Roots Definition and 978 Threads

How do I solve, 1: (√3*√3*√3)/(√3+√3+√3)? How do I simplify it? I'm confused on how to shorten √x+√x+√x, I just don't get it. Also if: √700 = 26.46, then how is √70000 = 264.6? Shouldn't it be 2646?

Homework Statement Let ##G=G_{12}##, ##H_1=G_3##, ##H_2=G_2##. Decide if there are groups ##K_1##, ##K_2## such that ##G## can be expressed as the internal semidirect product of ##H_i## and ##K_i##.The Attempt at a Solution Suppose I can express ##G_{12}## as an internal semidirect product...

Let $P(x)=x^2+6x+k$ for all real $x$, where $k$ is some real number. For what values of $k$ does $P(P(x))$ have exactly 3 distinct real roots?

Proposition 5.2.1 in Artin states that: THEOREM. Let $p_k(t)\in \mathbf C[t]$ be a sequence of monic polynomials of degree $\leq n$, and let $p(t)\in \mathbf C[t]$ be another monic polynomial of degree $n$. Let $\alpha_{k,1},\ldots,\alpha_{k,n}$ and $\alpha_1,\ldots,\alpha_n$ be the roots...

Hello I didn't know in which forum to put this... I solved a linear algebra question, and my answer was: {1}^{1/3} which to my understanding is 1. In the book however, they said it is equal to cis 120k k=0,1,2,... where 120 is degrees. I tried taking the complex number 1+0i and turn it into...

I'm trying to solve this problem and got stuck. Find the roots of $\sqrt{-j}$ converting $0-j$ into polar form $r=\sqrt{0^2-1^2}=1$ $\theta=\tan^{-1}\left(\frac{-1}{0}\right)$ I got stuck on this part. please help.

Homework Statement What the title says. There's a b part to the problem, but of course I can't move on to it until I understand what is going on here. Homework Equations A third degree polynomial is of the form f(x) = ax3 + bx2 + cx + d This information was not given in the...

This is problem 13 from section I 4.7 of Apostol's Calculus Volume 1: Prove that 2(\sqrt{n+1}-\sqrt{n})<\frac{1}{\sqrt{n}}<2(\sqrt{n}-\sqrt{n-1}) if n\geq 1. Then use this to prove that 2\sqrt{m}-2<\displaystyle\sum_{n=1}^m\frac{1}{\sqrt{n}}<2\sqrt{m}-1 if m\geq 2. I have proved the first...

Ive had this problem with the calculator since I bought it. It might be that the calculator does not have enough implemented functions to pull it off or I am missing something. It happens when I am solving for imaginery roots. Example: y^2 + y^1+y = 0. I go to mode, and select 6(equation)...

I am reading Chapter 2: Commutative Rings in Joseph Rotman's book, Advanced Modern Algebra (Second Edition). I need help with Exercise 2.47 on page 114. Problem 2.47 reads as follows: I need help with showing that f(x) has a root \alpha \in \mathbb{F}_4 . My work on this part of the problem...

This is one of my weakness in Math, to prove an existing fact. please Tell how to go about doing these problem. 1. Prove that when the discriminant of a quadratic equation with real coefficients is negative, the equation has two imaginary solutions. 2. Prove that when the discriminant of a...

If x^5 has 5 roots, if x^3 has 3 roots and if x^10 has 10 roots, so how many roots has x^3.14 ?

Hello everyone. How to find the 4th root of -4? I know it's just plugging in the number into the formula but how since n=4, how can we calculate that without calculator? And how to draw it? Here I attached what I have done so far.

say we have gone through the steps and have... ##(\lambda - 2)^{2}(\lambda ^{2}-9) = 0## which we can write as... ##(\lambda - 2)(\lambda - 2)(\lambda ^{2}-9) = 0## we have value for lambda of 2, 2, 3, -3 because we have a repeated root. now, say we have ##(\lambda^{2} -...

can you show me a way of solving this problem without considering the discriminant. Find the roots of equation subject to the given condition. $(2m + 1)x^2-4mx = 1-3m$ has equal roots. I solved it using discriminant but I want to know other way of solving it. Thanks!

Homework Statement Find the arclength of the parametrized path x(t) = (t^2)/2 , y(t) = (t^3)/3 for 1<t<3. Homework Equations Arc Length Formula The Attempt at a Solution x'=t and y'=t^2. Putting them into the arc length formula, I get sqrt(t^2 + t^4) inside. I'm confused...

Hi, I am just having a little trouble with differential equations. I have y'' - 6y' + λy = 0 I know I need complex roots and setting e^\alphax gives \alpha= 3+/-sqrt(9 - λ). Then I don't understand why set -ω^2= 9-λ. How do you know if it is -ω^2 or w^2. Thanks for the help.

form quadratic equation $ax^2 +bx+c=0$ in parametric form such that a,b,c are integers in AP and it has got rational roots

4) Consider the equation H(t) = 16(2)^2t - 10(2)^t + 1. What are its roots? (HINT: Does this look like a quadratic? Perhaps, at least at first, it should be treated like one). 5) Do the same for Y(x) = 2sin^2x - 3sinx - 2. What is wrong with your solutions? Even with the hint, I'm not really...

Problem: Show that the polynomial $x^8-x^7+x^2-x+15$ has no real root. Attempt: I am not sure what should be the best way to approach the problem. I thought of defining $f(x)=x^8-x^7+x^2-x$ because $f(x)+15$ is basically a shifted version of $f(x)$ along the y-axis. So if $15$ is greater than...

Hello, I have a polynomial of order n and I want to find all it's roots with bisection method. Is it possible? I already wrote an algorithm to find a root and it's works nice for finding one of it's roots, but what about others? Nikola

I have the following problem: \lim_{x\rightarrow 4}\frac{\sqrt{2x+1}-3}{\sqrt{x-2}-\sqrt{2}} If I multiply by the conjugate of the denominator I get \lim_{x\rightarrow 4}\frac{\sqrt{(2x+1)(x-2)}+\sqrt{2(2x+1)}-3\sqrt{x-2}-3\sqrt{2}}{x-4} but am not sure where to go from here. Any...

If \alpha and \beta are simple roots, then \alpha-\beta is not. This means that E_{-\vec{\alpha}}|E_{\vec{\beta}}\rangle = 0 Now, according to the text I read, this means that q in the formula \frac{2\vec{\alpha}\cdot \vec{\mu}}{\vec{\alpha}^2}=-(p-q) is zero, where \vec{\mu} is...

Homework Statement I have a simple problem with roots and absolute values. When is the root of a number both negative and positive? Is only the equation of a number say f(x) = √x both the negative root and the positive root? Homework Equations If a = 1; b = -2, och x = a2√(ab-b2+2) Why is x...

Homework Statement Hi, I am currently studying for a exam and I have noticed I have difficulty with squares and roots. I decided to take a problem from an exam so that I can illustrate the problems I am having with it. Homework Equations If f(x) = √(x+1)2 - √(x-1)2 (a) f(x) = 2; (b) f(x) =...

What is the maximum number of roots of a multivariate polynomial over a field? Is there a multivariate version of the fundamental theorem of algebra?

Hello.:) Find the 6 reals roots: P(x)=x^6-25x^5-192x^4+7394x^3-48936x^2+113304x-79488 Regards.

Okay, So I have attached a screenshot of my two graphs of a particle shot from a cannon. The blue one has had an air resistance constant of 0.1 applied to it and, as you can see, has 'shrunk'. For the particular question I am investigating a range of answers are plausible ( ie the x-intercepts...

My mind has gone blank and I've suddenly forgotten basic algebra, please could someone give me direction on how to make P the subject of this equation? E = (P^2 C^2 + M^2 C^4)^1/2 + (P^2 C^2)^1/2 thanks for any help

If $P(x)=x^4+mx^3+nx^2+px+q$ is a polynomial whose roots are all negative integers, and given that $m+n+p+q=2009$, find $q$.

So, this is probably really simple...but I keep getting the wrong answer when trying to simplify this: 3\sqrt{\frac{(10x^3)^2}{(10x^6)^{-1}}}Could someone show the steps to simplifying it? Thanks so much. (:

Homework Statement Hello everyone, In this problem, I was to mark all the sixth roots of 1 in the complex plane. Then, I was to figure out what the primitive root W6 is. However, I am stuck by the question: "Which power of W6 is equal to 1/W6?" Homework Equations See Below The Attempt at a...

Homework Statement For the series x^n - x^(n-1) - x^(n-2) ... - x^(0) the roots seem to be x = 2 and the circle around the complex plane with radius i or 1 I'm not sure how you would say it as n approaches infinity. Here's an image of the roots where n = 15...

Hello. I open this 'thread', in number theory, but he also wears "calculation". I've done a little research, I share with you. Let \ r_1, r_2, \cdots, r_n, roots of the polynomial. P(x)=p_0x^n+p_1x^{n-1}+ \cdots+p_{n-1}x+p_n Let \ Q(x)=q_0 x^n+q_1x^{n-1}+ \cdots +q_n, such that its roots...

Hello. Find the 6 complex roots: x^6+10x^5+70x^4+288x^3+880x^2+1600x+1792 Regards.

By pythagorean identity, ##\sin(x)^2 + \cos(x)^2 = 1##, so ##\sin(x) = \sqrt{1 - \cos(x)^2}##; also, ##\sinh(x)^2 - \cosh(x)^2 = - 1##, therefore ##\sinh(x) = \sqrt{\cosh(x)^2 - 1}##. Happens that the last equation is incorrect, here is a full list of the correct forms for the hyperbolics...

Homework Statement problem in a pic attached Homework Equations The Attempt at a Solution i solved i and ii a , when it came to b , i just said that every one of the 3 roots will be squared having 2 roots 1 + and 1 - but then i read the marking schemes ( also attached) , and i got...

Homework Statement *Find the distance between 1 and the various n-th roots of unity - denoted d(k) *Find a formula for the sum of distances between 1 and each of the n-th roots of unity - denoted S(n) *Find the limit as n->infinity of (1/n).S(n) Homework Equations *The n-th roots...

Here are the questions: I have posted a link there to this thread so the OP can view my work.

If $p,\,q,\,r$ are roots of the equation $x^3+ax^2-4x+3=0$, find the value of $\dfrac{1}{p^2}+\dfrac{1}{q^2}+\dfrac{1}{r^2}$ in terms of $a$.

The equation $x^3+ax+b=0$ has three distinct real roots. Show that $a<0$.

Can I write the parametric equations for the graphs in the following case: on the x-axis, I want to plot a real number 'b'. On the y-axis, I want to plot the roots (all real roots) for x of the equation (7+b2)x3+(6-b)x2+9x-6=0. e.g. when b=1, I plot 1 on the x-axis and x=0.46124674 (the real...

Solve for real roots of the equation $(x-3)^4+(x-7)^4=24832$.

I don't understand why roots of unity are evenly distributed? Every time when we calculate roots of unity, we get one result and then plus the difference in degree, but I think this follows the rule of even distribution and I don't understand that, it is easy to be trapped in a reasoning cycle...

This is a simple question. The problem I'm facing is A cube plus B cube = 22 C cube A cube plus B cube over 22 = C cube At this junction I like to ask if I want to cuberoots both sides, will the 22 be cube root as well? I'm...

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 Provide an example of a function such that f(x) has two and only two real roots and f'(x) has two and only two real roots, where f is defined for all real numbers and differentiable everywhere on its domain. Homework Equations The Attempt at a Solution I know that if a...

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 +...

What is the last three digits of the product of the positive roots of $\large\sqrt{1995}x^{\log_{1995} x}=x^2$.

Let $p, q, r, s \in \mathbb{N}$ such that $p \ge q \ge r \ge s$. Show that the function $f(x)=x^4-px^3-qx^2-rx-s$ has no integer root.