Questions about the set of values at which a given function evaluates to zero. For questions about "square roots", "cube roots", and such, consider using the (arithmetic) tag. For questions about roots of Lie algebras, use the (lie-algebra) tag instead.

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3
votes
2answers
74 views

Showing how the roots of this complex polynomial are different.

I want to show that the complex polynomial $p(z) = z^5 + 6z - 1$ has four different roots in the annulus $\{z \in \mathbb{C} : \frac{3}{2} < |z| < 2 \}$. I used Rouché's theorem to proof that ...
3
votes
1answer
128 views

Roots of $z^{2n} + \alpha z^{2n -1} + \beta ^2$

I've been looking at a problem available here. The problem is: Let $n$ be a natural number, and $\alpha$, $\beta$ nonzero reals. Show that the number of roots of $p(z) = z^{2n} + \alpha z^{2n -1} + ...
1
vote
1answer
269 views

Do the false position method really need that there exists only one root inside $[a; b]$?

I'm studying the False Position Method for finding zeroes of real functions and in the book I'm reading the author says that it is required that only one root of $f$ is contained inside the initially ...
4
votes
3answers
278 views

Prove $x^{n}-5x+7=0$ has no rational roots

This question arises in STEP 2011 Paper III, question 2. The paper can be found here. The first part of the question requires us to prove the result that if the polynomial ...
0
votes
1answer
72 views

number of solutions in homogeneous system

What is the maximum possible number of solutions of homogeneous system $N \times N$ ($N$ variables, $N$ equations) of degree $2$, where in each equation we have linear terms in $x_i$ and quadratic ...
4
votes
1answer
817 views

Calculating the Roots of Sine

Aside from the obvious knowledge that the roots of $\sin x$ are all integer multiples of $\pi$, is there a formal, algebraic method to calculate the roots of trigonometric functions similar to the ...
2
votes
1answer
113 views

How to find all zeros of a polynomial

Let $$f(x) = x^4 - 3x^3 + 2x^2 - 7x - 11. $$ I want to find the roots of $f(x)$. I know that there are $3$ or $1$ positive roots, $1$ negative root, and $2$ or $0$ imaginary roots but I can't figure ...
4
votes
1answer
204 views

Solving a transcendental equation consisting of a quadratic part and a part involving inverse Lambert W functions

Question statement I would like to solve the following equation in the two variables $x$ and $y$: \begin{gather} 0 = x^2 - a y^2 + i b [x y - W^{-1}(x)W^{-1}(y)] , \end{gather} where $a$ and $b$ are ...
2
votes
4answers
129 views

Finding the root of a degree $5$ polynomial

$\textbf{Question}$: which of the following $\textbf{cannot}$ be a root of a polynomial in $x$ of the form $9x^5+ax^3+b$, where $a$ and $b$ are integers? A) $-9$ B) $-5$ C) $\dfrac{1}{4}$ D) ...
0
votes
1answer
77 views

Write the 2nd degree equation which have the following roots

$y_1$=${(x_1+x_2\varepsilon+x_3\varepsilon^2)}^3$ $y_2$=${(x_1+x_2\varepsilon^2+x_3\varepsilon)}^3$ where $x_1,x_2,x_3$ roots for the $x^3+ax^2+bx+c=0$ and $\varepsilon$ = ...
2
votes
1answer
75 views

Solution to set of three equations

I have the following three equations: $$\cos\theta \left(\cos\psi - k_3\sin\psi\right) = k_1$$ $$\sin\phi\sin\theta\cos\psi - \cos\phi\sin\psi - k_3\left(\cos\phi\cos\psi + ...
2
votes
2answers
63 views

Finding Root of an Equation with Variables Dependent on Each other

Sorry for the title. I'm sure there is better terminology. I'd be interested to here what that terminology is haha. Here is my problem: ...
2
votes
4answers
162 views

Finding root of equation

This question was asked in one of the enterance test of mathematics in India which is For the equation $1+2x+x^{3}+4x^{5}=0$, which of the following is true? (A) It does not possess any real root ...
16
votes
2answers
646 views

Adriaan van Roomen's 45th degree equation in 1593

Adriaan van Roomen proposed a 45th degree equation in 1593(see this book, picture reference as follows): $$ \begin{gathered} f(x) = x^{45} - 45x^{43} + 945x^{41} - 12300x^{39} + 111150x^{37} - ...
2
votes
0answers
68 views

Overdetermined system - showing that there are no roots that satisfy the set of equations

We consider an overdetermined set of equations, consisting of two equations for one complex variable $x$. I want to show that there are no roots for $x$ in the complex unit disc but without the ...
6
votes
3answers
142 views

Approximating the roots of $\epsilon^{2}x^{3}+x+1$

I saw the following in my lecture notes, and I am having difficulties verifying the steps taken. The question is: Assuming $0<\epsilon\ll1$ find all the roots of the polynomial ...
1
vote
0answers
49 views

Rouche's theorem: $g(z)$ has no roots in $L$, $|g(z)| > |f(z)|$ for the contour $\partial L$. Does $f(z)$ have no roots in $L$?

Let $f(z)$ and $g(z)$ be analytic functions. Let $L$ be the complex unit disc and its contour is $\partial L$, the complex unit circle $|z| = 1$. If $g(z)$ has no roots in $L$, e.g. $g(z) = z + 2$, ...
2
votes
2answers
56 views

Determining the sign of a polynomial given its factorization

Is there a quick way of determining where a polynomial is positive/negative without actually plugging values? Say you have a polynomial $$1) f(x)=(x+a)(x+b)$$ or $$2) f(x)=(x-a)(-x+b)(x-c)$$ ...
3
votes
6answers
541 views

Solving $\sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2}$

Where do I start to solve a equation for x like the one below? $$\sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2}$$ After squaring it, it's too complicated; but there's nothing to factor or to ...
13
votes
3answers
146 views

$\sum_i x_i^n = 0$ for all $n$ implies $x_i = 0$

Here is a statement that seems prima facie obvious, but when I try to prove it, I am lost. Let $x_1 , x_2 \dots x_k$ be complex numbers satisfying: $$x_1 + x_2 \dots + x_k = 0$$ $$x_1^2 + x_2^2 ...
2
votes
0answers
90 views

A question about cubic equation.

I'd like to share my doubt on cubic equation. Step 1: $ax^3+bx^2+cx+d=0$, Step 2: We can substitute $x=y-\frac b {3a}$ to get $y^3+py+q=0$ where $p,q$ are something. Step 3: By Vieta's ...
4
votes
5answers
455 views

Polynomials - The sum of two roots

If the sum of two roots of $$x^4 + 2x^3 - 8x^2 - 18x - 9 = 0$$ is $0$, find the roots of the equation
1
vote
1answer
192 views

no. of real roots of exponential equation in three questions

How Can i calculate no. of real roots of exponential equation in three questions (1) $2^x = 1+x^2$ (2) $2^x+3^x+4^x = x^2$ (3) $3^x+4^x+5^x = 1+x^2$ My Try:: (1) Let $f(x) = 1+x^2-2^x$ now ...
7
votes
2answers
140 views

If $\mathbb f$ is analytic and bounded on the unit disc with zeros $a_n$ then $\sum_{n=1}^\infty \left(1-\lvert a_n\rvert\right) \lt \infty$

I'm going over old exam problems and I got stuck on this one. Suppose that $\mathbb{f}\colon \mathbb{D} \to \mathbb{C}$ is analytic and bounded. Let $\{a_n\}_{n=1}^\infty$ be the non-zero zeros of ...
6
votes
3answers
184 views

How can I find all the solutions of $\sin^5x+\cos^3x=1$

Find all the solutions of $$\sin^5x+\cos^3x=1$$ Trial:$x=0$ is a solution of this equation. How can I find other solutions (if any). Please help.
6
votes
1answer
171 views

Bounding the roots of the sum of two monic polynomials with real coefficients.

Let $P_1(z)$ and $P_2(z)$ be monic polynomials with real coefficients and roots $\{z_1^{(1)},z_1^{(2)},...\}$ and $\{z_2^{(1)},z_2^{(2)},...\}$, respectively. Are there any results relating the ...
2
votes
2answers
123 views

Condition For No Existence Of Real Root

$2x^{4}+5ax^{3}-2bx^{2}+1=0$ has no real root in $(5,2014)$ Find the conditions for $a$ and $b$ I am suspicious of even the existence of its solution and at a loss.
3
votes
3answers
105 views

Roots of a cubic equation

I have the following equation: $s^3+as+b=0$ Now I want the values for a and b for which the given equation has the following complex roots: $c \pm di$ I don't really care about the remaining root. ...
0
votes
1answer
27 views

Relationship between 2 Dimensional Quadratic systems and roots

Given four points $(x_1, y_1) (x_2, y_2) (x_3, y_3) (x_4, y_4)$ How does one construct a system of two equations: $a_1x + a_2x^2 + a_3y + a_4y^2 + a_5xy = c_1$ $b_1x + b_2x^2 + b_3y + b_4y^2 + ...
1
vote
1answer
122 views

Rouche's theorem for two functions that have the same number of roots

I hope this is not too long. Thanks in advance! Edit: I edited it for a great deal, most of the information was unnecessary. Let us define a function $h(z) = f(z) + g(z)$. We know that $f(z)$ has ...
-4
votes
1answer
185 views

Root of a quadratic equation that has modulus $1$

Let us suppose $\alpha \in \mathbb C$ and $|\alpha|=1$ and $\alpha$ satisfies a monic quadratic equation. Then prove that $\alpha^{12} =1$. Show me the right way to solve this. Thanks in advance.
4
votes
3answers
588 views

Solve $\sin(z) = z$ in complex numbers

Show that $\sin(z) = z$ has infinitely many solutions in complex numbers. Little Picard theorem should help, but using big Picard theorem is undesirable. Thanks a lot!
0
votes
1answer
115 views

Complex numbers and absolute values

If i have equation: \begin{align} P = \left|\psi\right|^2 \end{align} where $P$ is a probability and we know there is no negative probability. This means $P$ must belong to $\mathbb{R}$. If i want ...
2
votes
2answers
155 views

Conditions that Roots of a Polynomial be Less than Unity

Is is the case that Samuelson's result is a more specific result of Rouche's Theorem, or the Routh–Hurwitz stability criterion? Is it not the goal for a polynomial to be stable that all of its roots ...
1
vote
0answers
72 views

Fixed Point Iteration Scheme

I have been asked to "Find a fixed point iteration scheme for minimising $f(x) = e^{cos (x)}$". Does anybody know what a fixed point iteration scheme actually is? I know it's not Fixed Point ...
1
vote
1answer
95 views

Solution of a polynomial in interval $(0,1)$

Let $\displaystyle a_0 + \frac{a_1}{2} + \frac{a_2}{3} + ... + \frac{a_n}{n+1} = 0$, where $a_i$'s are some real constants. How can we prove that the equation $a_0 + a_1x + a_2x^2 + ... +a_nx^n = 0$ ...
6
votes
3answers
416 views

Minimum degree of a polynomial passing through points

If $P(x)$ is a polynomial such that $P(a_{1})=b_{1}, P(a_{2})=b_{2}, \ldots , P(a_{k})=b_{k}$, how can I find the polynomial which has minimum degree and for whom the relations above are true?
0
votes
0answers
40 views

How to effciently solve a radical equation of the form $0=\sum_{j=1}^n a_j\sqrt{|b_j-x|}$?

Given a radical equation of the form $$0=\sum_{j=1}^n a_j\sqrt{|b_j-x|}$$ where $b_j>0$ and the sign of $a_j\in\mathbb R$ matches that of $b_j-x$, is there any more efficient (analytical?) solution ...
5
votes
3answers
138 views

roots of the polynomial equations and relation among the coefficients

If the equation $x^4 + ax^3 + bx^2 + cx + 1 = 0$ ($a,b,c$ are real numbers) has no real roots and if at least one root is of modulus one, then what is the relation between $a,b$ and $c$?
2
votes
2answers
157 views

Roots of cubic polynomial lying inside the circle

Show that all roots of $a+bz+cz^2+z^3=0$ lie inside the circle $|z|=max{\{1,|a|+|b|+|c| \}}$ Now this problem is given in Beardon's Algebra and Geometry third chapter on complex numbers. What might ...
3
votes
5answers
446 views

How to find the number of real roots of the given equation?

The number of real roots of the equation $$2 \cos \left( \frac{x^2+x}{6} \right)=2^x+2^{-x}$$ is (A) $0$, (B) $1$, (C) $2$, (D) in finitely many. Trial: $$\begin{align} 2 \cos \left( ...
2
votes
4answers
159 views

Interception with $x$-axis - not so trivial?

I want to find the interception with the x-axis of the following function: $f(x) = \frac{1}{4}x^4-x^3+2x$. So putting $0 = \frac{1}{4}x^4-x^3+2x$ I would get $0 = x(\frac{1}{4}x^3-x^2+2)$ but how to ...
3
votes
1answer
132 views

Rouché's Theorem on $z^{10} + 10z + 9$

Please note: this question was asked before, but the solution provided does not work as far as I know; see How to find the number of roots using Rouche theorem? We have $f(z) = z^{10} + 10z + 9$ and ...
0
votes
0answers
98 views

How to solve an equation in three variables fixing two of the variables?

Also, I have the following equation, I want to solve it for $b$ keeping $a$ and $c$ fixed. $5b^5+(60-5a)b^4+(125+50c-80a)b^3+(594c-445a-775)b^2+(2324c-1005a-3270)b+3000c-750a-3000=0.$ Also how to ...
2
votes
3answers
75 views

why if x in 1/n power >(<) y in 1/m power then x in c/n power >(<) y in c/m power?

As you might guess this is one more stupid question from non-matematician, and you are right. I found this exercise in "Algebra and trigonometry book": $7^{1/2}$ or $4^{1/4}$. After some googling I ...
2
votes
1answer
134 views

Location of Complex Roots

Here is a problem I think dealing with Rouche's theorem: How many roots does the equation $$ \frac{1}{2}e^z+z^4+1=0 $$ have in the left half plane $Re(z)<0$ I see that in order to have a root in ...
2
votes
0answers
207 views

Can a linear combination of even Legendre polynomials have common real root(s) with a linear combination of odd Legendre polynomials?

I am using the following definition of Legendre Polynomials: $P_0(x)=1$, $P_1(x)=x$ and $$P_{k+1}(x)=\left(\frac{2k+1}{k+1}\right)xP_k(x)−\left(\frac{k}{k+1}\right)P_{k−1}(x)$$ Let ...
0
votes
2answers
164 views

Solve an equation of 3rd order [duplicate]

What is the simplest method to solve an equation of 3rd degree. For example: $$-x^{3} + x^{2} + x - 1 = 0$$ Please I don't want the resolution of this equation I just want the simplest method to use ...
4
votes
1answer
358 views

How to solve a polynomial with power fractions like $a-ax+x^{0.8}-x^{0.2}=0$

I have something like $a-ax+x^{0.8}-x^{0.2}=0$ with parameter a>0 and variable x>0. I know by trial and error that the equation has three real roots for parameter a greater than certain value, ...
3
votes
2answers
1k views

Finding the Number of Zeros of a Function in a Given Annulus

Consider $z^6 - 6z^2 + 10z + 2$ on the annulus $1<|z|<2$. By Rouche's Theorem $|f(z) + g(z)| < |f(z)|$ implies that both sides of the inequality have the same number of zeros. I understand ...