Roots Definition and 978 Threads

#85

As a preface to this theorem stated in my text, it states that: "If all the coefficients of a polynomial ##P(x)## are real, then ##P## is a function that transforms real numbers into other real numbers, and consequently, ##P## can be graphed in the Cartesian Coordinate Plane." It then goes on...

Hi. I have coefficient of x2 as in an expression that looks like this * calculator shows little yellow triangle because 'x' is not defined. If I can write the coefficient of x2 as - 0.091372213746554 then why did the author write coefficient of x2 like this shown below? Thanks.

Does anyone know of any resources on questions on primitive roots and order of a modulo n? They need to be suitable for elementary number theory course. (These could be interesting results and challenging ones).

I am teaching elementary number theory to first year undergraduate students. How do introduce the order of an integer modulo n and primitive roots? How do I make this a motivating topic and are there any applications of this area? I am looking at something which will have an impact.

I am teaching elementary number theory to first year undergraduate students. How do introduce the order of an integer modulo n and primitive roots? How do I make this a motivating topic and are there any applications of this area? I am looking at something which will have an impact.

<Moderator's note: Moved from a technical forum and thus no template.> Hi everyone, I have encountered a partial differential equation with square roots which I don't have a clue in solving it. After letting z=F(x)+G(y), I can't really figure out the next step. I tried squaring both sides but...

Hi everyone. One of my pet hobbies/interests is in history, and I've done some background reading on early American and Canadian history, including the history of immigration to the US. One of the questions that I have is what percentage of white Americans are descended from those Europeans...

Homework Statement Find roots of $$ -\lambda ^3 +(2+2i)\lambda^2-3i\lambda-(1-i) = 0 $$ Homework EquationsThe Attempt at a Solution I tried my old trick I tried to separating the 4 terms into 2 pairs and try to find a common factor in the form of ##\lambda + z## between them, $$ -\lambda ^2...

Hello all, I was trying to find derivatives of two functions containing square roots. I got answers which I believe should be correct, however, the answers in the book differ significantly. The first answer of mine was checked in MAPLE and found correct. My guess that the author made some...

One of the solutions to x4-2x3+kx2+px+36 = 0 is x = 3i Prove that this polynomial has no real solutions (roots) and find the real values of k and p. ------------------------------------------------------------------------------------------------------------------- So far the only progress...

Here a, b, c > 0, and a > bc. Can anyone find the solution of k as a function of (a, b, c)? Thanks.

Given the equation $$x^n + a_1x^{n-1}+a_2x^{n-2}+…+a_n = 0$$ - with real coefficients, and $a_1^2 < a_2$. Show that not all the roots are real.

Let $r_1,r_2, …,r_7$ be the distinct roots (one real and six complex) of the equation $x^7-7= 0$. Let \[p = (r_1+r_2)(r_1+r_3)…(r_1+r_7)(r_2+r_3)(r_2+r_4)…(r_2+r_7)…(r_6+r_7) = \prod_{1\leq i<j\leq 7}(r_i+r_j).\] Evaluate $p^2$.

I am so sorry for having posted this challenge/puzzle with a serious typo: The roots of the equation should be functions of $a, d$ and $e$. In my old version I wrote $a, b$ and $e$. I will see to, that future challenges are properly debugged before posting.For $e \ne 0$, determine the roots...

Given that $\alpha$ and $\beta$ are the roots of a quadratic equation, evaluate $\frac{1}{\alpha^2}+\frac{1}{\beta^2}$. I find this question to be interesting.

I can't find the zeros to 4x^5-10x^4-14x^3+49x^2-28x+4 I found my positive zeros, 2, 1/2 using synthetic division and possible zeros. But from there I'm stuck.

Homework Statement Prove that if ##p(x) \in \Bbb{Q}[x]## is an irreducible polynomial, then ##p(x)## has no repeated roots in ##\Bbb{C}##. Homework Equations I will appeal to the theorem I attempted to prove here...

Homework Statement Let ##f(x) = (x-a_1)...(x-a_n) \in k[x]##, where ##k## is a field. Show that ##f(x)## has no repeated roots (i.e., all the ai are distinct elements in ##k##) if and only if ##gcd(f,f')=1##, where ##f'(x)## is the derivative of ##f## Homework Equations ##(x-a)^2 |f(x)##...

Homework Statement [/B] $$\int \left ( \frac{-1}{2*\sinh(x)*\sqrt{1-e^{2x}})} \right ) dx$$ or http://www.HostMath.com/Show.aspx?Code=\int \left ( \frac{-1}{2*\sinh(x)*\sqrt{1-e^{2x}})} \right ) dx Homework Equations the sinh identity, which is (e^x-e^-x)/2 The Attempt at a Solution Tried...

Homework Statement 4(d2x/dt2) +3x = t*e-3tsin(5t) Homework EquationsThe Attempt at a Solution So I know how to take the Laplace transform and find the function for the Laplace domain: X(s) = 10(s+3)/(((s+3)2+25)2)(4s2+3) + (10s/(4s2+3)) + (2/(4s2+3)) But trying to convert...

The square root of any integer that is not a square number is always an irrational number. I find this fact rather spectacular, but my question is why is this true? I have seen the formal proof for the irrationality of root 2 so I could vaguely see how one could prove that all (apart from sq...

Homework Statement Prove that the roots of trigonometric polynomials with integer coefficients are denumerable. Homework EquationsThe Attempt at a Solution The book does not define what a trig polynomial is, but I am assuming it is something of the form ##\displaystyle a_0 + \sum^N_{n=1}a_n...

Prove that polynomials of the form:\[P_n(x)=x^{2n}-2x^{2n-1}+3x^{2n-2}-...-2nx+2n+1, \: \: n = 1,2,...\]- have no real roots.

If the square root as two coordinate axes in the complex plane, does the cubic root has 3 coordinate axes and so on for nth root? @vanhees71 Can you please explain this?

Hello I have a third order polynomial, for example y(x) = -60000x^3 - 260x^2 + 780x + 0.6 I need to know what is x at y = 28 and/or y= 32. I can goto MATLAB and find the roots ( x = - .1158, -.0007, and .1122 ) or I can go to http://www.wolframalpha.com and it also finds the roots and...

Homework Statement 1)The value of k, so that the equations 2x2+kx-5=0 and x2-3x-4=0 have one root in common 2)The value of m for which one of the roots of x2 is double of one of roots of x2-x+m=0 3)If x2-ax-21=0 and x2-3ax+35 have a root in commom Homework EquationsThe Attempt at a Solution I...

Hello. Assume that I have two polynomials of degree 2, i.e., Quadratic Equations. 1. Assume that the Quadratic Equation is: x2 + 7x + 12 = 0 The roots of the Quadratic Equation is -3 and -4. 2. Assume that there is another Quadratic Equation: x2 + 8x + 12 = 0 The roots of the Quadratic...

Section 3.1 Question 2dFind the domain of the function. Let CR = CUBE ROOT s = CR{3t + 12} Because it is a cube root, I say the domain is ALL REAL NUMBERS.

√(2-√(2^(2)-1))+√(4-√(4^(2)-1))+√(6-√(6^(2)-1))+...+√(80-√(80^(2)-1)) How the find it's value

i need to conduct an experiment on root of a plant that has alkaloids in it, but i don't know what plants does. can you guys suggest any plants to me? it'll be better if the plant is common to find thanks in advance

When solving radical equations, we must check for extraneous roots. What are extraneous roots?

In a certain type of problem, a quadratic equation is formed with the square root of energy being the variable to be found ex: (a*sqrt(E)^2+b*sqrt(E)+c=0). Then they claim since energy (E) is real and positive, only solutions to the quadratic equation in sqrt(E ) being real and positive are...

How come a+a- ψn = nψn ? This is eq. 2.65 of Griffith, Introduction to Quantum Mechanics, 2e. I followed the previous operation from the following analysis but I cannot get anywhere with this statement. Kindly help me with it. Thank you for your time.

Are there practical uses for the formulas for the sum and product of quadratic roots? I have only seen the topic for these sum and product formulas in one section of any college algebra and intermediate algebra books, and then nothing more. I'm just curious if people, ... scientists or...

Homework Statement I have a problem with lines in analytic geometry, and I solved it in a certain way (parallel lines interceptions) which gives the correct result, and I'm happy with that. There was another method I thought I could use to solve it though, which went through the formulas of...

Show that the product of the roots of the equation x^2 + px + q = 0 is q. I need help with the set up. Must I use the discriminant here?

Show that the sum of the roots of the equation x^2 + px + q = 0 is -p. I need help with the set up. Is the discriminant involved here?

Homework Statement Find roots of the EQN: r^3-r^2+1=0 Homework Equations none The Attempt at a Solution r^2(r-1)+1=0 from there i solved, r^2=-1 and r-1=-1 to find the following roots: r=+i,r=-i, r=0 Is my method correct? Also, I don't think that synthetic division would work here since my...

Homework Statement Given that ##f(x) = x^2 - bx + 1##, find the values of b for which at least one of the roots are positive Homework EquationsThe Attempt at a Solution So first I used the quadratic equation to find the roots: ##\displaystyle x = \frac{b \pm \sqrt{b^2 - 4}}{2}##. Now, given...

How does the value of ##\displaystyle \sqrt[a]{-1}## vary as ##a## varies as any real number? When is this value complex and when is it real? For example, we know that when a = 2 it is complex, but when a = 3 it is real. What about when ##a = \pi##, for example?

1) the problem I understand Newton's method and I was able to find all the real roots of the function.However, I don't understand how to find the complex roots. I know that z=x+yi, and that I can plug in z for the formula. However I, don't know how to change the function (...

The number of real roots of the equation $$2cos \left( \frac {x^2 + x} {6} \right)=2^x + 2^{-x}$$ Answer options are : 0,1,2,∞ My approach : range of cos function is [-1,1] thus the RHS of the equation belongs to [-2,2] So, we have -2 ≤ 2x + 2-x ≤ 2 solving the right inequality, i got 2x...

In this Khan Academy video they say that it is ok to break the square root ##\sqrt{a\cdot b}##, with ##a, b \in \mathbb{R}##, into the product of two square roots ##\sqrt{a}\cdot \sqrt{b}##, only when: (1) both are non negative, (2) one of the two is negative and the other is possitive. I...

I'm using this method: First, write the polynomial in this form: $$a_nx^n+a_{n-1}x^{n-1}+...a_2x^2+a_1x=c$$ Let the LHS of this expression be the function ##f(x)##. I'm going to write the Taylor series of ##f^{-1}(x)## around ##x=0## and then put ##x=c## in it to get ##f^{-1}(c)## which will be...

Homework Statement [/B] For which values of c, cER, will the equation j(x) = c have real roots? Homework Equations j(x) = 2x^2 - 8x + 5 The Attempt at a Solution [/B] I understand I need to get this into the form of b^2 - 4ac, yet I do not understand why this is important and such. From my...

Homework Statement Algebra - I.M. Gelfand, Problem 164. Prove that a polynomial of degree not exceeding 2 is defined uniquely by three of its values. This means that if P(x) and Q(x) are polynomials of degree not exceeding 2 and P(x1) = Q(x1), P(x2) = Q(x2), P(x3) = Q(x3) for three different...

Homework Statement z2-(3+i)z+(2+i) = 0 Homework EquationsThe Attempt at a Solution [/B] Does the quadratic formula work in this case? Should you deal with the real and complex parts separately?

How do you prove the identity 3 = \sqrt{1 + 2\sqrt{1 + 3\sqrt{1+4\sqrt{1 + \cdots}}}} with a real proof that actually proves the convergence? I know there are "proofs" that "prove" the identity with some trickery that ignore all the convergence issues, and I'm not interested in those trickeries.

Can someone help me distinguish when Solve[polynomial==0,x] returns exact solutions as opposed to solutions in terms of Roots? For example, if I code: myroots = x /. Solve[1/2 + 1/5 x + 9/10 x^2 + 2/3 x^3 - 1/5 x^4 + 3/11 x^5 == 0, x] this cannot be solved in terms of radicals so Solve...