Polynomial Definition and 1000 Threads

Homework Statement Let F be a field, F[x] the ring of polynomials in one variable over F. For a \in F[x], let (a) be all the multiples of a in F[x] (note (a) is an ideal). If b \in F[x], let c(b) be the coset of b mod (a) (that is, the set of all b + qa, where q \in F[x]). F[x]/(a), then is the...

a is a root of order k of the polynomial p provided that k is a natural number such that p(x)=[(x-a)^k]r(x), r is a polynomial and r(a) not equal to 0. Prove a is a root of order k of the polynomial p iff p(a)=P'(a)=...=[p^(k-1)](a)=0 and [p^(k)](a) not equal to 0. Note: [p^(k-1)](a) :=...

Let V =Mn(k),n>1 and T:V→V defined by T(M)=Mt (transpose of M). i) Find the minimal polynomial of T. Is T diagonalisable when k = R,C,F2? ii) Suppose k = R. Find the characteristic polynomial chT . I know that T2=T(Mt))=M and that has got to help me find the minimal polynomial

Power Series Representation of a Function when "r" is a polynomial Homework Statement Find a power series representation for the function and determine the radius of convergence. f(x)=\stackrel{(1+x)}{(1-x)^{2}} Homework Equations a series converges when |x|<1...

Let f(x)=\frac{1}{x^2+x+1} Let f(x)=\sum_{n=0}^{\infty}c_nx^n be the Maclaurin series representation for f(x). Find the value of c_{36}-c_{37}+c_{38}. After working out the fraction, I arrived at the following, f(x)=\sum_{n=0}^{\infty}x^{3n}-\sum_{n=0}^{\infty}x^{3n+1} But I dun get how to...

What's the difference between polynomials (as elements of a ring of polynomials) and polynomial functions??

Homework Statement Find the Taylor Polynomial of degree 3 for the function f(x) = ln3x about a = 1/3Homework Equations NoneThe Attempt at a Solution I have found up to the fourth derivative of f(x) along with the values of the derivatives at x = 1/3. At this point i get Σ{(-1)kk!fk(1/3)}, but...

Hi there, I have over the last couple of month worked my way through Algebra Demystified and College Algebra Demystified (well almost), and I just completed the chapter on polynomial functions and I’m stuck at one of the questions in the chapter review where I’m given a complex root and the...

Homework Statement x^4 + 4x^3 - x^2 + 16x - 12 I know that with some higher order polynomials you can substitute say x^4 as a = x^2 thereby making it easier to break the thing apart and find its factors. I know I am looking for 4 roots, but my little substitution method doesn't really work...

f(x) = (x^2)/100 + (x^3)/1000 + (x^4)/10000 + ... till the power infinity

I have to find the complex zeros of the following polynomial: x^6+x^4+x^3+x^2+1 This evidently doesn't have any real solution so I tried to facto it with long division and by guessing I came up with: (x^2-x+1)(x^4+x^3+x^2+x+1) How can I factor the 4th degree polynomial now? Or how can I...

the textbook says that: "a non-constant analytic polynomial cannot be real-valued, for then both the partial derivative with respect to x and y would be real and the cauchyriemann equation cannot be satisfied." why??there's no explanation in the book and this sentence is written as an example...

Homework Statement Show that there is a polynomial function f of degree n such that: 1. f('x) = 0 for precisely n-1 numbers x 2. f'(x) = 0 for no x, if n is odd 3. f'(x) = 0 for exactly one x, if n is even 4. f'(x) = 0 for exactly k numbers, if n-k is odd Homework Equations The...

Homework Statement It seems that I'm a little bit lost about this exercise. It says: Find the taylors polynomial of third degree centered at the origin for z=\cos y \sin x. Estimate the error for: \Delta x=-0.15,\Delta y=0.2. So, I did the first part (the easy one), the taylors polynomial for...

Homework Statement If f(x)=(x+1)p(x) where f(x)=x^{2n}+2nx+2n-1, what is p(x)? Answer given: x^{2n-1}-x^{2n-2}+...-x^2+x+2n-1 Homework Equations The Attempt at a Solution I tried the long division and managed to some terms correct. Is there any other methods of finding this...

Homework Statement Find the inverse of the function. f(x)=2x^{3}+5Homework Equations Possibly the quadratic equation.The Attempt at a Solutionf(x)=2x^{3}+5 y=2x^{3}+5 -2x^{3}=-y+5 x^{3}= \frac{-y+5}{-2} x= \pm\sqrt[3]{\frac{-y+5}{-2}} y= \pm\sqrt[3]{\frac{-x+5}{-2}}So the solution is two...

Factor a 4th order polynomial (Solved) Homework Statement Find the roots of: x^5-1=0 Homework Equations Polynomial long division. The Attempt at a Solution x^5-1 = (x-1)(x^4+x^3+x^2+x+1) = 0 x^4+x^3+x^2+x+1 = (x^2+1)^2+x^3+x-x^2 (x^2+1)^2+x^3+x-x^2 = (x^2+1)^2+x(x^2+1)-x^2...

Here is a potentially neat problem. Let x(t),y(t) (for all t\in \mathbb{R}) be polynomials in t. Prove that for any x(t),y(t) there exists a non-zero polynomial f(x,y) in 2 variables such that f(x(t),y(t))=0 for all t. The strategy is to show that for n sufficiently large, the polynomials...

Are lx3l or lxl3 polynomials? If not, what would be a good example of a cubic polynomial function (R \rightarrow R) that doesn't cover all real numbers in its codomain?

I can prove that data follows a curve of the form y=Ax^n and y=Ae^x by using log log and natural log transformations. I have some data that I believe is more complex, something of the form y=anx^n+an-1x^n-1+...+a1x+a0, in other words a polynomial function. Is there any way I can prove that it...

Hi everybody f(x) = aX0 is the form of any constant polynomial... right?? eg: f(x) = 3 is actually f(x) =3X0 where X belongs to R... ok?? since 00 is an unspecified quantity.. therefore on graphing a constant.. it should exists a hole on y-axis... and the y-intercept should not satisfy...

Homework Statement Let P_4(\mathbb{R}) be the vector space of real polynomials of degree less than or equal to 4. Show that {{f \in P_4(\mathbb{R}):f(0)=f(1)=0}} defines a subspace of V, and find a basis for this subspace. The Attempt at a Solution Since P_4(\mathbb{R}) is...

Hello everyone. I was going through my Linear Algebra Done Right textbook that threw me off. I hope this forum is appropriate for my inquiry; while there is no problem I'm trying to solve here, I don't know whether just asking for clarification would belong to the homework forum instead. If...

Hi all, I've been set some holiday work by my study director which is meant to be teaching us all about algorithms and a few other mathematical bits and bobs - unfortunately I've come unstuck on one of the bobs, and was hoping for some help! I've asked for help elsewhere but was given very...

Homework Statement Let f: C -> C be a holomorphic function such that there is a constant R such that |z| > R implies |f(z)| > R. Show that f is a polynomial. Homework Equations Not sure, I pulled this randomly from a complex analysis qualifying exam. The Attempt at a Solution So...

Homework Statement This is probably an easy question, but using the rational zero theorem I have not found any roots for this cubic polynomial. Factor the Following 6x^3-37x^2-8x+12 Homework Equations The Attempt at a Solution I have used all my knowledge of factoring and...

Homework Statement find a polynomial P(x) which has nonnegative coefficients. If P(1)=1 and P(5)= 426, then wast is p(3) Homework Equations P(1)= 6 P(5)= 426 P(3)= xThe Attempt at a Solution I have tried to use guess and check. I can't find a way to solve algebraically.

Homework Statement Let X be the vector space of polynomial of order less than or equal to M a) Show that the set B={1,x,...,x^M} is a basis vector b) Consider the mapping T from X to X defined as: f(x)= Tg(x) = d/dx g(x) i) Show T is linear ii) derive a matrix...

Homework Statement Evaluate the integral when x > 0: indefinite integral of ln(x2+19x+84)dxHomework Equations I know I need to use some form of integration by parts: integral of u*dv=uv-(integral of(du*v)) The Attempt at a Solution I began by making u=ln(x2+19x+84) and dv=dx. Thus, (after...

Homework Statement Show that a root of the equation x3 - 3x - 5 =0 lies in the interval [2,3], and then find the root using linear interpolation correct to one decimal place. Homework Equations n/a The Attempt at a Solution This is my first ever time using interpolants ( well at...

Homework Statement For I = [a,b], define: P3(I) = {v: v is a polynomial of degree ≤ 3 on I, i.e., v has the form v(x) = a3x3 + a2x2 + a1x + a0}. How to show v is uniquely determined by v(a), v'(a), v(b), v'(b). Homework Equations The Attempt at a Solution I'm not exactly sure...

The polynomial is x^n + A1x^(n-1) + A2x^(n-2) + ... + A2x^2 + A1x + 1. Where An (integer) is not zero for all n and n is even. For example: x²+x+1; x^4+2x^3+3x^2+2x+1. I'm looking for a method to say if that kind of polynomial is irreducible over racionals... Or when it is. Thx!

Hello, I'm trying to find the inverse polynomial of y = ax^2 + bx + c in the form of x = dy^2 + ey + f. I'm able to approximate this using Excel, but would prefer a more elegant solution. Any suggestions? Steve

Homework Statement Factor the equation (without complex numbers) a^{10} + a^{5} + 1 This is a olympiad question The Attempt at a Solution I substituted a^{5} = x getting a quadratic eqation. But when I factored the quadratic equation I get complex roots and this is against the question...

Hi can anyone help me please or give me a strong hint? I have to prove this: If f(x) is a function from natural to natural Numbers and f(f(f(x))) is stricly increasing a polynomial than f(x) is also.

Homework Statement Well, this problem is quiet similar to the one I've posted before. It asks me to approximate to the e number using taylors polynomial, but in this case tells me that the error must be shorter than 0.0005 Homework Equations...

Homework Statement Solve the following equation: z^4+z^3+z^2+z+1 = 0 z is a complex number. 2. The attempt at a solution I was trying to factorize it to 1st degree polynomial multiplied by 3rd degree polynomial: (z+a)(z^3+bz^2+cz+1/a) = 0 I discovered that I need to solve 3rd...

Homework Statement I am trying to get the roots of: (x-a)^3 - (x-a)b^2 - 2b^3 -2b^2(x-a) which I know they are x = a-b and x = a+2 b the problem is, how can I reach that solution ? The Attempt at a Solution At first I thought of separating the independent term (the one...

Homework Statement Prove that if a0+a1x+a2x2+a3x3+...+anxn=0 then a0=0, a1=0 ... an=0 Homework Equations none The Attempt at a Solution I think I can do this for n up to 2 in the following way (please tell me if you see any gaps in my logic here): f(x)=a0+a1x+a2x2=0 (from the...

Homework Statement Turn y = y(0) * e^(-kt) into a polynomial. Homework Equations The Attempt at a Solution I have no idea of how I would go about doing this. I know you can use taylor series to approximate it, but is there any other way? Thanks, Darthxepher

Homework Statement If we divide f(x)= x^3+qx^2-x-2 by x+1, we get the same remainder as if we divide it by x-2. Determine the value of q Homework Equations f(x)= x^3+qx^2-x-2 The Attempt at a Solution I tried to plug in f(-1) into the equation, and then f(2) into the equation...

Hello, (quick backgroun info) : I am a physics student who has gone through pre quantum type material and a little of quantum mechanics. I am working in a lab with fortan code based on Quantum field theory. Anyway I am working to change some pieces of this code to attempt to solve a...

Homework Statement Find the indicated value of the polynomial using the Remainder Theorem p(x)=2x^3-2x^2+11x-100; find p(3) Homework Equations p(x)=2x^3-2x^2+11x-100 The Attempt at a Solution Synthetic division 3] 2 -2 11 -100 6 12 69 2 4 23 [-31 answer: p(3)=-31 im not...

1. Homework Statement [/b] Let V be the space of all polynomials of degree less than or equal to 2 over the reals. Define the transformation, H, as a mapping from V to R[x] by (Hp)(x)=\int^x_{-1}p(t)dt\\. a) Find the monic generator, d, which generates the ideal, M, containing the range of H...

Hello I am appealing to the computer savvy bunch here and asking for a graph. I need the graph to show a function that is of the form (Acos(x)+Bsin(X)) it can be simple... just anything harmonic I dont' even need the function you can just draw it. and then I need a polynomial that has the...

What is the most general solution to an equation of the form: a_1 p_1 + \ldots + a_n p_n =0 where p_i are given polynomials in several (N) variables with no common factor (i.e. their GCD is 1) and a_n are the polynomials we are looking for (again in the same N variables). Of course, I'm asking...

Homework Statement Use the subspace theorem to decide which of the following are real vector spaces with the usual operations. a) The set of all real polynomials of any degree. b) The set of real polynomials of degree \leq n c) The set of real polynomails of degree exactly n...

Does anyone know how to prove the following identity: \Sigma_{k=0}^{n}\left(\stackrel{n}{k}\right) H_{k}(x)H_{n-k}(y)=2^{n/2}H_{n}(2^{-1/2}(x+y)) where H_{i}(z)represents the Hermite polynomial?

Hi Could someone see if I have done the following limit right please? By the way, where is the limit symbol in the latex reference as I couldn't find it :( Anyway the limit is as x-> infinity (I won't keep writing that out) of \frac{-x-1/2}{2x^4}

I'm revising complex numbers and having trouble with this question... Question: Verify that 2 of the roots of the equation: z^3 +2z^2 + z + 2 = 0 are i and -2. Find any remaining roots Attempt at solution: i^3 +2 i^2 + i + 2 = (-1)i + 2(-1) +i + 2 = -i -2 + i +2 =0...