Polynomial Definition and 1000 Threads

Hi, To calculate the intersection of two straight lines the cross product of the line vectors can be used, i.e. when the lines start in points p and q, and have direction vectors r and s, then if the cross product r x s is nonzero, the intersection point is q+us, and can be found from...

Homework Statement Let t_j=j/100, a_j=j, b_j=-j, for j=0,1,...,99. Define f(t)=\sum\limits_{k=0}^{99} (a_k\cos(2\pi kt)+b_k\sin(2\pi kt)) Determine the values of c_l, d_m for l= 0,...5, m=1,...,4, so that P(t)=\frac{c_0}{2}+\sum\limits_{k=1}^4 (c_k\cos(2\pi kt)+d_k\sin(2\pi kt))+c_5\cos(10\pi...

Homework Statement . Let ##f:\mathbb R^2 \to \mathbb R## / ##f \in C^2##, ##f(0,0)=0## and ##\nabla f(0,0)=(0,1)## ##Df(0,0)=\begin{pmatrix} 1 & 1\\ 1 & 2\\ \end{pmatrix}## Let ##g:\mathbb...

Homework Statement So, we are supposed to find the tenth taylor polynomial of sinx. On wolfram, I get a final term with x^11 at the end. How does that make sense?! According to the formula, n=10 so the maximum degree should be 10... Homework Equations Taylor polynomial formula. The...

Homework Statement Find the number of zeroes of p(z) = z^5 + 10z - 1 inside |z| < 1 Homework Equations The Attempt at a Solution Let f(z) = z^5 g(z) = 10z-1 On |z| = 1: 10|z| - 1< |10z-1| < 10|z| + 1 (is this true...?) 9 < |g(z)| < 11 |f(z)| = 1 So |g| >...

I am reading R.Y. Sharp: Steps in Commutative Algebra. Lemma 1.13 on page 7 (see attachment) reads as follows: -------------------------------------------------------------------------------------- 1.13 LEMMA. let R be a commutative ring, and let X be an indeterminate; let T be a commutative...

Homework Statement S = {1, x, x^2} Find ||1||, ||x||, and ||x^2||.Homework Equations ##\sqrt{v \cdot v}##The Attempt at a Solution I don't know the components of each vector, so how can I perform the dot product?

Hello! Tchebysheff polynomials are often defined with trigonometric functions: T_m (x) = \begin{cases} \cos(m \arccos (x)) & -1 \le x \le 1\\ \mathrm{cosh} (m \mathrm{arccosh} (x)) & x > 1\\(-1)^m \mathrm{cosh} (m \mathrm{arccosh} |x|) & x < 1 \end{cases} But they are also polynomials, and...

How would I go about solving an Nth degree polynomial function such that N>=5? Ax^{n}+Bx^{n-1}+...+Z = 0

I have given polynomial: $$x^3-8x^2+19x-12$$ I know how to find the roots with Horners method,i am just wondering if there is an easier and quicker way to find them? Thank you!

Dear! Is possible to solution a polynomial of kind y = Ax^a + Bx^b ? Thx!

I think the solution to the radial schrodinger equation includes a form of the Laguerre polynomials, the polynomial v(ρ). Does anyone know what this v(ρ) polynomial is called? The only information my book gives is: "The polynomial v(ρ) is a function well known to applied mathematicians."...

Hi guys! I'm kind of stuck in my review here in Quantitave on the Polynomial part :confused: Homework Statement A. The first problem I had is Problem 7 on the link I will provide which has this. (x + y)3 + (x-y)3 = ? B. The second problem I had is Problem 8 on the same link I will...

I need help understanding why the ln (x) taylor polynomial is (x-1)-1/2(x-1)^2... + etc. I cannot grasp the concept..

I wonder what are the tecniques,or what is the easiest way to simplify given polynomial: $$x^3-9x^2+27x-27$$ If possible,without Horners algorithm. Thank you!

Recently, I've developed a habit of trying to separate the idea of a function from the idea of the image of the function. This has mostly just confused me, but I am adamant about sticking to it. I think the two terms, "ring of polynomials" and "ring of polynomial functions," are not...

why does a nth degree polynomial has atleast one root and a maximum of n root...? In my book it's given, it's the fundamental theorem of algebra. Is there a proof...? Thank's for help. (In advance)

Hi, I'm looking for a program that will take [1 + (1+p)]*p and return the unchanged polynomial. i.e. 2p + p^2 I know that MATLAB allows u to do polynomial convolution but for my purposes a program that returns the polynomial in the above format will be much much easier to work with...

$\displaystyle x^4+2x^3-8x^2+18x+9$ this is what i tried, $x^4+2x^3-8x^2 = (x^2+4x)(x^2-2x)$ then, $a(x^2+4x)+b(x^2-2x)=18x$ where ab=9 did i set up my solution correctly? can you tell me where I'm wrong.

The roots of $x^4-x^3-x^2-1=0$ are $p, q, r, s$. Find $f(p)+f(q)+f(r)+f(s)$, where $f(x)=x^6-x^5-x^3-x^2-x$.

Given the following expression \[ \mathcal{P}_{n}(0) = \left.\frac{1}{2^{n}n!}\frac{d^{n}}{dx^{n}} \sum_{k = 0}^{n}\binom{n}{k}(x^2)^k(-1)^{n - k}\right|_{x = 0}, \qquad (*)...

Homework Statement Show that if ##f:\mathbb{R}\to \mathbb{R}## is a polynomial function of odd degree, then ##f(\mathbb{R}) = \mathbb{R}##. The attempt at a solution How can I rigorously prove this? What is the most direct and concise way to prove this? What I have is the following...

Given $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$, where $a_0, a_a,\cdots,a_n$ are all smaller than 4 and are integer, $a_n \in (0, 1, 2,\cdots)$. Given that $f(4)=2009$, find $f(1)$.

Homework Statement Homework Equations The Attempt at a Solution How did they go from the first step in the blue to the second step in the blue? I tried integration by parts but that didn't work.

I have to find all solutions for X when: x4-2x3-25x2+50x I have done it so,but I am not sure if this is ok: x(x3-2x2-25x+50) = x(x2(x-2)-25(x+2) = x(x2-25)(x-2) =x(x-5)(x+5)(x-2) Now i see that root/zeroes are +5,-5 and 2. I know that this polynomial has another zero that is 0,but how do i...

Homework Statement A car is moving with speed 10m/s and acceleration 1 m/s^2 at the given instant. Using a second-degree Taylor polynomial, estimate how far the car moves in the next second. Homework Equations The Attempt at a Solution I don't get how you're supposed to apply...

The no. of real solution of the equation $\displaystyle 1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!} = 0$

Hello guys, I'd like to ask you how to efficiently factorize complicated polynomial like this one for example: $$\frac{128t^4+128t^3+192t^2+32t+40}{(4t^2+1)^4}$$ I've spend more than a hour trying to decrypt and decompose the polynomial, but to no avail. For simple cubic polynomial I know...

Here is the question: I have posted a link there to this topic so the OP can see my work.

Hi guys, somehow after a couple years of not doing math I got a bit rusty... How do I solve x^3+3x^2-4=0 ? I'm kinda stuck? I figured factorizing but I can't seem to find any good factors :/

Homework Statement Was given a matrix To find the eigenvalues I set up the characteristic equation [-1-x | 7 | -5 ] [-4 | 11-x | -6 ] [-4 | 8 | -3-x] With some dirty work I got this bad boy out, which I'm having trouble factoring -x3+7x2-15x+9Homework Equations...

Hi everyone, :) This is one of the thoughts that I got after thinking about finding the minimal polynomial of a matrix. I know that the minimal polynomial is easy to find when a matrix is diagonalizable. Then the minimal polynomial only consist all the distinct linear factors of the...

any hints on how to start this problem? $12x^4+19x^3-26x^2-61x-28$

Hi everyone, :) Here's another question that I solved. Let me know if you see any mistakes or if you have any other comments. Thanks very much. :) Problem: Prove that the eigenvector \(v\) of \(f:V\rightarrow V\) over a field \(F\), with eigenvalue \(\lambda\), is an eigenvector of \(P(f)\)...

how would i start factoring this $m^8-n^8-2m^6n^2+2n^6m^2$

this stunned me. i stare at these problem for a while and tried my best to factor it but no success. please help me solve this. 1. $x^3+x^2y^2+x^2+xy+y^3+4y-3xy^2-12x$ 2. $3x^2+7xy-3xz-2yz+4y^2-6z^2$

can you guys help me factor this polynomial. $\displaystyle 2x^2-4xy+2y^2+5x-3-5y$ $6x^2-xy+23x-2y^2-6y+20$ by the way this is what i tried in prob 1 $2(x-y)^2+5(x-y)-3$ -->> I'm stuck here. and in prob 2 i have no idea where and how to start. thanks!

If I could use any polynomial up to degree ∞, then can I get a close fit to any continuous function? I know that with a 4th degree polynomial you can get a pretty close fit to the sine function between 0 and 2pi...

Homework Statement Given the matrix 2 0 0 0 0 0 0 1 2 0 0 0 0 0 0 1 2 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 2 What is the minimal polynomial? Homework Equations - The Attempt at a Solution This is the Jordan form, so I guess the solution is just...

In order to challenge a broader section of the forum, there is a part a and a part b to this challenge - if you feel that part b is an appropriate challenge, then I request you do not post a solution to part a as part a is a strictly easier question than part b. The new challenge: Are there any...

In order to challenge a broader section of the forum, there is a part a and a part b to this challenge - if you feel that part b is an appropriate challenge, then I request you do not post a solution to part a as part a is a strictly easier question than part b. The challenge: Prove that the...

Homework Statement Determine the least possible degree of the function corresponding to the graph shown below. Justify your answer. Homework Equations The graph is attached. I remade the graph using google grapher, but the graph I got in the test have exactly the same x-intercepts (-2 of order...

so I was reading my textbook and was showing steps on applying the quotient rule to the function: y=ex/(1+x2) it went from (1+x2)(ex)-(ex)(2x)/(1+x2)2 to ex(1-x)2/(1+x2)2 I understand the first step, but don't get how they got to ex(1-x)2 in the numerator. can someone please explain the...

Not sure how this can be done. can anyone help?

Homework Statement 1. Let ##p(x) = a_{0} x^{n} + a_{1} x^{n−1} + ... + a_{n} , a_{0} \neq 0 ##be an univariate polynomial of degree n. Let r be its root, i.e. p(r) = 0. Prove that ## |r| \leq max(1, \Sigma_{1 \leq i \leq n} | \dfrac{a_{i} }{ a_{0} } | )## Is it always true that? ## |r| \leq...

Homework Statement Find a specific number δ>0 such that if x2 + y2 = δ2, then |x2+y2+3xy+180xy5 < 1/10 000. Answer: Choose δ < 0.002 Homework Equations ε-δ def'n of limit: lim (x,y) → (a, b) f(x) = L if for every ε > 0 there exists a δ > 0 such that 0 < √(x-a)2+(y-b)2, |f(x) - L| < ε...

In Dummit and Foote Chapter 13: Field Theory, the authors give several examples of field extensions on page 515 - see attached. In example (3) we read (see attached) " (3) Take F = \mathbb{Q} and p(x) = x^2 - 2 , irreducible over \mathbb{Q} by Eisenstein's Criterion, for example" Now...

Hopefully, anyone who has studied polynomial long division understands the link between it and regular long division. If you divide $58$ into $302985$, you could follow the usual long division procedure and obtain the answer. Alternatively, if you divide $x+4$ into $x^{3}+2x^{2}+6x+7$, you could...

Hello Everyone, Actually i would like to have a clear Explanation about Runge Phenomenon. Could you help me Please.

Homework Statement You have a fixed payment loan, you know the quantity you need to pay every year, the years until maturity and (I suppose to) the loan value and you need to calculate which is the yield to maturity. Homework Equations Loan Value =Fp/(1+i) + Fp/(1+i)2+...+ Fp/(1+i)n...