What is Roots: Definition and 976 Discussions

The Roots are an American hip hop band, formed in 1987 by Tariq "Black Thought" Trotter and Ahmir "Questlove" Thompson in Philadelphia, Pennsylvania, United States. The Roots serve as the house band on NBC's The Tonight Show Starring Jimmy Fallon, having served in the same role on Late Night with Jimmy Fallon from 2009 to 2014.
The Roots are known for a jazzy and eclectic approach to hip-hop featuring live musical instruments and the group's work has consistently been met with critical acclaim. ThoughtCo ranked the band #7 on its list of the 25 Best Hip-Hop Groups of All-Time, calling them "Hip-hop's first legitimate band."In addition to the band's music, several members of the Roots are involved in side projects, including record production, acting, and regularly serving as guests on other musicians’ albums and live shows.

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  1. anemone

    MHB Prove Polynomial Roots: a(b) of x^6+x^4+x^3-x^2-1

    If a,\;b are roots of polynomial x^4+x^3-1, prove that a(b) is a root of polynomial x^6+x^4+x^3-x^2-1.
  2. I like Serena

    MHB AM-GM inequality for sum of 3 square roots

    Let $a,b,c$ be positive real numbers with sum $3$. Prove that $√a+√b+√c≥ab+bc+ca$.
  3. J

    MHB No. of real roots of Quadratic equation

    The no. of Real Roots of the equation $\displaystyle \frac{\pi^e}{x-e}+\frac{e^\pi}{x-\pi}+\frac{\pi^{\pi}+e^{e}}{x-\pi-e} = 0$ My try:: Let $\displaystyle f(x) = \frac{\pi^e}{x-e}+\frac{e^\pi}{x-\pi}+\frac{\pi^{\pi}+e^{e}}{x-\pi-e}$ Now we will take a interval $x\in \left(e\;,\pi\right)$...
  4. H

    Solving Roots of Unity with De Moivre's Theorem

    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...
  5. Fernando Revilla

    MHB Roots of p ( z ) in IR [ z ] (Lava's question at Yahoo Answers)

    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.
  6. C

    How can complex roots and equations be solved using different methods?

    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...
  7. Petrus

    MHB Calculating Derivatives and Finding Roots in Math

    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...
  8. C

    Slope fields:Can y converge on two roots?

    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...
  9. Wes Tausend

    Discarding some quadratic roots - ok?

    ... 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...
  10. Fernando Revilla

    MHB Jesusluvsponies's question at Yahoo Answers (Real and rational roots)

    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.
  11. D

    Forgotten how to solve for square roots

    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}...
  12. T

    Solving for the roots of unity of a complex number

    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 =...
  13. K

    Closed form expression of the roots of Laguerre polynomials

    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...
  14. 1

    Finding the roots of a high degree polynomial equation

    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?)
  15. B

    Nth order linear whose auxiliary has repeated roots

    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...
  16. melese

    MHB No Rational Roots of $x^n+\cdots+1=0$

    (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.
  17. H

    Evaluating difficult integral involving square 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!
  18. Fernando Revilla

    MHB Ann's question at Yahoo Answers (At most two roots)

    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.
  19. I

    Use the inverse function theorem to estimate the change in the roots

    Homework Statement 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...
  20. r-soy

    MHB What are the steps to find the roots of a cubic equation?

    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... ?
  21. I

    MHB Use the inverse function theorem to estimate the change in the roots

    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...
  22. Albert1

    MHB Finding 2d-2c from Equation with 4 Roots

    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
  23. Sudharaka

    MHB Marie's Question from Facebook about Square Roots

    Marie on Facebook writes:
  24. Q

    Does Organic Chemistry Have Strong Roots in Computational Chemistry?

    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...
  25. D

    Does anyone know a database for nth roots of unity

    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.
  26. J

    Formal Derivative and Multiple Roots

    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 ×...
  27. anemone

    MHB Find the real roots of an equation.

    Hello to all members of the forum, Problem: Find the real roots of the equation $\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 even start to work with this problem, could anyone please help me? Thanks.
  28. anemone

    MHB Rationalizing a denominator involving the sum of 3 cube roots

    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...
  29. Saitama

    Finding the Value of m in a Complex Number Equation

    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...
  30. S

    Max number of roots of radical equation

    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...
  31. J

    Finding roots to a recursively defined polynomial of degree n

    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...
  32. Fernando Revilla

    MHB Prove that this matrix equation has no roots

    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...
  33. T

    Why does n^(c/n) approach 1 as n approaches infinity?

    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?
  34. P

    Finding extrema when derivative has no rational roots.

    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...
  35. J

    Rational roots - standard form of equation

    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...
  36. L

    Solving equation with two imaginary roots

    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
  37. P

    Roots of derivative of polynomial.

    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...
  38. D

    How Does the Value of 'a' Affect the Integral on the Unit Circle Using Residues?

    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...
  39. M

    Prove that cos(2∏/7), cos(4∏/7), cos(6∏/7) are the roots of this equation.

    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...
  40. R

    Finding Roots of b - tan(b) = 0

    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
  41. A

    The Cartan matrix in order to find the roots.

    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...
  42. DryRun

    Homogeneous Linear ODE with complex roots

    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...
  43. marellasunny

    Talor series expansion of roots of algebraic equation

    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...
  44. E

    Roots of linear sum of Fibonacci polynomials

    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 +...
  45. C

    What is the Sum of nth Roots of Unity and How Can It Be Proven?

    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...
  46. E

    Find an effeciant way to deduce the roots of the equatioon

    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 ...
  47. W

    Analysis: Potentially flawed proof in book describing roots of multiplicity m

    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...
  48. 5

    Expressing geometrically the nth roots of a complex number on a circle

    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...
  49. V

    Roots of a third degree polynomial

    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) =...
  50. Jameson

    MHB Finding roots of a quadratic (Lindsay's question from Facebook)

    Lindsay on Facebook writes: I think you mean you got -48 instead of 48, but either way let's go through solving this. x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} Looking at $3x^2+12x+8$ we see that $a=3$, $b=12$ and $c=8$. Plugging that into the above quadratic equation yields. x = \frac{-12 \pm...
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