Polynomial Definition and 1000 Threads

Hello everyone, I'm new to this forum. I have this Linear Algebra question that I have no clue how to solve. Any help would be much appreciated. :) The question goes as follows: The polynomial p(x) = x3 + kx + (3 - 2i) where k is an unknown complex number. It is given to you that if p(x) is...

Hello! :) The interpolating polynomial that interpolates at the following data: $f(5)=?,f(8)=14,f(12)=214$ is $4,125x^{2}-32,5x+10$. The corresponding interpolating polynomial in the Newton form is $p_{2}(x)=a_{0}+a_{1}(x-5)+a_{2}(x-5)(x-3)$.Which is the value of $a_{2}$? Is it 4,125 because...

Prove that if x and y are not both , then x^4+x^3y+x^2y^2+xy^3+y^4 > 0 I have no idea how to start this proof, can anyone give me an idea?

Hi! Say that we wish to approximate a function f(x), \, x\in [0, 2\pi] by a trigonometric polynomial such that f(x) \approx \sum_{|n|\leq N} a_n e^{inx} \qquad (1) The best approximation theorem says that in a function space equipped with the inner product (f,g) = \frac{1}{2...

Homework Statement M: V -> V linear operator st M^2 + 1_v = 0 find the POSSIBILITIES for min. pol. of M^3+2M^2+M+3I_v Homework Equations The Attempt at a Solution using M^2 = -1_v, i rewrote the operator(?) as M^3 + M + I_v i don't know what to do. i guessed min poly to...

i understand how to find minimal poly. if a matrix is given. i am curious if you can find the matrix representation if minimal polynomial is given. i'm not exactly sure how you could since you can possibly lose repeated e-values when you write minimal polynomial. how can u create a n...

I'm doing a calculation which finds the characteristic polynomial of a matrix, HH, with rather complex entries and then determines the discriminant of that polynomial. For smaller matrices up to around 7x7 it finishes evaluating the Discriminant command within a few hours, but at a 10x10, which...

hi i have this homework question and I am not sure if my thought process is valid. The Question: let a, b and c be roots of the polynomial equation: x^3+px+q=0 and S(n)=(a^n)+(b^n)+(c^n) now prove: that for S(n)= -p(S(n-2))-q(S(n-3)) for n>3my attempt: ------------- first off...

Homework Statement z^3 + (-5+2i)z^2 + (11-5i)z -10+2i =0 has a real root, find all the solutions to this equation. The Attempt at a Solution I have only solved imaginary number polynomials with a given root, but this has no given root, how do I find the real solution? that I can then...

The polynomial z^4 + 2z^3 + 9z^2 - 52z + 200 = 0 has a root z=-3+4i. Find the other 3 roots. Since the given root is complex, one of the other roots must be the complex conjugate of the given root. So the 2nd root is z=-3-4i. To find the other roots, I divided the polynomial by z^2 + 6z +...

Homework Statement Solve the following equation: x^4 + 12x^3 + 46x^2 + 60x + 20 = 0 Homework Equations Well, I know how to solve simpler equations, in which the unknown dosen't appear at a power higher than 3. I tried to factor this polynom but I didin't suceed. The Attempt...

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 $f(x)=x^6-x^5-x^3-x^2-x$.

Homework Statement So my professor wanted us to convert the statement: [ (p -> q) & (q -> r) ] -> (p -> r) into a Boolean polynomial Homework Equations The Attempt at a Solution 1. (p -> q) & (q -> r) = (1 + p + pq)(1 + q + qr) = (1 + p + q + pq + qr) 2. [ (p...

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...