Polynomial Definition and 1000 Threads

Calculate MacLaurin-polynom of grade 3 to function $$\cos(\ln(1+2x-3x^2))$$if i make Taylor expansion in that ln first is this correct $$\ln(1+2x-3x^2)=2x-3x^2-\frac{(2x-3x^2)^2}{2}+\frac{(2x-3x^2)^3}{3}...$$ Is that correct? Regards, $$|\pi\rangle$$

A little explanation here. My professor assigned a homework question without attempting the problem herself. When we were assigned this problem, we were forbidden to use the notion of a Taylor series in our proof (at least not without proving Taylor's Theorem on our own) as we had not covered...

What are some examples of functions such that f(nx) = a_{k}f(x)^{k}+...+a_{1}f(x)+a_{0} for some integers n, k, and integer coefficients in the polynomial? The only example I can think of is cos(x), for which \cos(2x) = 2\cos(x)^{2}-1 and there are similar relations for n = 3, 4...

Find real solution(s) to the equation $$(x^2-9x-1)^{10}+99x^{10}=10x^9(x^2-1)$$

I am trying to solve an equation: arccos(Y) = arctan(Y), where Y = 1/x This turns into a quartic equation: x4 - x2 - 1 = 0 It looks simple enough to simplify further; however, I must be having a brain-fart because I can't do it. I'd like to avoid resorting to a numerical solution...

Hello Firstly apologies for what seems like an extremely fundamental question, it's been a while since I've done any calculus! I'm currently using a program to fit data with a two dimensional 3 degree polynomial curve( which outputs the fit in the following format) with the aim of...

Hello MHB, Find all roots to $$z^6-2z^3+2=0$$ I can se we there will be 6 roots. I start with subsitute $$z^3=t$$ so we got $$t^2-2t+2=0$$ and we get $$t_1=1+i$$ and $$t_2=1-i$$ what shall I do next? Shall I go to polar form? Regards,

Assume that \sqrt[M]{N} is irrational where N,M are positive integers. I'm under belief that X^M-N is the minimal polynomial of \sqrt[M]{N} in \mathbb{Q}[X], but I cannot figure out the proof. Assume as an antithesis that p(X)\in\mathbb{Q}[X] is the minimal polynomial such that \partial p <...

This claim is supposed to be true. Assume that p\in\mathbb{F}[X] is an irreducible polynomial over a field \mathbb{F}\subset\mathbb{C}. Also assume that p(X)=(X-z_1)\cdots (X-z_N) holds with some z_1,\ldots, z_N\in\mathbb{C}. Now all z_1,\ldots, z_N are distinct. Why is this claim true...

Homework Statement Here's a screenshot of the problem: http://puu.sh/2Bta5 Homework Equations The Attempt at a Solution As can be seen by the screenshot, the answer's already given, but I'm not sure how to go about getting it. This one has me stumped since e-4x and sin(5x) are...

Homework Statement y(6) - 3y(4) + 3y''-y = 0 Homework Equations The Attempt at a Solution The characteristic equation of that differential equation is: r^6 - 3r^4 + 3r^2 - r = 0 But how am I expected to solve such a high degree polynomial (and thus the DE?)

Homework Statement Given field extension C of Q, Find the minimal polynomial of a=sqrt( 5 + sqrt(23) ) (element of C).Homework Equations The Attempt at a Solution I may be complicating things, but let me know if you see something missing. Doing the appropriate algebra, I manipulated the above...

Homework Statement a. Compute the second order Taylor polynomial centered at 2, P2(x), for the function ln(x). b. Estimate the maximum error of the answer to part a for x in the interval [1,2].Homework EquationsThe Attempt at a Solution For part a, I'm thinking that when it says "second...

Homework Statement Let V:= ℝ_{2}[t] V \in f: v \mapsto f(v) \in V, \forall v \in V (f(v))(t) := v(2-t) a) Check that f \in End(V) b) Calculate the characteristic polynomial of f. Homework Equations The Attempt at a Solution a) Is it sufficient to check that (f+g)(t)=f(t)+g(t) ...

I am trying to understand the proof of Gauss's Lemma as given in Dummit and Foote Section 9.3 pages 303-304 (see attached) On page 304, part way through the proof, D&F write: "Assume d is not a unit (in R) and write d as a product of irreducibles in R, say d = p_1p_2 ... p_n . Since p_1...

Homework Statement Show that the polynomial function: P(x)=x6+2x4+3x2+4 has six nonreal zeros Homework Equations -none- The Attempt at a Solution I tried using synthetic division with all the possible values it could have(p/q) but none of them worked. I was just wondering...

I would like help to get started on the following problem: Determine all the ideals of the ring $$ \mathbb{Z}[x]/<2, x^3 + 1> $$ Appreciate some guidance. Peter

A polynomial of degree ≤ 2 ? what does this mean. Would it just be a + bt + c t^2 = f(t) Or at^2 + bt + c = f(t) Is there even a difference between the two equations considering the fact that a,b, and c are unknown?

I am trying to get a good understanding of the structure of the rings \mathbb{Z}[x]/<x^2> and \mathbb{Z}[x]/<x^2 +1> . I tried to first deal with the rings \mathbb{R}[x]/<x^2> and \mathbb{R}[x]/<x^2 +1> as they seemed easier to deal with ... my thinking ... and my problems are as...

Here is the question: Here is a link to the question: Let P(x) = x4 + ax3 + bx2 + cx + d. Need to find the values of the variable that satisfy the guidelines below? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

I am reading Dummit and Foote Section 9.2: Polynomial Rings Over Fields I I am having some trouble understanding Example 3 on page 300 (see attached) My problem is mainly with understanding the notation and terminology. The start of Example 3 reads as follows. "If p is a prime, the ring...

Here is the question: Here is a link to the question: The remainder of f(x)/(x^2+x+1) and f(x)/[(x+1)^2] are x+5 and x-1 respectively.? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Folks, I have attached a picture illustrating the labelling of the linear reactangle element which can be represented by the following equation ##u(x,y)=c_1+c_2 x +c_3 y +c_4 xy## (1) ##u_1=u(0,0)=c_1## ##u_2=u(a,0)=c_1+c_2a## ##u_3=u(a,b)=c_1+c_2a+c_3b+c_4ab##...

Let V be the linear space of all real polynomials p(x) of degree < n. If p ∈ V, define q = T(p) to mean that q(t) = p(t + 1) for all real t. Prove that T has only the eigenvalue 1. What are the eigenfunctions belonging to this eigenvalue? What I did was T(p)= (lamda) p = q (Lamda) p(t+1) =...

Hey, This isn't really a homework question, per se, as I am relearning some pre calculus for kicks. But I figured his would be the place to ask. For whatever reason, this very simply factoring issue has got my head spinning. I'm not exactly sure what I am doing wrong to factor this equation...

Hello, for homework I was given $$x^{3}-27$$ to factor and my answer can be written this way: $$(x - A)(x^{2} + Bx + C)$$ I am being asked to find the value of A, B and C. I am sure this is simpler than I think but it's perhaps the Bx that is confusing me as I haven't practiced a lot of these...

Here is the question: Here is a link to the question: Help with math question please!? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Could anyone help me out with this: Which of the following statements are true and which are false? Justify your answers. iii) There exists a polynomial P such that |P(x) - \cos(x)| \leq 10^{-6} I've tried to thinking about it, and it seems as though it is false, because |cos(x)|...

Found this on a test for an integrated algebra 2 high school math class! Factor completely. 6x3 - 3x2 + 12 The Attempt at a Solution 3( 2x3 - x2 + 4) eq.1 At this point I checked for rational roots using the rational roots theorem and synthetically dividing. I got nothing...

Hi, I need to rearrange an equation: y = ax^2 + bx + c to the form of: x = ? I'm not entirely sure how to go about this and the examples I've found require the equation to be in a different form. Any tips or a point in the right direction would be great! Thanks in advance

hi i need some help solving this equation for x y=0.0001x^3-0.0409x^2+4.6716x+280.32 can someone please help me, i really need to solve this for x! full layout would be great many thanks!

Homework Statement Consider the followign function f(x) = x^-5 a=1 n=2 0.8 \leq x \leq 1.2 a) Approximate f with a tayloy polynomial of nth degree at the number a = 1 b) use taylor's inequality to estimate the accuracy of approximation f(x) ≈ T_{n}(x) when x lies in the interval...

Homework Statement The energy density of electromagnetic radiation at wavelength λ from a black body at temperature T (degrees Kelvin) is given by Planck's law of black body radiation: f(λ) = \frac{8πhc}{λ^{5}(e^{hc/λkT} - 1)} where h is Planck's constant, c is the speed of light, and...

How to "Offset" a polynomial Suppose I have a function for a curve, for example y=x2. I want to find a function to "offsets" it by 2 units. That is, I want a larger parabola that is exactly 2 units away from my original parabola. What I have in mind is the offset command in AutoCAD. Is...

I think I solved it a week ago, but I didn't write down all the things and I want to be sure of doing the things right, plus the excersise of writing it here in latex helps me a loot (I wrote about 3 threads and didn't submited it because writing it here clarified me enough to find the answer...

Hi all, I have been stopped by a sextic (6th degree) polynomial in my research. I need to find the biggest positive root for this polynomial symbolically, and since its impassible in general, I came up with this idea, maybe there is a way to approximate this polynomial by a lower degree...

Homework Statement Find the Taylor polynomial approximation about the point ε = 1/2 for the following function: (x^1/2)(e^-x)The Attempt at a Solution I'm trying to get a taylor polynomial up to the second derivate i.e.: P2(×) = (×^1/2)(e^-x) + (x-ε) * [(e^-x)(1-2×)/2(×^1/2)] +...

Here is the question: Here is a link to the question: Write a function for the polynomial that fits the following description.? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Recently I am studying about theorems regarding to polynomial equations and encounter the lower and upper bounds theorem. Which states that if a<0 and P(a) not equals 0, and dividing P(x) by (x-a) leads to coefficients that alternate signs, then a is a lower bound of all the roots of P(x)=0. The...

Here is the question: Here is a link to the question: Math help: factoring? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

The question is: Using the chain rule to prove that a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ which is polynomial in the coordinates is differentiable everywhere. (The chain rule is for the use under function composition circumstances, how to apply it here to prove that the function $f$...

This is not actually a homework question, but it seemed appropriate to put it here. In an old exam from 1921 I found the following problem. I never learned how to solve this type of thing and I haven't been able to figure it out, so: how does one solve this? Homework Statement Solve for...

Homework Statement obtain the number r = √15 -3 as an approximation to the nonzero root of the equation x^2 = sinx by using the cubic Taylor polynomial approximation to sinxHomework Equations cubic taylor polynomial of sinx = x- x^3/3!The Attempt at a Solution Sinx = x-x^3/3! + E(x) x^2 =...

the error of a taylor series of order(I think that's the right word) n is given by \frac{f^{n+1} (s)}{n!} (x-a)^n I think this is right. The error in a linear approximation would simply be \frac{f''(s)}{2} (x-a)^2 My question is what is s and how do I find it. Use linear...

So I'm in a college algebra class and I know how to do polynomial long division. I'm curious as to why polynomial long division works. I've looked at some proofs, but they use scary symbols that I don't understand (I am quite dumb). Do I need very high-level math to comprehend why polynomial...

See attached PDF for details: How do I calculate errors on the fit parameters, p?

Homework Statement Give an example of a cubic polynomial, defined on the open interval (-1,4), which reaches both its maximum and minimum values. Homework Equations - The Attempt at a Solution I can see that I would need a function such that there is some f(a) and f(b) in...

Homework Statement Let T:P_m(\mathbb{F}) \mapsto P_{m+2}(\mathbb{F}) such that Tp(z)=z^2 p(z) . Would a suitable basis for range T be (z^2, \dots, z^{m+2}) ?

Let A ∈ Mn×n(F ) Why dim span(In, A, A2, A3, . . .) = deg(mA)?? where mA is the minimal polynomial of A. For span (In,A,A2...) I can prove its dimension <= n by CH Theorem but what's the relation between dim span(In,A,A2...)and deg(mA)

I encounter a problem on the fitting ability of a special class of multi-variable polynomials. To be specific, I need find whether a special class of multi-variable polynomials, denoted by p(m), where m is the number of variables, can universally and exactly fit all member in another special...