Roots Definition and 962 Threads

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.[/color] ================================= I proved part a by first showing that each of 1^k...

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...

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...

Hello, My calculus book says that readers who are writting \sqrt{9} as \pm3 must stop doing that, because it is incorrect. The question is: why is it incorrect? Regards

It's been a while since I studied calculus and basically I have a review sheet for a course I'm taking, but not a graded assignment. So, I was hoping if anyone knew a resource to point me in the right direction with a couple of problems: \int_0^\theta x^a dx Where a is not an element...

I am having trouble to find a straight forward method for finding coefficients using partial fraction expansion with repeated or complex roots. My study notes arent too clear so I am finding the differentiation method hard to follow for repeated roots. As for complex roots I can find the roots...

Hai, I wanted to find the three cube roots of -1+i And since the questions says `three` n = 3 So it should be something like this \sqrt{2}cos(\frac{3\pi}{4} + i sin(\frac{3\pi}{4}) But the key says that the answer is on the form 2^{1/6} why? clearly n was 3?

Hi I'v got a maths exam on Tuesday for my 2nd year of chemial engineering. Been going through a past paper and have been going over 2nd order homogenous DE's Im at the stage of calculating the roots (wether repeated or 2 distinct roots) I take the easy path like so: E.g m2 + 4m + 4 =...

This is probably very easy, but anyway.. I just don't really understand the question, therefore don't know what answer to give.. Homework Statement The question reads... For what values of n will one root of the equation (n - 2)x^2 + (n + 2)x + 2n + 1 = 0 be the reciprocal of the...

I've been looking at some practise exams for the University I would like to apply to, I have to sit the exam on 4th November. We have never done finding the roots of a cubic equation before and I cannot figure it out from looking on the internet, the formulas are all horrible to understand...

This has been bugging me for a few days now. Usually when I have a question about math I can answer it myself, but this, the answer is evading me. I know that the answer to the negitive square root of 100 is .1. How is this number obtained? Is there a graphical representation? Does the...

Is it possible to find a root of f(x), given just f(x_1)==y and f ' (x)? if so, how would one go about it? If this is in the wrong forum can a mod please move it? thanks.

1: Which of the following is incorrect a cos2x = 2-(sin^2) x b (sin^2)x + (cos^2)x = 1 c cos x = sin ( pi/2 -x) d sin2x = 2sinxcosx I know b is correct, fairly sure c is correct because i know cos is out of phase by pi/2 have no bloody idea how to determine which of these is...

Derivative of a sum of functions What would you interperate this as? It is one of the subsections of Differentiation from the syllabus of a University entrance Exam in November but I cannot think what it is referring to. Differentiation: Derivative of xa, including for fractional...

could someone tell me where the i^i=e^(pi/2) came from??

Say I have a monic polynomial, x^3 + ax^2 + bx + c with a=-2.372282, b=1.862273, c=-0.483023 The discriminant is given by a^2 b^2 - 4 b^3 - 4 a^3 c - 27 c^2 + 18 ab c which is < 0, indicating 1 real root and 2 complex conjugates. A method for solving a general cubic using the...