Roots Definition and 962 Threads

how does $$\frac{5700}{\sqrt{15,300}}$$ turn into $$\frac{570}{\sqrt{153}}$$ ??

Determine the sum of real roots of the equation $q^4-7q^3+14q^2-14q+4=0$.

Homework Statement p prime, If p=1 ( mod 3) then Zp contains primitive cube roots of unity. Now I am considering which p does Zp contains primitive fourth roots of unity. opposite way? I mean if p=1(mod4) then Zp contains primitive fourth roots of unity?? 2. The attempt at a solution I...

Homework Statement If [sinx]+[√2 cosx]=-3, x belongs to [0,2∏] (where [.] denotes the greatest integer function) then x belongs to 1)[5∏/4,2∏] 2)(5∏/4,2∏) 3)(∏,5∏/4) 4)[∏,5∏/4] The Attempt at a Solution If I put x=5∏/4, LHS=-2 which does not satisfy. 2∏ and ∏ also does not...

why does a nth degree polynomial has atleast one root and a maximum of n root...? In my book it's given, it's the fundamental theorem of algebra. Is there a proof...? Thank's for help. (In advance)

Hi, I am new here and I don't know if anyone is going to answer to this post, but if you do so thank you very much. I have been frowning on these kind of problems! I have been trying to solve some exercices from my homeworks. However, I don't know if I am doing them correctly. Here is one...

I have stumbled upon an approximation to the average of integer square roots. \sum^{n}_{k=1}{\sqrt{k}/n} \approx \sqrt{median(1,2,...,n)} Sorry I am not very good at LaTeX, but I hope this comes across okay. Could anyone explain why this might be happening? In fact, I just discovered that...

Homework Statement Hello guys, I need to prove that the sum of complex roots are 0. In the Boas book, it is actually written 'show that the sum of the n nth roots of any complex number is 0.' I believe it's equivalent. The Attempt at a Solution I have managed to obtain this summation. It is...

this is not a homework question, but rather I feel like there is a contradiction in the theorem and just want clarity. I know the theorem is correct so I am looking for help in where the mistake is in my logic. take f(x) = x^3 + x^2 - 4x- 7 the rational roots theorem says if there are any...

I have to find all solutions for X when: x4-2x3-25x2+50x I have done it so,but I am not sure if this is ok: x(x3-2x2-25x+50) = x(x2(x-2)-25(x+2) = x(x2-25)(x-2) =x(x-5)(x+5)(x-2) Now i see that root/zeroes are +5,-5 and 2. I know that this polynomial has another zero that is 0,but how do i...

Hi, I'm somewhat new here, only posted a few times, and would like some help from you guys here if possible I'm stuck with a problem on the topic mentioned. x'=Ax A is a 2*2 matrix A = [-5 1] [-1 -3] Now I managed to find the eigenvalues which is -4, repeated twice (multiplicity 2) And the...

Let $f(x)=x^3-3x+1$. Find the number of distinct real roots of the equation $f(f(x))=0$.

Checking solutions -- textbook wrong about roots? If I have the equation sqrt(3x + 1) = x - 3 and I need to solve for x, by squaring both sides then solving the resulting quadratic, I get the solutions x = 1, 8 However, since I squared the equation, I need to check if the solutions are...

Homework Statement Let f(x) be a non-constant twice differentiable function defined on R such that f(x)=f(1-x) and f'(1/4) =0 then what is the minimum number of real roots of the equation (f"(x))^2+f'(x)f'''(x)=0. The Attempt at a Solution f'(x)=-f'(1-x) f"(x)=f"(1-x)...

Hi MHB, This is the second headache problem that I wish to get some insight from MHB today... Problem: It is known that the equation $$ax^3+bx^2+cx+d=0$$ has three distinct real roots. How many real roots does the following equation have? $$4(ax^3+bx^2+cx+d)(3ax+b)=(3ax^2+2bx+c)^2$$...

Homework Statement 23 - 5x - 4x2 = 0 find (\alpha - \beta)2 Homework Equations In previous parts of the question I've calculated \alpha + \beta, \alpha\beta, 1/\alpha + 1/\beta and (\alpha+1)(\beta+1) but I can't think of any rules I know to help me solve the problem. The...

Let $p$ be the sum and $q$ be the product of all real roots of the equation $x^4-x^3-1=0$. Prove that $q<-\dfrac{11}{10}$ and $p>\dfrac{6}{11}$.

Can someone show me step by step guide,how to find all possible solutions for example?

Hello everyone, What is the square root of a square of a negative number equal to? For example: \sqrt{-1}^{2} It seems there are two possible ways of doing this, the problem is that I am getting two different answers using these two approaches i.e; We can first take the square of -1 and then...

I just got a new Ti-nspire Cx CAS calculator and I am having trouble with being able to express a Radical in simplified form when there are exponents and variables of x and y. My problem is that this calculator will not show the simplified form correctly. I have taken others advice in setting...

Hi MHB, I have encountered a problem recently for which I couldn't think of even a single method to attempt it, and this usually is an indicator that a problem really isn't up my alley. That notwithstanding, I don't wish yet to concede defeat. Could someone please show me at least some...

Solve the following quadratic equation. Use factorisation if possible. X2 - 4X - 8 = 0 Normally I wouldn't have trouble factorising a quadratic, but I have just been introduced to a new way to do it and I want to use this way to answer the question. Here's how far I get, then I'm unsure what...

Homework Statement I have the limit: lim [ (x+h)^2/3 - x^(2/3) /h ] How would I further simplify and evaluate this limit. 2. The attempt at a solution I have tried using a change of variable and using this in the sum of cubes formula (i.e. (x+h)^(2/3) = a, x^(2/3) = b, and then plugging...

Homework Statement 1. Let ##p(x) = a_{0} x^{n} + a_{1} x^{n−1} + ... + a_{n} , a_{0} \neq 0 ##be an univariate polynomial of degree n. Let r be its root, i.e. p(r) = 0. Prove that ## |r| \leq max(1, \Sigma_{1 \leq i \leq n} | \dfrac{a_{i} }{ a_{0} } | )## Is it always true that? ## |r| \leq...

Find the four roots of the equation $(x-3)^4+(x-5)^4+8=0$.

I think a lot a users have vague concepts about the roots of unity. I try to post a link to WolframAlpha, which calculates all the second roots of unity http://www.wolframalpha.com/input/?i=sqrt%281%29 There you can see the input \sqrt{1} and the plot of all roots in the complex plane...

I found that the cartan subalgebra of ##sp(4,\mathbb{R})## is the algebra with diagonal matrices in ##sp(4,\mathbb{R})##. Now to find out the roots I need to compute: ##[H,X]=\alpha(H) X## For every ##H## in the above Cartan sublagebra, for some ##X \in sp(4,\mathbb{R})## Now, I know that...

Here is the question: I have posted a link there to this topic so the OP can see my work.

How do I reduce u^4 + 5u^3 + 6u^2 + 5u + 1 = 0 to v^2 + 5v + 4 = 0 by using v = u + 1/u ?

The product of two of the roots of the equation ax^4 + bx^3 + cx^2 + dx + e = 0 is equal to the product of the other two roots. Prove that a*d^2 = b^2 * e

Obtain the sum of the squares of the roots of the equation x^4 + 3x^3 + 5x^2 + 12x + 4 = 0 . Deduce that this equation does not have more than 2 real roots . Show that , in fact , the equation has exactly 2 real roots in the interval -3 < x < 0 . Denoting these roots α and β , and the other...

This is a well-known result in complex analysis. But let's see what people come up with anyway: Challenge: Prove that there is no continuous function ##f:\mathbb{C}\rightarrow \mathbb{C}## such that ##(f(x))^2 = x## for each ##x\in \mathbb{C}##.

The roots of the equation x^3 - x - 1 = 0 are α β γ and S(n) = α^n + β^n + γ^n (i) Use the relation y = x^2 to show that α^2, β^2 ,γ^2 are the roots of the equation y^3 - 2y^2 + y - 1 =0 (ii) Hence, or otherwise , find the value of S(4) . (iii) Find the values of S(8) , S(12) and S(16)I have...

Homework Statement I was trying to figure out whether or not ##\zeta_5 + \zeta_5^2## and ##\zeta_5^2 + \zeta_5^3## were complex (where ##\zeta_5## is the fifth primitive root of unity). Homework Equations The Attempt at a Solution ##\zeta_5 + \zeta_5^2 = \cos(2\pi/5) + i\sin(2\pi/5) +...

How many real and non-real roots does $$z^5 = 32$$ have? $$z^9 = -4$$? For $$z^5 = 32$$: $$z^5 = r^5 ( \cos 5v + i \sin 5v )$$ and $$32 = 32 ( \cos 0 + i \sin 0 )$$ yields $$r = 2 \\ 5v = n \cdot 2\pi \iff v = n \cdot \dfrac {2\pi}5$$ So all roots are given by $$z = 2 \left( \cos \left( n...

Homework Statement Find all complex solutions of z^6 + z^3 + 1 (z^3 + 1)/(z^3 - 1) = i Homework Equations The Attempt at a Solution I am going crazy with trial and error with these, there must be some systematic method or tricks that I am oblivious of. For the second question I...

Here is the question: I have posted a link there to this topic so the OP can see my work.

Find the sum of the squares of the roots of the equation x^3 + x + 12 = 0 and deduce that only one of the roots is real . The real root of the equation is denoted by alpha . Prove that -3< alpha < -2 , and hence prove that the modulus of each of the other roots lies between 2 and root 6 . I...

Hi, I'm getting a bit confused about the adjoint representation. I learned about Lie algrebras using the book by Howard Georgi (i.e. it is very "physics-like" and we did not distinguish between the abstract approach to group theory and the matrix approach to group theory). He defines the...

Homework Statement Let ##ζ_3## and ##ζ_5## denote the 3rd and 5th primitive roots of unity respectively. I was wondering if I could write the product of these in the form ##ζ_n^k## for some n and k.Homework Equations The Attempt at a Solution We know that ##ζ_3## is a root of ##x^3=1##, and...

Hi there, I was wondering if it is possible to find the roots of the following polynomial P(x)=x^n+a x^m+b or at least can I get the discriminant of it, which is the determinant of the Sylvester matrix associated to P(x) and P'(x). Thanks

Hello everyone, I am a bit confused about definitions rules. I can have more questions but for now I want to ask only one question: Let us say I have a number: \sqrt[6]{3x3x3x3x3x3} 3x3x3x3x3x3 is equal to both 27^2 and (-27)^2. But If I write these two expressions separately I can get...

Homework Statement Let ##a,b,c## and ##m \in R^{+}##. Find the range of ##m## (independent of ##a,b## and ##c##) for which at least one of the following equations, ##ax^2+bx+cm=0, bx^2+cx+am=0## and ##cx^2+ax+bm=0## have real roots.Homework Equations The Attempt at a Solution I don't really...

Hi, I have a very basic question that suddenly hit me regarding square roots. Why this is equal \[\sqrt[3]{(1+x^{3})^{2}}=(1+x^{3})^{^{\frac{2}{3}}}\] but this isn't \[\sqrt{(x-2)^{3}}\neq (x-2)^{\frac{3}{2}}\] (well according to Maple it isn't) I understand why the first one is correct...

x^3-5x-6=0 i've tried the p/q calculations in accordance with the rational roots theorem but I've yet to find the answers...

Without the use of a calculator, and showing your work, simplify: $$\frac{1}{2}\left(\left(239+169\sqrt{2}\right)^{ \frac{1}{7}}-\left(29\sqrt{2}-41\right)^{ \frac{1}{5}}\right)$$ edit: My apologies...I was careless in my first statement of the problem...(Nod)

Homework Statement Solve for z^4 +16=0Homework Equations The Attempt at a Solution What I first did was square rooted both sides to get z^2 = ±4i, but I don't how to continue from there. I'm guessing we have to find the roots from z^2=4i and then from z^2=-4i separately any help will be much...

If $$a,\;b$$ are roots of polynomial $$x^4+x^3-1$$, prove that $$a(b)$$ is a root of polynomial $$x^6+x^4+x^3-x^2-1$$.

Let $a,b,c$ be positive real numbers with sum $3$. Prove that $√a+√b+√c≥ab+bc+ca$.

The no. of Real Roots of the equation $\displaystyle \frac{\pi^e}{x-e}+\frac{e^\pi}{x-\pi}+\frac{\pi^{\pi}+e^{e}}{x-\pi-e} = 0$ My try:: Let $\displaystyle f(x) = \frac{\pi^e}{x-e}+\frac{e^\pi}{x-\pi}+\frac{\pi^{\pi}+e^{e}}{x-\pi-e}$ Now we will take a interval $x\in \left(e\;,\pi\right)$...