^p + x^{p-1} + ... + x - 1 is irreducible over Z_p where p is a prime.Homework Equations The Attempt at a Solution I know f(x) has no roots in Zp, but other than that, I'm stuck. Thanks for the help.

I am not sure how to format in LaTeX; I apologize for that. The Hermite polynomials Hn(x) (physicist's version) satisfy the recurrence relation, H_{n+1}(x) - 2xHn(x) + 2nH_{n-1}(x) = 0; H0(x) = 1 and H1(x) = 2x: Use this to derive the generating function for the Hermite polynomials...

I cannot for the life of me remember where I learned to do this nor can I remember why this works, but I know a method for finding the simplest polynomial for a series. I had something half typed up and then I found this site: http://www.johansens.us/sane/technotes/formula.htm which explains a...

Homework Statement At how many points in the xy-plane do the graphs of y=x^{12} and y=2^{x} intersect? Homework Equations none The Attempt at a Solution I have no idea what to do. I thought of trying to narrow it down to some intervals where the graphs may cross, but, since they're...

Homework Statement if p is prime, prove that p divides A, where A satisfis 1+\frac{1}{2}+...+\frac{1}{p-1}=\frac{A}{\left(p-1\right)!} Homework Equations The chinese remainder theorem? Eulers theorem? The Attempt at a Solution So as the question marks imply, I'm at a loss as to...

Homework Statement Prove that for n>=2, n not prime, the following polynomial is reducible f_n(x) = x^{n-1} + x^{n-2} + ... + x + 1 Homework Equations The Attempt at a Solution If n is even, the polynomial has -1 as a root, so it is reducible. But when n is odd (and not prime) I'm not sure...

Hey guys, I really need some help please! I would really appreciate it if anyone can help out, if we have F16 = F2/(x^4+x+1). can anyone explain to me how can I compute the minimal polynomials and the characteristic polynomils over F2 of elements of F16 and to point out the primitive ones...

Hello, I am trying to solve a degree three polynomial, but unfortunately I am stuck t3+2t2+1=1 This is as far as I can get: t(t2+2t)+1=0 Where do I go from here? Thanks!

Homework Statement y = -5x3 + 4/x5 + 1.7pi Homework Equations Differentiate each Function. Derivatives The Attempt at a Solution I have done question like these before but this is the first time using pi. I used all derivative rules none of them give me the right answer. The answer I am...

Homework Statement Factor f(x) = 3x4 + 2 into a product of irreducible polynomials in Z5[x] Homework Equations The Attempt at a Solution I don't get it. I tried dividing it using the division logarithm, but then I can only get it to a point where it's like, 3(x-1)(..) <-...

Let p(x)=a0+a1x+a2x2\in Real Numbers and let z\in Complex Field. Show that p(conjugate of z)=conjugate of p(z)

Homework Statement Find the Taylor polynomial for f(x) = 1/(1-x), n = 5, centered around 0. Give an estimate of its remainder. The Attempt at a Solution I found the polynomial to be 1 + x + x2 + x3 + x4 + x5, and then tried to take the Lagrange form of the remainder, say, for x in [-1/2, 1/2]...

If the polynomial x^4-16x^2-25x+10 is divided by another polynomial x^2-2x+k, the remainder comes out to be x+a find k and a. My approach: I tried dividing the first polynomial by the second and then equating the remainder to (x+a). It didn't work out. Are my calculations wrong or am i on the...

Homework Statement Hi, I have been given the polynomial function P(z)=4z^4 -12z^2 +3z +19 I need to Establish the main technique/s required to solve the polynomial. Then I need to find the solution of the polynomial? Any help would be greatly appreciated because I have no idea where to...

Homework Statement Let R = Z[x] be a polynomial ring where Z is the integers. Let I = (x) be a principal ideal of R generated by x. Prove I is a prime ideal of R but not a maximal ideal of R.Homework Equations The Attempt at a Solution I want to show that R/I is an integral domain which...