Roots Definition and 962 Threads

Homework Statement Use De Moivre's Theorem to solve for the roots of unity 1, ω, ω2 Hence show that the sum of these roots is zero Homework Equations r(cosθ + isinθ) r(cos(θ + 2n∏)+isin(θ+isin∏)) The Attempt at a Solution I know the first root,1, is 1(cos 0 + i sin 0) but have no clue about...

Here is the question: Here is a link to the question: The equation z3 + az2 + bz + c = 0, where a, b, c are real, have a purely imaginary root (i.e. the real part o? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Question: http://gyazo.com/abca582e1109884964913493487ad8ae My solution: I got √6 + i√2 = √8(cos(pi/6) + isin(pi/6)) as they did below, then [itex] z^{\frac{3}{4}} = \sqrt{8}e^{i(\frac{\pi}{6} + 2k\pi)} [/tex] then took 4/3 of both sides and let k = 1, 0 etc to try and get the values of...

Hello MHB, I got one question, I was looking at a Swedish math video for draw graph and for some reason he did take derivate and did equal to zero and did calculate the roots and then he did take limit of the derivate function to the roots and it's there I did not understand, what does that...

For simple differential equations like y'=y(y-3) where there 2 roots, is it possible for y to approach y=root for both roots? Or must one diverge while the other converges? For multiple roots, can it only converge on one root, the others all diverging? If it can converge on multiple roots...

... I've recently retired and it has been a very long time since I was exposed to a classroom learning about quadratic equations. But now finally jobless, I have more time to satisfy some personal curiousities. For my difficulty in not being able to ask a more sensible question, I apologise...

Here is the question: Here is a link to the question: Polynomials, please help 10 points!? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement compute f'(x) using the limit definition. f(x) = \sqrt{x} Homework Equations f'(x) = \stackrel{lim}{h→0} \frac{f(x+h)-f(x)}{h} The Attempt at a Solution Plugging in the function values gives you f'(x) = \stackrel{lim}{h→0} \frac{\sqrt{(x+h)}-\sqrt{x}}{h}...

Homework Statement Find both square roots of the following number: -15-8i Homework Equations De Moivre's thm: rn(cos(n\sigma) + i sin(n\sigma) The Attempt at a Solution So to use De Moivre's I have to find the modulus and the argument. actually in this question r =...

The Laguerre polynomials, L_n^{(\alpha)} = \frac{x^{-\alpha}e^x}{n!}\frac{d^n}{dx^n}\left(e^{-x}x^{n+\alpha} \right) have n real, strictly positive roots in the interval \left( 0, n+\alpha+(n-1)\sqrt{n+\alpha} \right] I am interested in a closed form expression of these roots...

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?)

Suppose I am to solve an nth order linear homogenous differential equation with constant coefficients. I set up the auxiliary equation, find its roots, and then each root gives me a solution of the form e^{rx} to the ODE which is linearly independent from the others. But if there are repeated...

(BGR,1989) Prove that for any integer $n>1$ the equation $\displaystyle \frac{x^n}{n!}+\frac{x^{n-1}}{(n-1)!}\cdots+\frac{x^2}{2!}+\frac{x^1}{1!}+1=0$ has no rational roots.

Homework Statement Evaluate the following integral Homework Equations ∫ √(4-(√x)) dx The Attempt at a Solution I am having a mind block, I find this too challenging, help!

Here is a link to the question: Show that the equation x^4 +4x+c has at most 2 roots? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Hi, If we have this equation: $$m^3 - m^2 - 8m + 12 = 0 $$ how we can get the roots $$m_1,\,m_2,\,m_3$$? can someone please help... ?

Let $p(\lambda )=\lambda^3+a_2\lambda^2+a_1\lambda+a_0=(\lambda-x_1)(\lambda-x_2)(\lambda-x_3)$ be a cubic polynomial in 1 variable $\lambda$. Use the inverse function theorem to estimate the change in the roots $0<x_1<x_2<x_3$ if $a=(a_2,a_1,a_0)=(-6,11,-6)$ and $a$ changes by $\Delta...

a,b,c,d are four roots of equation : (x-2)(x+1)(x+4)(x+7)-19=0 and a<b<c<d find : 2d-2c

Marie on Facebook writes:

So I'm an undergraduate student in Chemistry in my junior year and I recently transferred schools for a better science program. The one I was at was very, well, easy. Like toddler easy. I never went to class and I aced everything. Here, they're far ahead and it's much more rigorous. I was...

I am typing up a latex document and I need to find roots of unity, lots of them, for numbers like say 42. I was just wondering if anyone knew of a database that had this stuff on hand rather than having to do it all by hand and worrying about having made some stupid algebra error.

Primitive roots of 1 over a finite field Homework Statement The polynomial x3 − 2 has no roots in F7 and is therefore irreducible in F7[x]. Adjoin a root β to make the field F := F7(β), which will be of degree 3 over F7 and therefore of size 343. The multiplicative group F× is of order 2 ×...

[FONT=times new roman] Hello to all members of the forum, [FONT=times new roman]Problem: [FONT=times new roman] Find the real roots of the equation [FONT=times new roman]$\displaystyle x^3+2ax+\frac{1}{16}=-a+\sqrt{a^2+x-\frac{1}{16}}\;\;\; (0<a<\frac{1}{4}) $ I really have no idea on how to...

Hi members of the forum, Problem: Rationalize the denominator of $\displaystyle \frac{1}{a^\frac{1}{3}+b^{\frac{1}{3}}+c^{\frac{1}{3}}}.$ I know that if we are asked to rationalize, say, something like $\displaystyle \frac{1}{1+2^{\frac{1}{3}}}$, what we could do is the following...

Homework Statement Let z be a complex number satisfying the equation ##z^3-(3+i)z+m+2i=0##, where mεR. Suppose the equation has a real root, then find the value of m.Homework Equations The Attempt at a Solution The equation has one real root which means that the other two roots are complex and...

I am doing some independent study and appreciate that a polynomial (in x) of integer degree (n) can have at most n roots; many proofs to this effect exist. My query concerns the number of roots of equations in which the powers of x are not integers (or rational numbers) but irrational...

Hello all, I have a series of polynomials P(n), given by the recursive formula P(n)=xP(n-1)-P(n-2) with initial values P(0)=1 and P(1)=x. P(2)=xx-1=x2-1 P(3)=x(x2-1)-(x)=x3-2x ... I am confident that all of the roots of P(n) lie on the real line. So for P(n), I hope to find these roots. I...

I quote an unsolved problem from another forum. The characteristic polynomial of the given matrix $M$ is $\chi (\lambda)=-\lambda^3-3\lambda^2-17\lambda-11$. The derivative $\chi'(\lambda)=-2\lambda^2-6\lambda-17$ has no real roots and $\chi'(0)<0$, so $\chi'(\lambda)<0$ for all...

My book is showing this as an intuitive step, but I'm not quite seeing the reasoning behind it. n**(c/n) → 1 as n → ∞, for, I think, any positive c. But why?

Homework Statement How may I find stationary points, increase/decrease intervals, concavity for f(x)=(x^3-2x^2+x-2)/(x^2-1)? Homework Equations The Attempt at a Solution I am familiar with how it should be done, except that here I get f'(x)=x^4-4x^2+8x-1 for the numerator of the...

Hi everybody! I've hit a blank with regards to this 1 equation on a old exam paper - think I've overloaded myself a bit and just feel a bit like a airhead at the moment! I understand the actual method and getting to the answer but it starts off with a equation which you then need to get to...

1. -xn2+2(k+2)x-9k=0 Has two imaginary roots, what are the values of k? Attempted to break it down and use the quadratic formula but wasn't able to do it. Would like a pointer in the right direction of where to begin to solve it. Thanks

Hi, Homework Statement I am asked to prove that given all roots of a polynomial P of order n>=2 are real, then all the roots of its derivative P' are necessarily real too. I am permitted to assume that a polynomial of order n cannot have more than n real roots. Homework Equations...

Homework Statement Calculate the integral ∫dθ/(1+acos(θ)) from 0 to 2∏ using residues. Homework Equations Res\underline{zo}(z)=lim\underline{z->zo} (z-z0)f(zo)*2∏i The Attempt at a Solution To start I sub cos(θ)=1/2(e^(iθ)+e^(-iθ)) so that de^(iθ)=ie^(iθ)dθ Re-writing in...

Homework Statement When cos(4θ)=cos(3θ). prove that θ=0, 2∏/7, 4∏/7, 6∏/7 Hence prove that cos(2∏/7), cos(4∏/7), cos(6∏/7) are the roots of 8x3+4x2-4x-1 Homework Equations The Attempt at a Solution I can do the first part, but i have some difficulty in solving the second...

How would you find the roots of: b - tan (b) = 0 please do not that i have to plot the graphs of y=b and y=tan b and then i should find the solution. I want to know how to do it the other way. thank you

Hi All, I've been following a group theory course which I am struggling with at the moment. I'm from 3 different books (Georgi, Cahn and also Jones). I'm trying to understand section 8.7 and 8.9 in the book by Georgi. I (think I) understand that any pair of root vectors of a simple Lie...

Homework Statement I'm trying to understand the simplification of the general solution for homogeneous linear ODE with complex roots. Homework Equations In my notes, i have the homogeneous solution given as: y_h (t)= C_1 e^{(-1+i)t}+C_2e^{(-1-i)t} And the simplified solution is given as: y_h...

I have a algebraic equation like so: x^2-1-εx=0 the roots are obviously- x=ε/2±√(1+ε^2/4) How can I expand the expression for the roots- as a taylor series? the answer is given as: x(1)=1+ε/2+ε^2/8+O(ε^3) I am assuming the author expanded the root 'x' in terms of ε before hand and...

For what complex numbers, x, is Gn = fn-1(x) - 2fn(x) + fn+1(x) = 0 where the terms are consecutive Fibonacci polynomials? Here's what I know: 1) Each individual polynomial, fm, has roots x=2icos(kπ/m), k=1,...,m-1. 2) The problem can be rewritten recursively as Gn+2 = xGn+1 +...

i'm trying to prove the sum of nth roots of unity = 0, but I don't really know how to proceed: suppose z^n = 1 where z ε ℂ, suppose the roots of unity for z are 1, ω, ω^2, ω^3 ... ω^n the sum of these would be S = 1 + ω, ω^w, ω^3 +...+ ω^(n-1) + ω^n from here I had an idea to do some...

Homework Statement Consider the fourth order equation x4 + 5x2 + 6 = 0. (a) Suggest an efficient way to find all roots of this equation. (b) List all the roots. Homework Equations The Attempt at a Solution -I plotted the graph. -I thought of iteration - Is that correct ...

Homework Statement In the book "Friendly introduction to analysis, 2nd Ed." by kosmala there is a definition of the root of a function and subsequent theorem and proof. Either the proof is not directly addressing certain important properties, or is flawed. The definition and theorem are as...

Homework Statement Let z \in \mathbb{C}. Prove that z^{1/n} can be expressed geometrically as n equally spaced points on the circle x^2 + y^2 = |z|^2, where |z|=|a+bi|=\sqrt{a^2 + b^2}, the modulus of z. Homework Equations // The Attempt at a Solution My problem is that I am...

Homework Statement Knowing that the equation: X^n-px^2=q^m has three positive real roots a, b and c. Then what is log_q[abc(a^2+b^2+c^2)^{a+b+c}] equal to? Homework Equations a + b + c = -(coefficient \ of \ second \ highest \ degree \ term) = -k_2 abc = -(constant \ coefficient) =...

Homework Statement z^2 = a + bi a = real number b = real number find all the solutions for z Homework Equations The Attempt at a Solution (x+y)^2 = a + bi ?

Homework Statement For what real values of 'a' do the roots of the equation x^2-2x-(a^2-1)=0 lie between the roots of the equation x^2-2(a+1)x+a(a-1)=0 Homework Equations The Attempt at a Solution The required conditions are \large D_1\geq0 \large D_2\geq0 \large...

Homework Statement Find the values of 'a' so that two of the roots of the equation (a-1)(x^2+x+1)^2=(a+1)(x^4+x^2+1) are real and distinct Homework Equations The Attempt at a Solution I am thinking of converting this equation in quadratic form so that I can find discriminant and make it...

$\beta m^5 + m^2 + 1 =0$ How do I find the roots?

Homework Statement z^4 - z^2 + 1 = 0 is an equation in ℂ. Which of the following alternatives is the sum of two roots of this equation: (i) 2√3; (ii) -(√3)/2; (iii) (√3)/2; (iv) -i; (v) i/2 Homework EquationsThe Attempt at a Solution All I know is that the sum of all roots should equal 0...