Roots Definition and 978 Threads

Show that a complex number, w exists such that the fifth roots may be expressed as 1, w, w^2, w^3 and w^4I am having trouble understanding what the question is asking of me. Could anyone please help? Thanks.

Hello all, I'm attempting to find in literature a method of determining from a Lie algebra's full root system in an arbitrary basis which roots are simple. It seems there are many books, articles, etc on getting all the roots from the simple roots but none that go the other way. My task is...

x^4 + 7x^2 + 6 =0 I know the answers are imaginary but I don't remember how to solve this equation.

Hi there I was wondering if there is a simple way to solve for the roots of a complicated summation of trig functions that can't be combined with any simple identities. I have an equation of the form: 0 = sin(8x-arctan(4/3))+3.2sin(16x+pi/2) where the two sines have different amplitudes...

Homework Statement If x = -1; 1 are the two roots of the equation x^2 +ax+b = 0; then find the values of a and b. The Attempt at a Solution x^2+ax+b=0 (x-1) (x+1)= 0 ,,,,,, by expanding x^2-1=0 ,,,,, so a=0 , b=-1 is it right or the question is asking for something eles,,,, ?

My "Computational World" class is supposedly an intro to computer science without programming, but the questions are all over the place. I'm completely stuck on this question: Show expressions for: (a). 3 cube roots of -1 (b). 4 fourth roots of -1 (c). In the general case, N nth-roots of...

The function y=e^x can be expanded using the power series, thus y=e^x=1+x+\frac{x^2}{2}+\frac{x^3}{3!}+... This is a polynomial of infinite degree, and the theorem that says a polynomial must have at least one root in the complex field, and thus this extends to a polynomial of nth degree having...

Not sure where this should go, but - How would you calculate the square root of a triangular number with a base other than 1? For example, 2, 6, 12, 20 (Base 2). Would rather have help to figure it out than the actual formula.

Homework Statement The equation x4-6x3-73x2+kx+m=0 has two positive roots, \alpha \beta, and two negative roots, \delta \gamma. It is given that \alpha\beta=\delta\gamma=4 (i) Find the values of k and m Homework Equations stated above The Attempt at a Solution m =...

Homework Statement If a and b are positive real numbers, and \lambda^{2} = ab, then \lambda = \pm \sqrt{ab}. Homework Equations None. The Attempt at a Solution This is more of a conceptual question that has always escaped me. I do not understand how the square root of two...

Is there a mathematical equation to find square/cubic/etc. roots of a number? Any help would be greatly appreciated - this is purely for my own interest. Also, I;m doing the C2 module of AS Maths so I may not understand more complex terms used in an explanation (if any are needed) - apologies...

Homework Statement The equation x3 − 3x2 + px + 4 = 0, where p is a constant, has roots α −β , α and α + β , where β > 0. (a) Find the values of α and β . (b) Find the value of p. how do i start off? all i know is that sigma a= -b/a and ab= c/a and ab(gamma) = -d/a . Would this be one of...

Let p be a prime. a) If gcd(k,p-1)=1, then 1^k, 2^k,..., (p - 1)^k form a reduced residue system mod p. b) If 1^k, 2^k,..., (p - 1)^k form a reduced residue system mod p, then gcd(k,p-1)=1. ================================= I proved part a by first showing that each of 1^k, 2^k,..., (p -...

Homework Statement One root of 2x2-kx+k=0 is twice the other. Find K. (assume k is not 0) Homework Equations not sure what this means. The Attempt at a Solution \alpha+\beta=\frac{-b}{a} \alpha\beta=\frac{c}{a} then i tried \alpha=2\beta then substituted in...

Homework Statement The problem is: y" + 2y' - 3y = x^2*e^x Homework Equations The Attempt at a Solution I know the roots are y1 = -3 and y2 = 1, becoming e^x. I'm not sure how to set my yp up though with the repeating e^x. My ideas are yp = x * (x^2*A*e^x) or x * (Ax^2 +...

Homework Statement Let f be a polynomial of degree n >= 1 with all roots of multiplicity 1 and real on R. Prove that f has at most one more real root than f' f' has no more nonreal roots than f Homework Equations We are given the Gauss Lucas theorem: Every root of f' is contained in...

1. Hi really struggling with this question any help would be great. Use Newtons method to find the root of 4sin^2x - x = 0 which lies closest to [B]x=2, correct to 3sf. Homework Equations The Attempt at a Solution

Example: f(x) = x2 + 2x +1 f(x)≡0 (mod 15) Find ALL roots mod 15. ====================== Solution: 15=3x5 Consider f(x)≡0 (mod3). mod 3: check 0,1,2. Only 2 solves it. Consider f(x)≡0 (mod5). mod 5: check 0,1,2,3,4. Only 4 solves it. So x≡2(mod 3) and x≡4(mod 5). Looking at...

Homework Statement These two questions are very similar: 1) Let c be a constant. If a and b are the roots of the equation x^2 + 2x - c = 0 then 2b-a^2 = ? 2) Let k be a constant. If a and b are the roots of the equation x^2 - 3x + k = 0 Then a^2 + 3b = ? Homework Equations...

I can't figure out how to get the answer in the back of the book. y''+6y'+13y=0 So far, I have... ert (r2 + 6r + 13) = 0 r2 + 6r + 13 = 0 r2 + 6r + ___ = -13 + ___ r2 + 6r + 9 = -13 + 9 (r+3)2 = -4 r = -3 +/- 2i Then we have a crazy-looking thing after assuming y = C1e(r1)(t)...

Hi, I have a question, which seems deceptively simple to me, but when I thought about it, I couldn't really come up with a rigorous proof. Here goes, Are the roots of a polynomial equation unique? Suppose we have a general monic polynomial equation: z^{n} + c_{1}z^{n-1} + c_{2}z^{n-2} +...

Homework Statement Let U1, U2, and U3 be independent random variables uniform on [0,1]. Find the probability that the roots of the quadratic U1x2+U2x+U3 are real. Homework Equations The Attempt at a Solution So we need to find P(U22>4U1U3), which involves evaluating some...

Homework Statement z is a complex number. Find all the solutions of (z+1)^5 = z^5 The Attempt at a Solution Of course one could expand (z+1)^5, but I remeber our professor solving this with roots of unity. Can anyone help?

Homework Statement Not so much a homework problem, but a problem that is annoying me because of its simplicity. Not all cubic polynomials with rational coefficients can be factorized by the rational root theorem (or is this false?). What I am finding hard to comprehend is how a cubic with...

my question is , given the Group G of symmetries for the equation x^{4} + a^{2}=0 for some 'a' Real valued i see this equation is invariant under the changes x \rightarrow -x x \rightarrow ix x \rightarrow -ix x \rightarrow -x x \rightarrow i^{1/2}x x...

In trying to understand the underlying mechanics of irreversible processes, I came up with four mechanical asymmetries that seemed relevant to describe energy transfer through the bulk motion of a frictionless piston that divides a cylinder with two gasses of same pressure but different...

Homework Statement (a) If n is even find a polynomial function of degree n with n roots. (b) If n is odd find one with only one root. Homework Equations N/A The Attempt at a Solution If by no roots, they mean no real roots then I guess: f(x) = x^n+1 would work for both even...

Find a primitive root modulo 101. What integers mod 101 are 5th powers? 7th powers? -I tested 2. -2 and 5 are the prime factors dividing phi(101)=100 so i calculated 2^50 is not congruent to 1 mod 101 and 2^20 is not congruent to 1 mod 101. -Therefore 2 is a primitive root modulo 101 I guess...

Matlab help state that the square root of X = \begin{pmatrix} 7 & 10 \\ 15 & 22 \end{pmatrix} are A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} , B = \begin{pmatrix} 1.5667 & 1.7408 \\ 2.6112 & 4.1779 \end{pmatrix} , C=-A and D=-B . When I used the MATLAB command...

Please prove that if x is quadratic nonResidue modulo 109 and x also cubic nonresidue modulo 109 than x is guaranteed to be primitive root modulo 109 thanks you very much

\frac{\sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2}}{\sqrt{\sqrt{5}+1}} What technique/method could I use to simplify that one fast? It was asked in a math contest here in our country and the question is only for 20 seconds. The answer is \sqrt{2}. You know any technique for that one guys?

hi. i have recently become very interested in the idea of the nth roots of unity. i have discovered how to calculate them (using eigenvalues), and i find it very fascinating that there are not n many nth roots of unity(unlike scalars). aparently in the case where the matrix is 2x2, there are...

So I found the characteristic equation of a matrix, and I know the roots of the equation are supposed to be the eigenvalues. However, my equation is: \lambda^3-2\lambda^2 I have double checked different row expansions to make sure this answer is correct. So don't worry about how I came to get...

Hi all, I've been staring at this question on and off for about a month: Suppose that p is an odd prime, and g and h are primitive roots modulo p. If a is an integer, then there are positive integers s and t such that a \equiv g^s \equiv h^t mod p. Show that s \equiv t mod 2. I feel as...

Let A be an n x n matrix. Show that det(A) is the product of all the roots of the characteristic polynomial of A.

Homework Statement For the quad equation x^2 - px + 9 = 0 1. Write down the sum of roots and product of roots 2. Find p IF twice the sum of the roots EQUALS the product 3. Find p IF the roots are unequal Homework Equations Sum = (a+b) = -b/a Product = (ab) c/a The...

Homework Statement Show that there is only one stationary point of the curve y=e^{x/2} - ln (x), where x>0 and determine the nature of the stationary point. My approach: dy/dx = 0.5e^{x/2} - 1/x When dy/dx=0 For stationary point. Thus, through algebraic manipulation...

Homework Statement solve the initial value problem y''+9y = 0, y(0) = 1, y'(0) = 1 Homework Equations gen solution form is y(t) = C1e^At*(cosBt) + C2e^At*(sinBt) where A is the real number and B is the imaginary number The Attempt at a Solution i just wanted to check if I am doing...

Homework Statement Prove that if f is a polynomial function of degree n, then f has at most n roots, i.e., there are at most n numbers a with f(a) = 0. Homework Equations N/A The Attempt at a Solution I know that I'm supposed to use induction on the degree of the polynomial. If...

hi i know its a little later in the day but I am having trouble working out the polar form off the seven roots. what i have got so far is that they are divided into 60 degrees around the 360 i also need the congurants which when i use the sin(60) sin (120) i have the right numbers but when i...

Hi All, I am using Solve[{f(x,y)==0,g(x,y)==0},{x,y}] to find "x,y" roots of "f" and "g" functions. I am only interested in positive "x" and "y" roots, ignoring all the other. Is there a way to use "Select" command to find all positive roots? Thanks.

What is the proof that states that if the square root of a natural number is not another natural number, it must be irrational? In other words, the square root of a natural number must be either natural or irrational.

If you know \sqrt{(a^2+b^2)} < \epsilon, do you know a < \epsilon and b < \epsilon? If so, how?

Homework Statement Show that, if \omega is an nth root of unity, then so are \overline{\omega} and \omega^{r} for every integer r. Homework Equations \omega=r^{1/n}e^{i((\theta+2\pi)/n)} The Attempt at a Solution I got the first part and for \omega^{r} I have it equals...

Hi all, What happens when we take the product of the imaginary parts of all the n-roots of unity (excluding 1)? I read somewhere that we get n/(2^(n-1)). How can we prove this? Thanks!

I am in a numerical methods class, which uses MATLAB and c to do methods like regular falsi and Newton raphson. I should know this, but why do we bother finding the value of x that makes our function evaluate to zero? Is it so that we have some basis as to where to start or stop a certain...

Homework Statement If c is any nth root of unity other than 1, then 1 + c + c^2 + \cdots + c^{n-1} = 0 The Attempt at a Solution This is what is done so far and I am at a dead stall for about 2 hours lol. Any ideas on what I should be thinking of next? Should I continue...

sin x. d(sin x)/dx = cos x. d(cos x)/dx = -sin x. d(-sin x)/dx = - cos x. d(-cos x)/dx = sin x. i. i^2 = -1. i^3 = -i i^4 = 1 i^5 = i. I know there is a relationship between trig, the complex numbers, and exponential functions. Is there a relationship between the pattern shown here?

Homework Statement I need to use MATLAB to solve these problems. http://users.bigpond.net.au/exidez/IVDP.jpg Homework Equations MATLAB The Attempt at a Solution a) R1=3.6; R2=R1; C1=33*10^-6; C2=22*10^-6; % defining the polynomial constants Vs=[R1*R2*C1*C2...

Hello, I am a first-year math grad student, and I just started teaching two sections of precalc! yikes! So I had a major embarrasing moment during office hours when I told 2 students the wrong answer because I completely forgot about the phenomenon known as "Extraneous roots"! Does anyone...