Roots Definition and 962 Threads

Homework Statement \alpha and \alpha^{2} are two roots of the equation x^{2} -12x + k = 0 Find 2 values for k. The Attempt at a Solution \alpha + \alpha^{2} = 12 \alpha^{3} = k I have no idea where to go from here. Any help appreciated.

I'm trying to prove the following, which is left unproven in something I'm reading on ruler-and-compass constructions: If ax^3+bx^2+cx+d is a polynomial over a subfield F of ℝ, and p+q\sqrt{r} is a root (with \sqrt{r}\notin F) then p-q\sqrt{r} is also a root. The theorem immediately before...

Ok, so obviously, given some polynomial P(x) of degree r, it has r roots in the complex numbers by the FTOA, and if these roots are u_1, u_2,... it can be written as \begin{array}{l} P(x) = (x - {u_1})(x - {u_2})(x - {u_3}) \cdots \\ P(x) = {x^r} - ({u_1} + {u_2} + {u_3} + \ldots ){x^{r - 1}}...

1. ei∏/3z3 = 1/(1+i) 3. Wasn't sure at all how to start...Attempted to rearrange, bringing the exponential to the right and expanding using Euler's theorem, but it didn't work. I'll be really grateful to anyone generous enough to help :) thanks

Homework Statement The equation kx2 - 3x + (k+2) = 0 has two distinct real roots. Find the set of possible values of k. Homework Equations Since the equation has two distinct real roots, b2 - 4ac > 0 The Attempt at a Solution b2-4ac>0 9-4(k+2)(k)>0 9-4(k2+2k) >0 9-4k2-8k>0 =...

Homework Statement Hi all, I was wondering if there is a procedure you can follow to calculate the complex roots of a cubic equation.Homework Equations For example the equation x3 - 1 = 0 has roots of x = 1 x = -0.5 + √3/2 i x = -0.5 - √3/2 i Admittedly, I got those solutions off wolfram...

I'm going over some things I didn't do too well on in my latest Algebra test. One question was: List all of the roots of x^{8}\:-1\:=0, and write them in the form a+bi. So I knew I had to list all the 8th roots of unity. In other places in the test they used the notation e^{iθ} and this was a...

Homework Statement (8x+1)^0.5 Homework Equations The Attempt at a Solution I tried using substitution but it clearly doesn't work because nothing in the brackets equals when derived. Anyone help me with the beginning steps?

I stumbled upon a math question, at the glance of it, seemed easy. One is supposed to find the exact value of this square root: √30*31*32*33+1 They are all under the square root operator and Fundamental BEDMAS/BIDMAS applies of course. The trick here is when the condition states that one...

(a^{\frac{1}{2n}}a^{\frac{1}{2n}})^n=(a^{\frac{1}{2n}})^n(a^{\frac{1}{2n}})^n=(a^{\frac{1}{2}}) (a^{\frac{1}{2}})=a=(a^{\frac{1}{n}})^n all above is just done by using that the order of the factors that you multiply does not matter we have proven that...

Homework Statement Find the intervals of all possible value of p which the equation equation: (p-1)x^2+4x+(p-4)=0 has two different roots. Homework Equations ax^2+bx+c>0 ?? The Attempt at a Solution (p-1)x^2+4x+(p-4)>0 ?? How would I go about solving this? Is two roots...

Homework Statement Use Newton's method to find ALL roots of e^x=3-3x Homework Equations The Attempt at a Solution I know how to use Newton's method, but how is it possible to use it to find ALL the roots of the function? Just by looking at the function however, I THINK that...

Homework Statement Let f(x)=x^4 - x - 1. Show that f(x)=0 has two real roots. Homework Equations None The Attempt at a Solution x(x^3 - 1 - 1/x) = 0 which gives x=0 and x^3 - 1 - 1/x=0, x^2 - 1/x - 1/x^2=0, but WolframAlpha says x~~0.724492 and x~~-1.22074. I kept dividing by x it but...

I've heard there's a proof out there of this, basically that (I think) you can use the intermediate value theorem to prove that an Nth-degree polynomial has no more than N roots. I'm not in school anymore, just an interested engineer. Does anyone know where I can find this proof or any...

Unique nth Roots in the Reals -- Rudin 1.21 In Principles of Mathematical Analysis 1.21, Rudin sets out to show that for every positive real x there exists a unique positive nth root y. The proof is rather long and I would like to zoom into the portion of it where it seems that Rudin takes too...

I'm slightly confused about why complex numbers are required to find all n roots of a number. Is it specifically because of the fact that you can represent complex numbers as a rotation of the plane? I understand why a number should have n roots, I'm just not sure which part of the definition of...

Is the discriminant, of the quadratic equations, the difference between the two roots? Or is it a special case?

Homework Statement Given 2 + 3i is one root of the equation: x^4 - 6x^3 + 26x^2 -46x +65 = 0 Find the remaining roots. The attempt at a solution I am thinking about this as a possible solution although it is too long to be plausible, unless I'm wrong. let x = a + bi and then replace in the...

Homework Statement Evaluate the following in x+iy form. i3+i Homework Equations i=rei\Theta The Attempt at a Solution i=rei\frac{pi}{2} i3+i = i3*ii (-i)(ei2\frac{pi}{2}) =-ie-\frac{pi}{2} =-.2079i I understand how it all works out except \frac{pi}{2}. I can't figure out how...

Homework Statement The purpose of this program is to calculate the approximate roots of the Sine function on given intervals. The intervals are input by the user, and then the do loop continues until the condition (m becomes very close to 0 or equals 0) is met. The Attempt at a Solution...

Homework Statement Prove that \Sigma^{n}_{k=1} wk = 0 and there has to be at least two phasors/exponentials Homework Equations complex analysis The Attempt at a Solution I tried writing out the sigma on the first line. Then I tried writing the same thing with n+1 on the...

I had this assignment in math class and it doesn't just add up. x + square(5x + 10) - 8 = 0 Nothing too difficult to solve. And the answer is: x1 = 18 if square(5 * 18 + 10) = -10 x2 = 3 if square(5 * 3 + 10) = 5 But teacher says that x1 is not correct. To me it is not logical...

Problem: find number of roots of z^n + a_{n-1}z^{n-1} + ... + a_0, |z| < 1 What is wrong with this argument: Let f(z) = z^n + a_{n-1}z^{n-1} + \cdots + a_0, and g(z) = - a_{n-1}z^{n-1} - ... - a_0. Then, |f| > |g| and f+g = z^n. by Rouche thm, number of roots of f is equal to number of roots...

Homework Statement Hey! I don't really face a problem at the moment but i'd like someone to act as another brain for me. I've solved - hopefully right - a quartic equation and i get all the 4 roots out but 2 of them turn out to be complex numbers. Now, I'm fine with that but i'd also need...

Homework Statement Hi. I actually understand most of this question, but not the parts in red. Question. http://img703.imageshack.us/img703/7237/2008testhphysf.jpg If above doesn't load, please go to [PLAIN]http://img703.imageshack.us/img703/7237/2008testhphysf.jpg Homework...

Dear Fellow, I need to find the roots of equation x^8- 2x^6+3x^4-7x^2+a=0 in MATLAB, in such a way that I need 4 non repeating roots. I have used the technique of p=[1 -2 3 -7 1] s=roots(p) x1=s(1) x2=s(2) x3=s(3) x4=s(4) where xi,i=1,2,3,4 are roots of...

In general, how many square roots does a complex number have?

Homework Statement Find all the roots of z^{6} = -2Homework Equations z = \sqrt[n]{r} (cos(\frac{\theta}{n} + k\cdot \frac{2\pi}{n}) + isin(\frac{\theta}{n} + k\cdot \frac{2\pi}{n}) where k = 0, 1, \cdots, n - 1The Attempt at a Solution I know that r = \sqrt{x^2 + y^2} = \sqrt{(-2)^2} = 2 but I...

Homework Statement Find roots of complex equation (1-x)^5 = x^5 Homework Equations Probably Euler and/or De Moivre. The Attempt at a Solution I know i need to have z^5 on one side and all the rest on the other side. But i need 5 roots, and I'm only getting one. (1-x)^5 = x^5...

[SOLVED] Homework Statement Let 4x^2-4(α-2)xα-2=0 (α\epsilon R) be a quadratic equation, then find the value of α for which both the roots are positive. Homework Equations The Attempt at a Solution the conditions will be 1) Discriminant D≥0 by this condition i got α (-∞,2][3,∞) 2)...

Hi, I recently bought an HP50G, and I'm having trouble figuring out how to get numbers from the stack into a complex number. Could anyone help? For example, for 1+2i, I know I'd enter it as (1,2). But when I have something like 6^0.5+2i, I don't want to type the numbers out. Thanks Edit...

Homework Statement If the equations ax2+bx+c=0 and x3+3x2+3x+2=0 have two common solutions then show a=b=c. Homework Equations The Attempt at a Solution first equation will be the factor of second. taking out common from first equation. how to show a=b=c?? please provide...

In general, if α(alpha) and ß (beta) are roots of eqn. ax^2 +bx +c=0 then for finding the equation whose roots are α+2 and ß+2 can be done by addition of roots (α+2+ß+2=-b/a) and product of roots (α+2)(ß+2)=c/a By solving this we get ax^2 -(4a-b)x + (4a-2b+c)=0 The problem is this...

I am trying to use MATLAB to find the roots of a quadratic by the standard iterative techniques. I am totally on top of all this work in theory and in practice when it comes to doing it with a calculator or Excel, but I have never used MATLAB before and I have been given the code below to use...

Homework Statement θ^3 - pθ^2 +qθ - r = 0 such that p and r do not equal zero If the roots can be written in the form ak^-1, a, and ak for some constants a and k, show that one root is q/p and that q^3 - rp^3 = 0. Also, show that if r=q^3/p^3, show that q/p is a root and that the product...

Hi, Below is a fortran program to calculate the roots of an equation by Newton's method. I compile the program with the free compiler g95: g95 -c Newton.f95 then g95 Newton.f95 -o Newton.exe When I run the program Newton.exe I get the error can't assign value INTENT(IN)::x to...

Why the roots of Eq. x^2 + a*x + b = 0 and of Eq. x + a*Sqrt[x] + b = 0 are not identically? How can I expand the second Eq. in simple fractions: x + a*Sqrt[x] + b = ... ? Thank you. Lucas

Why the roots of Eq. x^2 + a*x + b = 0 and of Eq. x + a*Sqrt[x] + b = 0 are not identically? How can I expand the second Eq. in simple fractions: x + a*Sqrt[x] + b = ... ? Thank you. Lucas

I've been doing some work and I keep running into polynomials of the following form: P(x,y,z) = ax^2 + by^2 + cz^2 + 2(exy + fxz + gyz) \mod p where a,b,c \in \mathbb{Z}_p/ \{0\} and d , e, f \in \mathbb{Z}_p . It would be great if I knew anything about the existence of roots of P ...

Does anyone know where I can review finding complex roots? For example, x3= 8 I know the roots are 2, -1+isqrt(3), and -1-isqrt(3), but I can't remember how to figure it out.

Homework Statement Show that p(x) = x^2 - \sqrt2 is irreducible in \mathbb Z[\sqrt 2] . The Attempt at a Solution I think I have this, but I just want to make sure my reasoning is correct. I'm sure there are other ways. Firstly, it is sufficient to show that p(x) is irreducible in the...

I have been seeing a few during in my practice questions which leaves me worrying. If it is a quadratic function, the irrational numbers can be easily obtained using the equation. However, I got a question today which eventually took this form: 28D3+36D2-41D2+4 = 0 (I reevaluated...

Homework Statement Find the general solution to y'''+y=0 Homework Equations The Attempt at a Solution y''+y=0 r^3+1=0 r^3=-1 r=(-1)^(1/3) --------------------- -1=-1+0i -1=cos(pi)+isin(pi)=e^(i*pi) -1=cos(pi+2xpi)+isin(pi+2xpi)=e^i(pi+2xpi) --------------------------...

Hi, I was looking at the derivation of the equation for a hyperbola on Wolfram Mathworld. In one step, the webpage instructs you to "complete the square". It starts with: \sqrt{\left(x -c\right)^{2} +y^{2}}-\sqrt{\left(x+c\right)^{2}+y^{2}} = 2a and then says, "rearranging and completing...

Casus irredicibilis describes a situation where a cubic polynomial with integer coefficients has three distinct real roots, but you can't express those roots using just real numbers and radicals. Instead, if you want to only use radicals, then you must use complex numbers, even though the roots...

Homework Statement if p is a prime of the form p=4k+1 and g is a primitive root of p, show that -g is a primitive root. I'm not sure if this is a decent proof or not. My final argument looks suspicious. Any thoughts? Thanks Tal The Attempt at a Solution First, notive that...

Hi, I'm currently doing a project and this topic has come up. Are there any known famous classes of polynomials (besides cyclotomic polynomials) that fit that description? In particular, I'm more interested in the case where the polynomials have odd degree. I know for example that the roots of...

Homework Statement This isn't really a question on its own, rather a step in the solution to another question: How would I prove that y= A\cos x + B\sin x (A, B arbitrary constants) has at least n zeroes in the interval [\pi , \pi (n+1)] where n\in\mathbb{Z}\;? (I don't need to be too...

i have this function f(x) = (1 − 6x^2)^-1 and I need 21 roots between [-1,1] equidistant (the points to be at same distance from one to another) can i find the roots with some function in matlab? i found out just the polyval function for polynom

Homework Statement Find the roots of y-y^3=1 Homework Equations The Attempt at a Solution Factor theorem doesn't help in this equation. Found from Wikipedia, a very complex formula is needed to solve cubic eq. Can someone show me the steps to find the root? Thank you ^^