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 Solving for $k$: When Does $P(P(x))$ Have 3 Real Roots?

    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?
  2. caffeinemachine

    MHB Theorem: If Polynomials Converge, Roots Also Converge

    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...
  3. Y

    MHB Understanding Cubic Roots of 1: Exploring cis 120k

    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...
  4. D

    MHB Finding roots of a complex number

    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.
  5. TheSodesa

    Form some third degree equation that has the roots -3, -1 and 2.

    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...
  6. R

    MHB Proving an inequality with square roots

    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...
  7. S

    Complex Roots (Imaginery Numbers) Answers With Sharp El-W519X

    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)...
  8. Math Amateur

    MHB Exercise 2.47 on Page 114: Showing a Polynomial Has Root in \mathbb{F}_4 - Peter

    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...
  9. D

    MHB Proving - properties of quadratic roots

    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...
  10. J

    How many roots does x^3.14 have?

    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 ?
  11. M

    Calculating and Graphing the 4th Root of -4

    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.
  12. R

    Quick question on repeated roots when solving differential equations

    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} -...
  13. paulmdrdo1

    MHB Finding the roots of quadratic equation

    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!
  14. M

    Simplifying Square Roots of a Parametrized Path

    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...
  15. D

    Understanding Complex Roots in Differential Equations

    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.
  16. kaliprasad

    MHB Quadratic equation with rational roots

    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
  17. E

    MHB Finding Roots of Quadratic Equations & Sinusoidal Functions

    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...
  18. Saitama

    MHB Show that the polynomial has no real roots

    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...
  19. N

    Bisection method and multiple roots

    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
  20. R

    MHB Limit with a lot of square roots

    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...
  21. S

    Why is q=0 for Simple Roots in Lie Algebras?

    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...
  22. B

    When is the root of a number both negative and positive?

    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...
  23. B

    Problem with squares and roots

    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) =...
  24. D

    Roots of multivariate polynomials

    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?
  25. mente oscura

    MHB Solve x^6 + 25x^5 + 192x^4 - 7394x^3 + 48936x^2 - 113304x + 79488=0

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

    Is it possible to have a plus/minus function that has range of roots?

    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...
  27. R

    Algebra, how to separate a term in the sum of two roots

    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
  28. anemone

    MHB Finding $q$ in a Polynomial with Negative Integer Roots

    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 $q$.
  29. A

    MHB Simplifying with square roots?

    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. (:
  30. F

    Roots of Unity: Finding Primitive Root W6 and Solving for 1/W6

    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...
  31. T

    Roots of the Following Series in the Complex Plane

    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...
  32. mente oscura

    MHB Polynomials and Roots: Properties and Analysis

    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...
  33. mente oscura

    MHB What are the 6 Complex Roots of this Polynomial?

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

    Roots, signs and abs

    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...
  35. A

    Roots of a squared polynomial ( complex numbers)

    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...
  36. B

    Distance between n-th Roots of Unity

    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...
  37. MarkFL

    MHB Kim's Questions on Babylonian Method for Estimating Square Roots

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

    MHB What is the value of the sum of reciprocals of the roots in a cubic equation?

    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 $a$.
  39. anemone

    MHB Prove Real Roots of $x^3+ax+b=0$ When $a<0$

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

    Parametric equations for roots

    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...
  41. anemone

    MHB Solve Real Roots of $(x-3)^4+(x-7)^4=24832$

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

    Understanding Roots of Unity: Proving Even Distribution with Math

    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...
  43. J

    MHB Cube Roots: Solve A+B=C Problem Easily

    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...
  44. B

    Polynomial roots & Mathematical induction

    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...
  45. P

    Same number of roots for derivative as function

    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...
  46. S

    Complex roots of a quartic polynomial

    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 +...
  47. anemone

    MHB What are the last three digits of the product of the positive roots?

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

    MHB Prove that a function has no integer roots

    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.
  49. J

    MHB Product of roots abcd in 4th degree equation

    If $a,b,c,d$ are distinct real no. such that $a=\sqrt{4+\sqrt{5+a}}\;,b=\sqrt{4-\sqrt{5+b}}\;,c=\sqrt{4+\sqrt{5-c}}\;,d=\sqrt{4-\sqrt{5-d}}$. Then $abcd=$
  50. S

    Nature of the roots of the equations

    Homework Statement Homework Equations The given equations are; Q.1) Find the no of + roots of the equation x^4 -4x+1=0 Q.2) Find the no. of negative roots of the eqn x^4-4x-1=0 Q.3) Find the no of complex roots of the eqn x^4-4x-1=0 The Attempt at a Solution
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