1
vote
1answer
54 views

I need help proving a statement about rational roots

I have no idea where to start...this is the statement: If a polynomial of degree not greater than 5 with rational coefficients has multiple roots, it has also a rational root, except in the case ...
6
votes
1answer
92 views

zeros of a polynomial

Given $P(z)=z^6+6z+10$, find how many roots are in each quadrant I have already seen that $P(z)$ has six different roots, and that none of them are real or of the form $ki$, $k\in \Bbb R$. Since ...
0
votes
2answers
55 views

$x^3+3x^2+4x+5=0$ and $x^3+2x^2+7x+3=0$, how many common roots they have?

My attempt, Equate both, at the end you will get $x^2-3x-2=0$ That means $x=-1$ and $x=2$. But what after that. Please provide solutions as well.
12
votes
3answers
166 views

Why is this polynomial a monomial?

Let $p$ be a polynomial of degree $n$ such that $|p(z)| = 1$ for all $|z| = 1$. Why is it that $p(z) = az^n$ for some $|a| = 1$? I've noticed that we could easily prove this by induction if we ...
-4
votes
2answers
119 views

Select the approximate values of x that are solutions to $f(x) = 0$, where $f(x) = -7x^2 + 6x + 9$? [closed]

These are the answer choices: $$\begin{align*}\\ A&~~\{–0.78, 0.67\}\\ B&~~\{-7, 6\}\\ C&~~\{–0.86, –1.29\}\\ D&~~\{–0.78, 1.64\} \end{align*} $$
1
vote
1answer
32 views

Constant function with maximum modulus [duplicate]

Suppose that $f$ is analytic on a domain $D$, which contains a simple closed curve $\gamma$ and the inside of $\gamma$. If $|f|$ is constant on $\gamma$, then I want to prove that either $f$ is ...
1
vote
3answers
27 views

Roots of Polynomial Equation?

$y=1/x$ so I plugged in $x=1/y$ into the equation above and got $y^{4}+y^{3}+y^{2}/c+y/4-1/2$, but apparently it's wrong, when I looked up the answer below. What am I missing?
3
votes
4answers
112 views

Find roots of $3^x+x^3=17$

Find $x$ in the following equation: $$3^x+x^3=17$$
2
votes
3answers
93 views

Determine the number of zeros in the first quadrant $f(z) = z^4- 3z^2 + 3$ [closed]

Determine the number of zeroes of the following function which are in the first quadrant: $$f(z) = z^4- 3z^2 + 3$$ Help please!!! I'm not that good at complex variables!
1
vote
3answers
60 views

A polynomial's roots

Let $Q_n(x) = (x^2-1)^n$ and $P_n(x) = Q_n^{(n)}(x)$. Using Rolle's theorem, prove that $P_n$ has exactly $n$ roots.
0
votes
1answer
20 views

Show that a Polynomial has certain factorization

$P(x)$ is a polynomial in $x$ of degree $\leq n-1$. Show that $P(x)$ has $n-1$ distinct roots and thus has the factorization $$k\Pi_{i=2}^n(x-a_i)$$, where the constant $k$ is the coefficient of ...
2
votes
5answers
205 views

How to solve $x^4-8x^3+24x^2-32x+16=0$

How can we solve this equation? $x^4-8x^3+24x^2-32x+16=0.$
3
votes
4answers
202 views

Find all roots of $x^{6} + 1$

I'm studying for my linear algebra exam and I came across this exercise that I can't solve. Find all roots of polynomial $x^{6} + 1$. Hint: use De Moivre's formula. I guessed that two roots are $i$ ...
1
vote
1answer
76 views

Find the solutions of the equation…

How can I solve this equation? $$ \begin{equation*} \sqrt[3]{x-2}+\sqrt{x-1}=5 \end{equation*} $$ Frankly, I just have no idea at all!!! Thank you in advance!
0
votes
2answers
70 views

How many iterations of the bisection method are needed to achieve full machine precision

Suppose that an equation is known to have a root on the interval $(0,1)$. How many iterations of the bisection method are needed to achieve full machine precision in the approximation to the location ...
1
vote
2answers
35 views

Separability of $f(x) = (x-1)^2(x-3)$ over $\mathbb{Q}$

This is an example in Ash, Basic Abstract Algebra, ch.3.4 page 73 at the bottom (or here on page 11). It states that $f(x) = (x-1)^2(x-3)$ over $\mathbb{Q}$ is separable. But, $f'(x) = ...
2
votes
2answers
140 views

Solve for p in $\frac1{20} = (1 - p)^{19}p$

I need help to solve for $p$, where $p$ is a probability, i.e. is between $(0,1)$. $\frac1{20} = (1 - p)^{19}p$ How would one solve for $p$? Thnx
2
votes
2answers
62 views

Numerically finding roots of function - converges?

Well this question was in my homework, I have difficulty to "proof" it (or more correctly: seeing how I would solve it). Consider a floating point system ($s \cdot b^e$ where $1\leq s \leq 10 - 1 ...
2
votes
1answer
31 views

Extend rolle's theorem to complex functions?

If $f(z)$ is a polynomial of degree n with n distinct real roots $r _1$<,..., <$r_n$, then there exists exactly one root of $f '(z)$ in between any consecutive root of $f(z)$. The context of ...
0
votes
2answers
75 views

Relationship between f and f' in terms of number of real/non-real roots

$f$ is a polynomial of degree $n\ge1$ and $\forall x,x\in \Bbb R \rightarrow f(x)\in\Bbb R$. Prove that: (a)$f$ has at most one more real root than $f'$ (b)$f'$ has no more non-real roots than $f$ ...
2
votes
1answer
94 views

Game of polynomials

Written on a blackboard is the polynomial $x^2+x+2014$.Calvin and Peter take turns alternatively (starting with Calvin) in the following game. During his turn, Calvin should either increase or ...
0
votes
2answers
63 views

Find roots of polynomial $f(X) = X^7 - 6 X^6 + 10 X^5 - 13 X^3 + 18 X^2 -22 X + 12 \in \mathbb Q[X]$

Find the roots of the polynomial $$ f(X) = X^7 - 6 X^6 + 10X^5 - 13 X^3 + 18 X^2 -22 X + 12 \in \mathbb Q[X] $$ in $\mathbb Q$, $\mathbb R$ and $\mathbb C$. We covered the factor-theorem in ...
2
votes
2answers
118 views

Find all solutions of $z^5+a^5=0$

The task is as follows: Find all solutions of $z^5+a^5=0$, where $a$ is a positive real number. My initial attempt (which leads nowhere) My guess is that i'll have to find the 5 5th roots of ...
0
votes
1answer
100 views

If f is n-times differentiable, and $f^n$ is never 0, then f has at most n zeros in R

Let $n \ge 0$, let $f:\mathbb{R} \rightarrow \mathbb{R}$ be n-times diff erentiable on $\mathbb{R}$, and assume that $f^{(n)}(x) \neq 0$ for all $x \in \mathbb{R}$. Show that $f$ has at most $n$ zeros ...
0
votes
1answer
82 views

roots of cubic - descartes and viete [closed]

Consider the equation $y^3 - 8y^2 - y + 8 = 0$. According to Descartes, how many roots does the equation have and how many are false roots? According to Viete, what is the product and what is the ...
0
votes
1answer
100 views

Find the minumum using Newton-Raphson

I have the following function: $f(x) = 100(x_2 - x_1^2)^2 + (1-x_1)^2$ I have to find the minimum of this function using the Newton Raphson method. The point where I have to start is $x = [1.2$, ...
2
votes
3answers
59 views

$\left(\frac1\alpha-\frac1\beta\right)^2$ for $p(x)=x^2+x-2$

If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $p(x)=x^2+x-2$, then $\left(\frac1\alpha-\frac1\beta\right)^2 is:$ A) $\frac94$ B) $\frac{-9}4$ C) $\frac25$ D) $\frac{-2}5$ This ...
3
votes
3answers
3k views

Prove using Rolle's Theorem that an equation has exactly one real solution.

So the question is; Prove that the equation $x^7+x^5+x^3+1=0$ has exactly one real solution. You should use Rolle’s Theorem at some point in the proof. And I have, Since $f(x) = x^7+x^5+x^3+1$ ...
0
votes
2answers
341 views

Finding polynomal function with given zeros and one zero is a square root

I've been having trouble with this problem: Find a polynomial function of minimum degree with $-1$ and $1-\sqrt{3}$ as zeros. Function must have integer coefficients. When I tried it, I got this: ...
3
votes
1answer
2k views

Find all real zeros of $f(x)=2x^3+10x^2+5x-12$

Hey guys I'm having a little trouble with one problem: Find all real zeros of $$f(x)=2x^3+10x^2+5x-12.$$ I got $x=-4,(2x^2+2x-3)$. I'm just having trouble using the quadratic formula to get ...
1
vote
4answers
223 views

Prove that the polynomial $x^6+x^4-5x^2+1$ has at least four real roots.

Prove that the polynomial $x^6+x^4-5x^2+1$ has at least four real roots. Talking analysis here, using the definition of continuity, intermediate value theorem, and extreme value theorem.
0
votes
3answers
86 views

What are the methods to find approximatly the 5th roots of an equations

By which method, can I find the nearest root of : $x^5−2x+1.1=0$ ? Thank you.
5
votes
2answers
155 views

Find asymptotics of $x(n)$, if $n = x^{x!}$

Find the asymptotic for $x(n)$, if $n = x^{x!}$. I've tried 1) to take a logarithm: $x! \log{x} = \log{n}$. 2) to find $n'(x)$, using gamma-function for factorial $\Gamma(z) = \int_0^\infty ...
1
vote
3answers
142 views

How to solve $e^{ax}+e^{bx}+e^{cx}+d=0$

How to solve an equation like $e^{ax}+e^{bx}+e^{cx}+d=0$ (i.e. to write $x=...$) where $a,b,c,d$ are fixed non-zero real numbers. I have tried assuming that $x=ln(y)$ for $y>0$ but it goes ...
0
votes
3answers
61 views

Find the roots of 2 equations

Show that the equation $e^{-x} = x^2$ has a root between $x=0.70$ and $x=0.71$. I think you have to use natural logs to get rid of the $e$ however after that, i'm not sure how to solve for $x$
4
votes
3answers
126 views

Show that the real part of the root of an equation is constant

I've been stuck for a while on the following question: Let $z$ be a root of the following equation: $$z^n + (z+1)^n = 0$$ where $n$ is any positive integer. Show that $$Re(z) = -\frac12$$ ...
0
votes
1answer
47 views

Trying to find the root of the derivative of the MLE for a simple linear regression model

I have a function $$l(\beta_0, \beta_1, \sigma^2) = -\frac{n \log(2\pi)}{2} - n \log \sigma - \frac{1}{2 \sigma^2} \sum_{i=1}^{n} (y-\beta_0 - \beta_1 x_i)^2$$ which is the log-likelihood function of ...
0
votes
1answer
111 views

Help with finding the roots of a function

I have the following problem: Find the smallest root of the function $e^{-x} = \sin (x)$ and focus the root with Newton's method to $8$ decimal accuracy. Any suggestions? :) Thank you for any ...
2
votes
3answers
76 views

In a numerical system of base $r$, the polynomial $x^2 − 11x + 22 = 0$ has the solutions $3$ and $6$. What is the base r of the system?

From Algebra, the statement is equivalent to say that $(x^2− 11x + 22)_{r}$ = $(x − 3)_{r} \cdot (x − 6)_{r}$. Doing operations we arrive at $3 + 6 = 11_{r} = r + 1$, and $(3)(6) = 22_{r} = 2 \cdot ...
3
votes
2answers
11k views

Quadratic equation - Alpha and Beta Roots

If α and β are the roots of the equation x² + 8x - 5 = 0, find the quadratic equation whose roots are α/β and β/α. My working out so far: I know that α+β = -8 and αβ = -5 (from the roots) and then i ...
1
vote
1answer
164 views

finding value of constant such that function has distinct root

we have the function cubic function $$ x^3 -12x +k =0 $$ it has distinct root in $$ [0,2{]} $$ that task given to us is to find the the value of k satisfying the above conditions I proceeded ...
1
vote
2answers
367 views

How to factorize $x^3 - 7x + 6$?

How do you factorize this polynomial: $\mathbf{x^3 - 7x + 6}$ Some online solver doesn't even work saying: using GCF method doesn't work, but sites like Mathway.com gave me the answer, is there a ...
1
vote
2answers
97 views

Prove that if a polynomial $P$ has no roots in the upper half plane, then so does $P'$

Prove that if a polynomial $P$ has no roots in the upper half plane, then so does $P'$ This is a part of an exam preparation and I would appreciate a hint. My approach was to use Rouche's theorem but ...
0
votes
3answers
91 views

Find the eigenvalues of the matrices.

The characteristic equations for the two matrices are: $x^3-8x-7=0$ and $x^3-6x^2+11x-6=0$ I know that in order to find the eigenvalues, I need to factor these two equations out. I'm just having a ...
2
votes
1answer
56 views

A problem in polynomials [duplicate]

Let c be a fixed number.Show that a root of the equation x(x+1)(x+2)...(x+2009)=c can have multiplicity at most 2.Determine the number of values of c for which the equation has a root of ...
0
votes
1answer
76 views

Show that $|f(p_n)|<10^{-3}$ whenever $n>1$ but that $|p-p_n|<10^{-3}$ requires that $n>1000$.

Let $f(x)=(x-1)^{10}$. The root of the equation , $p=1$. The approximates of the root, $p_n=1+\frac{1}{n}$ Show that $|f(p_n)|<10^{-3}$ whenever $n>1$ but that $|p-p_n|<10^{-3}$ requires ...
0
votes
1answer
77 views

Difficulty to solve the exercise of Bisection method.

Find an approximation to $ {25}^{\frac{1}{3}}$ correct to within $10^{-4}$ using the Bisection algorithm. How to solve it? Where are the function and interval here?
0
votes
3answers
300 views

I am not understanding what has asked to compute of the following exercise.

let $f(x)=(x+2)(x+1)x(x-1)^3(x-2)$. To which zero of $f$ does the Bisection method converges when applied on the interval $[-3,2.5]$ Have i asked to find the root of $f(x)$ ?
0
votes
1answer
67 views

Determine the number of iteration to find solutions accurate to within $10^{-2}$ for $f(x)=x^3-7x^2+14x-6=0$ on $[a,b]=[1,3.2]$

i got the number of iteration,$n$, to achieve the accuracy, $\epsilon=10^{-2}$ is $n=5.5\approx 6$ But in answer script, $n=8$. My procedure is $ \frac{(b-a)}{2^n}<\epsilon$ ...
2
votes
1answer
101 views

Correct answer of the following math related to Bisection Method.

Use the Bisection method to find $p_3$ for $$f(x)=\sqrt x-\cos(x)$$ on $[0,1]$ I have got the answer $p_3=0.875$ But in answer script , $p_3=0.625$ Which one is correct? let $[a,b]=[0,1]$ ...