Polynomials are expressions like $15x^3 - 14x^2 + 8$. Questions tagged with this concern common operations on polynomials, like adding, multiplying, polynomial long division, factoring and solving for roots.

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Linear Algebra: Diagonalisability

Problem Let A be the matrix $\Bigg(\begin{matrix} 0&0&1\\ 1&0&0 \\0&1&0 \end{matrix} \Bigg)$ Giving brief justifications, determine whether A is ...
3
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2answers
218 views

Linear Algebra: Minimum Polynomial Question

Problem Let $V$ be a finite-dimensional vector space over a field $K$ and let $T$ be a linear transformation of $V$ to itself. Define the minimum polynomial of $T$ to be $m(x)$. Show ...
7
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4answers
278 views

Irreducibility issue [duplicate]

This is a homework question. Given $f(x)=x^{p-1}+x^{p-2}+\cdots+x+1$, where $p$ is any prime. Prove that $f(x)$ is irreducible over $\mathbb{Z}[x]$? Any idea, hint, etc? Hint given by my book was ...
4
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1answer
244 views

Find number of roots in some area (Rouché's theorem)

The task is to find number of $ {z^4} + {z^3} - 4z + 1 = 0$ in the area $1 < \left| z \right| < 2$. (this task is in the Rouché's theorem paragraph) I used this theorem many times, but I ...
3
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1answer
2k views

Finding inverse of a difficult function

Hi I'd like to find the inverse of: $$ y=(1/3)(x^{4} + 4x^{3}) $$ I have learned to do inverses using the following example: $$y=2x-1$$ $$x=2y-1$$ $$x+1=2y$$ $$(x+1)/2=y$$ $$f^{-1}(x)=(x+1)/2$$ ...
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3answers
147 views

Linear Algebra: Minimum Polynomial

Problem Let $V$ be a finite-dimensional vector space over a field $K$ and let $T$ be a linear transformation of $V$ to itself. We define the minimum polynomial $m(x)$ of $T$. Suppose that ...
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2answers
113 views

Efficient computation of the trajectory of roots of a parameterized polynomial

Let $N(s)$ and $D(s)$ be two polynomials in $s \in \mathbb C$ of degrees $m$ and $n$, respectively, with $m<n$. Consider the polynomial equation $$P(s) = N(s) + kD(s) = 0,$$ where $k > 0$. For ...
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236 views

Root of polynomial

The problem is: Suppose that you have $f(z)=z^2-10 \in \mathbb{Q}(z)$ and denote by $f^3=f\circ f\circ f$. Define $$\phi(z)=\frac{f^3(z)-z}{f(z)-z}.$$ If $z$ and $w$ are roots of $\phi$, then ...
2
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1answer
201 views

Compute an operator norm

Consider the operator $M$ acting on the space $\mathbb{R}[X]$ of real polynomials by $Mp(x)=xp(x)$. We equip $\mathbb R[X]$ with the $L^2$ norm $$ \|p\|^2=\int p(x)^2d\mu(x), $$ where $\mu$ is a ...
37
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4answers
995 views

AM-GM-HM Triplets

I want to understand what values can be simultaneously attained as the arithmetic (AM), geometric (GM), and harmonic (HM) means of finite sequences of positive real numbers. Precisely, for what points ...
8
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2answers
437 views

Non-trivial solutions for cyclotomic polynomials

I am seeking algebraic expressions which solve a polynomial equation, in particular an arbitrary cyclotomic polynomial. Let us agree we are not talking about expressions such as $e^{2\pi/7}$. My ...
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1answer
271 views

Can the rational roots theorem always find a root?

If a polynomial has only integer roots, is it always possible to find a root using the rational roots theorem?
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2answers
415 views

Counting Irreducible Polynomials

I'm investigating irreducible polynomials over finite fields at the moment, and I wanted to know if there is a formula for the number of irreducible polynomials of degree n over a fixed finite field ...
0
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2answers
107 views

What are some applications of smoothing a piecewise polynomial?

What are some applications of smoothing a piecewise polynomial? For example, I am interested in learning from you: 1) In what future areas of my math studies will this be useful? and 2) Are there ...
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2answers
637 views

Cyclotomic polynomials explicitly solvable??

I don't know why I'm having trouble with this, but I can't quite see whether the cyclotomic polynomials are considered solvable. Obvioulsy we can write the solution of the nth cyclotomic polynomial as ...
3
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4answers
4k views

Factoring 4 term polynomial

Trying to figure this one out but I see no logical approach to this at all. $x^3-3x^2-4x+12$ I know that it will be 3 parts most likely and that each will start with x but beyond that I will just ...
2
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1answer
155 views

Primitive roots as roots of a $q$-multinomial.

If $n$ is divisible by $m$, why is it the case that the $m$th primitive roots of unity are also roots of $\binom{n}{k}_q$ if and only if $m$ does not divide $k$? I'm viewing $\binom{n}{k}_q$ as a ...
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2answers
181 views

Inequality in Complex Plane

In continuation to my previous post : Inequality in Complex Plane I'm still having a small problem with a similar inequality : For $z$ such that: $|z|> 1$ I wish to prove: $$1+|z|+\dots+|z^{n-1}| ...
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2answers
163 views

Inequality in Complex Plane

I'm studying numerical analysis and in the book I'm reading there is a theorem thats find a raduis such that all the roots of a polynomial $P$ (with coefficient in $\mathbb{C}$) are in the open disk ...
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1answer
477 views

Discriminant of derivative of cubic equation being a perfect square

Is it possible for the discriminant of the first derivative of a cubic polynomial (x+a)(x+b)(x+c), where a, b and c are distinct non-zero integers (i.e. Discriminant[d((x+a)(x+b)(x+c))/dx] in ...
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5answers
353 views

How would I know if $f(x)=x^5-2x+10$ has a root at the interval $[-2, 2]$?

Unsure on the procedure on this one and then how to explain it. I don't think this function has any rational roots, right?
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3answers
126 views

Property of a polynomial $f\in\mathbb{Q}[X]$ such that $f(n)\in\mathbb{Z}$ for all $n\in\mathbb{Z}$?

We can always view $\binom{x}{k}$ as a polynomial in $x$ of degree $k$. With this in mind, why is it so that a polynomial $f\in\mathbb{Q}[x]$ is such that $f(n)\in\mathbb{Z}$ for all $n\in\mathbb{Z}$ ...
4
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3answers
294 views

Coefficients of $(1+x+\dots+x^n)^3$?

Consider the following polynomial: $$ (1+x+\dots+x^n)^3 $$ The coefficients of the expansion for few values of $n$ ($n=1$ to $5$) are: $$ 1, 3, 3, 1 $$ $$ 1, 3, 6, 7, 6, 3, 1 $$ $$ 1, 3, 6, 10, 12, ...
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2answers
106 views

How to give the answer of this equation in fraction: $-3x^2 + 2x + 4 =0$

I found the root of this equation is: $ x = \frac{1}{3} (1 \pm \sqrt{13}) $. How can I convert this result to fraction?. Sorry for my ignorance, I don't practice math for a long time.
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0answers
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How to find the recurrence polynomial?

Problem: If $f$ is a polynomial with unknown roots $x_1,-x_1,x_2,-x_2\ldots,x_b,-x_b \quad (b \in \mathbb{N})$ and can be expressed as (known expansion): $$f(x)=\sum_{a=0}^{2b}f_ax^a\quad(b \in ...
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1answer
653 views

Linear Algebra: Dual Basis Problem

Problem Let $V$ be the vector space of all polynomial functions $p$ from $\mathbb{R}$ to $\mathbb{R}$ which have degree two or less. Define three linear functionals on $V$ by ...
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2answers
721 views

How do I factor a polynomial function with a degree higher than 2 without guessing numbers of $\frac{p}{q}$?

I have an equation $f(x)=x^4+4x^3+2x^22-x+6$. In the past I was taught to factor it by getting the zeros by getting $p/q$, and start guessing zeros, and plugging them into the function. Once I got one ...
7
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1answer
833 views

Sum of squares of roots of a polynomial $P(x)$

Well, I recently proved a formula (at least, I think) to the sum of the inverse of the roots $x_{1}, x_{2}, x_{3},\ldots, x_{n} \in \mathbb{C}$, and $\neq 0$. It starts: Let a polynomial $P(x) = ...
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3answers
72 views

Complex Numbers Question

1) let $Z_0$ be a solution of $Z^{13}-13Z^{7}+7Z^{3}-3Z+1=0$, Is it true that $Z_0$'s conjugate is also a solution? 2) let $Z_0$ be a solution of $Z^{2}+iZ+2=0$, Is it true that $Z_0$'s conjugate is ...
2
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0answers
113 views

Elementary symmetrical polynomial equations, whose solutions are known to be natural numbers.

Let $n_1,n_2,\dots,n_k$ be natural numbers (excluding 0), and for each $1\leq i\leq k$ let $\sigma_i(n_1,n_2,\dots,n_k)$ be the elementary symmetrical polynomial consisting of the sum of all products ...
5
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2answers
208 views

Irreducibility of a cubic in $\mathbb{Q}[X]$

Problem Show that $$X^3-2008X^2+2010X-2009$$ is irreducible in $\mathbb{Q}[X]$. Progress I considered applying Eisenstein's Theorem, but there are no primes $p$ such that $p|2008$, $p|2009$ ...
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0answers
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For any prime $p \equiv 1\pmod{5}$ what integers $\{a_0, \dots, a_4\}$ satisfy $(\sum_{i=0}^{4}{a_ig^i})(\sum_{i=0}^{4}{a_ig^{-i}})=p^2$?

For any prime $p \equiv 1 \pmod{5}$ do there exist 5 integers $\{a_0, \dots, a_4\}$, each of absolute value less than $p$, satisfying $\sum_{i=0}^{4}{a_i}=p$, ...
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0answers
114 views

How do I transform the coefficients of a solved polynomial curve fit?

This all pertains to a piece of software I am writing but figured I'd get a better answer here than in Stackoverflow. I have no problem migrating the question if needed. Disclaimer: I am a software ...
2
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1answer
125 views

Find and prove an upper bound on the number of intersections on two distinct polynomials

Find and prove an upper bound on the number of times that two distinct polynomials of degree $d$ can intersect. What if the polynomials' degrees differ? My attempt: let $p(x)$ and $q(x)$ be two ...
3
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3answers
209 views

Do these special power functions generate all homogeneous symmetric polynomials?

Over rational numbers, the set of all power functions up to a certain degree generate all symmetric polynomials in that degree. My question is as follows. To be succinct, let's say we have four ...
0
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3answers
133 views

Evaluating $\binom{100}{i}a^i(1-a)^{(100-i)}$ in GMP-GNU [closed]

I want to calculate $\binom{100}{i}a^i(1-a)^{(100-i)}$ for different $i$ with $a=0.001$ using GMP-GNU. How can this be done?
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1answer
303 views

What is shortcut to this contest algebra problem about polynomial?

The polynomial $P(x)=x^4 + ax^3 + bx^2 +cx + d$ has the property that $p(k)=11k$ for $k=1,2,3,4$. Compute $c$. The answer is $-39$.
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1answer
79 views

How to change Hermite function to have a same shape with moving extremum point and zeros

If somebody has a experience with polynomials. How to set this Hermite function to have a general minimum where I want on $x$ axis, for example in $0.5$. Is it possible to be in analytic form ...
0
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1answer
250 views

Sum of the polynomial roots raised to a power. How to prove?

Problem: If we have a polynomial $f$ with a derivative $f\,'$ and quotient $q$ function defined as: $$q(x)=\sum_{i=1}^{\infty}a_ix^{-i}=\frac{f\,'(x)}{f(x)},$$ and the roots of $f$ are ...
3
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0answers
373 views

Galois group of solvable quintic is subgroup of Fr20

Why is it true that any solvable quintic polynomial in has a Galois group that is a subgroup on the Frobenius group of order 20? Thanks in advance.
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1answer
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Relation betwen coefficients and roots of a polynomial [duplicate]

Possible Duplicate: Create polynomial coefficients from its roots I am reading the first chapter titled Numerical Solutions Of Equations And Interpolation by K.A. Stroud (Advanced ...
3
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3answers
117 views

Factorizing a polynomial $f$ in $A[x]$ (with $A$ commutative), where $f$ has a zero in its field of fractions

Let $A$ be a commutative ring and $S$ a multiplicative subset of $A$ generated by $s\in A$ which is not a zero-divisor. Consider the polynomial ring $A[x]$. Given a polynomial $f\in A[x]$, ...
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1answer
154 views

polynomial division and field extensions

I would like to verify the validity of the following line of thought: Let $K \subset E$ be a field extension. Let $K[x]$ be the polynomial ring over $x$ and denote $K(x)$ its field of quotients (the ...
2
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2answers
346 views

Prove that the line $y=2x$ intersects the cubic curve $y = x^3 - x + 1$ in at least three different points

Prove that the line $y=2x$ intersects the cubic curve $y = x^3 - x + 1$ in at least three different points This is a homework question and I don't know where to begin, how would I go about ...
15
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5answers
2k views

Prove that the equation $x^{10000} + x^{100} - 1 = 0$ has a solution with $0 < x < 1$

Prove that the equation $x^{10000} + x^{100} - 1 = 0$ has a solution with $0 < x < 1$. This is a homework question. I know I could probably find a solution that would complete the proof, ...
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1answer
180 views

Inseparable polynomial over a non-perfect field

Assume that $F$ is a field and $\operatorname{char}(F)=p$. Let $a$ be an element in $F$ without $p$th root, then the polynomial $$x^{p^n}-a$$ is irreducible and inseparable over $F$ for all $n$. ...
3
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5answers
404 views

How to compute the characteristic polynomial of $A$

The matrix associated with $f$ is: $$ \left(\begin{array}{rrr} 3 & -1 & -1 \\ -1 & 3 & -1 \\ -1 & -1 & 3 \end{array}\right) . $$ First, I am going to find ...
3
votes
2answers
216 views

Completing the square

How might I find linear combinations $$\begin{align*} A&=a_1x+a_2y+a_3z\\ B&=b_1x+b_2y+b_3z\\ C&=c_1x+c_2y+c_3z \end{align*}$$ Such that I can transform the two polynomials ...
12
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2answers
740 views

Sum of derivatives of a polynomial

Let $p(x)$ be a polynomial of degree $n$ satisfying $p(x)\geq 0$ for all $x$. That is, for all $x$, $p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \geq 0$, $a_n\neq 0$. Show that ...
3
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0answers
957 views

Computation of coefficients of Lagrange polynomials

For our homework we should write a program, that creates Lagrange base polynomials $L_k(x)$ based on a few sampling points $x_i$. Now i am eager to develop a formula to be able to compute the ...