Roots Definition and 978 Threads

If $a,b,c,d$ are distinct real no. such that $a=\sqrt{4+\sqrt{5+a}}\;,b=\sqrt{4-\sqrt{5+b}}\;,c=\sqrt{4+\sqrt{5-c}}\;,d=\sqrt{4-\sqrt{5-d}}$. Then $abcd=$

Homework Statement Homework Equations The given equations are; Q.1) Find the no of + roots of the equation x^4 -4x+1=0 Q.2) Find the no. of negative roots of the eqn x^4-4x-1=0 Q.3) Find the no of complex roots of the eqn x^4-4x-1=0 The Attempt at a Solution

Homework Statement Use the rational roots theorem to prove 31/2-21/3is irrational. The Attempt at a Solution My teacher strongly hinted to us that this problem had something to do with the fact that complex roots come in conjugate pairs, and all we had to do was, "flip the sign"...

I have given polynomial: x^3-8x^2+19x-12 I know how to find the roots with Horners method,i am just wondering if there is an easier and quicker way to find them? Thank you!

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

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 Attempt: I...

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 \cdot \frac {2\pi}5 \right)...

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...