Roots Definition and 962 Threads

\sqrt{2+\sqrt{3}}+\sqrt{4-\sqrt{7}}=\sqrt{5+\sqrt{21}}

[SOLVED] Complex Numbers: Eigenvalues and Roots Below are some problems I am having trouble with, the computer is telling me my answers are wrong. It may be the way I am inputting the numbers but as my final is in a week and a half I would like to be sure. Thanks,

I remember learning an iterative method that gives the answer to trigonometric polynomials such as sin(x)-0.7-0.611cosx = 0 where x is the angle in degrees. The person who I learned this method from called it the method for solving transcendentals. Now I can't seem to find any...

Considering the roots of a cubic polynomial(ax^3+bx^2+cx+d),\alpha,\beta,\gamma \sum \alpha=\frac{-b}{a} \sum \alpha\beta=\frac{c}{a} \sum \alpha\beta\gamma=\frac{-d}{a} If I have those sums of roots..and I am told to find \alpha^9+\beta^9+\gamma^9[/tex] is there any easy way to find...

Homework Statement find all complex roots of z^8=81i Homework Equations The Attempt at a Solution let the angle=x z^8=r^8(cis8x) we know 81i=81 (cis pi/2) threfore z^8=81(cos pi/2 + i sin (pi/2) ) 8x= pi/2 + 2kpi x = pi/16 + kpi/4 kEz therefore...

Homework Statement Given that the roots of x^2+px+q=0 are \alpha and \beta, form an equation whose roots are \frac{1}{\alpha} and \frac{1}{\beta} b) Given that \alpha is a root of the equation x^2=2x-3 show that i)\alpha^3=\alpha-6 ii)\alpha^2-2\alpha^3=9Homework Equations...

Does anyone know how to sum a*r^(1/n) for all n?

For n a nonnegative integer, what (in terms of n) is the sum of the n-th powers of the roots of the polynomial x^6 - 1 ?

Homework Statement Show that f(x) = x^4 + 4x + c = 0 has at most 2 roots. Homework Equations The Attempt at a Solution I'm not really sure how to approach this problem, I think I have to use the IMVT / Rolle's Theorem / MVT. Any help to even get me started would be greatly...

Homework Statement Show f and g are inverse functions or state that they are not. f(x)= cube root of -8x-6 g(x)= -(x^3+6)/(8) Homework Equations You find inverses by plugging the equations into each other, if they are inverses then once you simplify the composed equation, it will equal x.The...

[SOLVED] Mean value theorem First I just want to say that my professor hasn't gotten up to teaching us this so I may be a little slow in understanding this material and want to thank you for being patient with me. The question asks to show that the equation X^4 -4X + c = 0 has at most two...

[SOLVED] Compex roots Homework Statement state the number of complex roots of each equation, then find the roots and graph the related function. x^2 + 25 = 0 Homework Equations The Attempt at a Solution x^2 + 25 = 0 so there are 2 complex roots. Once I have established that...

Homework Statement x^2\frac{d^2y}{dx^2}+ax\frac{dy}{dx}+by=0 show that if there is one real double root of the aux. eq'n show that the G.S. is given by y=c_1x^{n_1}+c_2x^{n_1}ln(x) Homework Equations Assume the trial solution y=x^n The Attempt at a Solution y=x^n...

Homework Statement Find an integer c such that the equation 4x^3 + cx - 27 = 0 has a double root. Homework Equations Ax^3+Bx^2+Cx+K = 0 Sum of Roots = -B/A Product of Roots = (-1)^n * k/a etc. The Attempt at a Solution I tried using P/Q with synthetic division to find a...

Homework Statement Write a fourth degree polynomial that has roots of 3 and 1-i. There is more than one correct solution Homework Equations The Attempt at a Solution I'm extremely lost as to where this problem is going, I know that to be a fourth degree its simply x^4, but how in...

Hello friends, I am studying in 10th class. Actually I have a question and I’m unable to solve this question. My question is: How can we find the square root of a number by hand? How about cube roots? If anybody can solve my question I will grateful. Thanks in advance!

y'' + 2y' + 5y = 0 (*) OK, what I have done is computing the two roots y1 = exp(-x)*cos2x and y2 = exp(-x)*sin2x. However, when I compute the derivatives of these two, and substitute into (*), the eq. doesn't equate 0. Are my roots wrong?

Homework Statement In the equation x^3+ax^2+bx+c=0 the coefficients a,b and c are all real. It is given that all the roots are real and greater than 1. (i) Prove that a<-3 (ii)By considering the sum of the squares of the roots,prove that a^2>2b+3 (iii)By considering the sum of the cubes of...

Homework Statement find the four fourth roots of -2\sqrt{3}+i2 i don't have any attempt for a solution because i don't know what to do.. im really lost.. i regret sleeping in class

Homework Statement Is it there a method to find out if a polynomial has no integer roots? The Attempt at a Solution I tried the division of polynomials, as well as the Horner's Method, but no luck.

Hi,.. using a Sturm or other sequence, could we find how many integer roots have the Polynomial K(x)= \sum_{n=0}^{d} a_{n}x^{n} where all the 'a_n' are integers (either positive or negative)

given a Polynomial or a trigonometric Polynomial K(z)= \sum_{n=0}^{N}a_{n}x^{n} and H(x)= \sum_{n=0}^{N}b_{n}e^{inx} is there a criterion to decide or to see if K(z) or H(x) have ONLY real roots

Does anyone know whether the graphical solution of cubic equations with real roots by means of intersecting a circle and a parabola or hyperbola (or just a parabola and hyperbola) is known or not? That solution has to give the equations for the circles, parabolas and hyperbolas involved and not...

Hi! Who knows: can any cubic root like \sqrt[3]{x} with x real be written as a form in which only square roots (real or complex) are involved?

please see my question i can't dfind its imaginary roots .the equ is X2 –3X +C,here 2 is the power of X and Cis constant we have to show that there exixts no reak number C for which the givev equation has two distinct rootss in [-1,1] i solve this by quadic formula but i got its real...

Let be a open curve on R^2 so x^{n}-c-ky=0 where k,n and c are integers, are there any methods to calculate or at least know if the curve above will have integer roots (a,b) so a^{n}-c-kb=0 ?? or perhaps to calculate the number of solutions as a sum (involving floor function) over integers of...

I don't get this problem and why the answer is what the book states that it should be: If f(x)=\sqrt{x^{2}} then f(x) can also be expressed as: l x l The answer I chose was simply x , but I don't know why it is wrong.

Homework Statement How to find the conditions on the coefficients of a quadratic equation for the roots to be outside the unit circle eg bx^2 + x - 1 = 0 where b is a constant How do we find the condition(s) that b must satisfy such that the roots of the quadratic lie outside the unit circle...

Hi I noticed that multiplication of all primitive roots modulo p ,p>3, congruent to 1 modulo p... I have tried some examples (13,17,19...) but i couldn't prove the general case (let g1...gk be primitive roots modulo p,p>3 ==> g1*g2*...*gk=1(p)) I need help to prove or disprove...

Excuse my typography - I'm new here... a, b, and c are rational numbers. I want to prove that * IF S = root(a) + root(b) + root(c) is rational THEN root(a), root(b) and root(c) are rational in themselves. Now I have done as follows: I reverse the problem and try to show that: * IF...

How do you do these two problems? 1. Find the sum of the 6th roots of unity. 2. Find the product of the 6th roots of unity.

Here is the question: ----------- Suppose n has a primitive root g. For which values of a (in terms of the primitive root g) does the equations x^2 \equiv a \ \text{(mod n)} have solutions? ----------- I really don't have much of an idea of how to even begin this one. Let g be a primitive...

Homework Statement We have this theorem: Let f(x)\in F[x] Then f(x) has multiple roots if and only if gcd(f(x),f'(x))=d(x) and d(x)\geq 1 We went BRIEFLY over the proof and we are supposed to be able to apply it on an upcoming exam. I'm not exactly sure how it works or what I'm...

I am trying to derive a version of Euler's criterion for the existence of cube roots modulo p, prime. So far, I have split the primes up into two cases: For p = 3k+2, every a(mod p) has a cube root. For p = 3k+1, I don't know which a it is true for, but I did a few examples and noticed...

Here is the question from the book: ------------ Determine a primitive root modulo 19, and use it to find all the primitive roots. ------------ \varphi(19)= 18 And 18 is the order of 2 modulo 19, so 2 is a primitive root modulo 19, but I am not sure of how to use that to find all...

If a is a perfect cube, a= n^3, for some integer n, and p is a prime with p is congreunt to 1 mod 3, then show that a cannot be a primitive root mod p, tat is ep(a) is not equal to p - 1

Homework Statement Let f(x) be a pllynomial of degree n, an odd positive integer, and has monotonic behaviour then the number of real roots of the equation f(x)+ f(2x) + f(3x)...+ f(nx)= n(n+1)/2 Homework Equations The Attempt at a Solution This seems like the summation of...

Homework Statement How do I find the roots of 4x^3+x+5 = 0? It doesn't appear to be in a nice form like many equations in the textbook?

Hi. I only just recently found out about an algorithm for calculating the square roots of a number. Lets say i want to evaluate \sqrt {n}. I can make an approximation by inspection, and say \sqrt n \approx \frac{a}{b}. Now, using this approximation, i can write: \left[...

1.Solve the following initial value problem d^2y/dt^2 + 4y = { 1, 0<= t <=Pi/2 } {sin(t), Pi/2 < t} y(0) = 2 y'(0) = -1 I used y(t) = e^rt 3. For y'' + 4y' =1 y(t)= e^rt Yp(0)=1 r^2+4 = 0 r = 2i y1(t) = C1 Cos(2t) +...

Homework Statement I need to find the real and imaginary roots of z^4 = -1. The Attempt at a Solution The polar coordinates of -1 are at (-1, pi), (-1, 3pi) etc so if I assume the solutions take the form z = exp[i n theta] then n theta = pi + 2npi This dosen't seem to give the...

Homework Statement Heres the question I'm tackling: <b> the height of a prism is x-1, width x-2, length x+3 and the volume is 42cm cubed. Find the dimensions</b> Homework Equations Quadratic formula, common factor, family of functions The Attempt at a Solution I know that 7, 2...

I want to show that every polynomial of degree 1, 2 and 4 in Z_2[x] has a root in Z_2[x]/(x^4+x+1). Any ideas? Ps. How can I use latex commands in my posts?

can a non-factorable function has rational real roots?

Homework Statement Find all roots of the equation cos(z)=2 (z is a complex number)Homework Equations The Attempt at a Solution What do they mean find the roots of this equation? We're just going over trig functions and it doesn't say anything about roots so I'm not sure what they're...

I'm having trouble following one step in a proof I'm studying. I'm sure I'm missing something obvious, but I just can't get it to work out (it supposed to be "obvious" which is why they left out the details). Anyway, it's part of a proof showing that if you have a monic polynomial with all...

Hi all, is there a general way of proving that sqrt(r1) + sqrt(r2) + sqrt(r3) + ... + sqrt(rn) is irrational, given that none of r1, r2, r3, ..., rn is the square of a rational number? (or is this statement even true in general?) for the case when n = 2, the proof is quite...

Homework Statement Show that for negative c (a,b,c - real) equation x^3+ax^2+bx+c=0 has at least one positive root. 2. The attempt at a solution Considering the equivalent form of the equation above for large |x|: x^3(1+\frac{a}{x}+\frac{b}{x^2}+\frac{c}{x^3})=0 we can conclude that there...

Homework Statement Find all the roots of sin h(z) = 1/2 2. The attempt at a solution sin h(z) = [1/2](e^z - e^-z) = 1/2 => e^z -e^-z = 1 => e^2z - e^z - 1 = 0 {multiplied e^z bothsides} this is a quadratic equation in e^z using quadratic formula, e^z = [1+- sqrt(5)]/2 taking 'ln' on...

Is there a website that given the coefficients of a order 4 polynom, it will give me all 4 roots of this polynom? Thanks.