This tag is used for both basic and advanced questions on polynomials in any number of variables. Including, but not limited to: solving for roots, factoring, checking for irreducibility. This tag is rarely used as the only tag for a question.

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1answer
21 views

Finding coefficients of two polynomials

Let $n$ be a natural number. Let $f(x)=\prod_{i=-n}^{n}(x-i)$. If $k$ is an even integer, then the coefficient of $x^k$ is zero. The coefficient of $x^{2n-1}$ is ...
2
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1answer
21 views

Roots of a quartic polynomial

If $a, 3a, 5a, b, b + 3,$ and $b+5$ are all roots of a fourth-degree polynomial equation where $0<a<b$, compute all possible values of $a$. By the fundamental theorem of algebra, the polynomial ...
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1answer
75 views

A stronger form of Rolle's Theorem in the direction of number of roots of $f'(x)$

Today I read an interesting generalization of the Rolle's Theorem for Polynomials in $E. 28$ of E. J. Barbeau's book on Polynomials. It says that if $a, b$ are two consecutive zeroes of polynomial ...
3
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0answers
29 views

Solutions of an equation of degree $n>4$

I know that the Abell-Ruffini theorem prove that we cannot solve a general equation of degree $n>4$ with radicals. But I've read that quintic equations can be solved by means of elliptic modular ...
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2answers
36 views

Polynomial roots. Finite Extensions.

If $a,b,c$ belong to $\mathbb Q$ and $ω^3=1$ is a third root of unity, then prove that if $$(a+b\sqrt[3]{2}+c\sqrt[3]{4})^3=1+2\sqrt[3]{2}-\sqrt[3]{4}$$ holds we also have $$ 1+2ω\sqrt[3]{2}-\bar ...
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1answer
22 views

Proving a polynomial splits over a certain field extension

If $K$ is the splitting field of $f\in F[x]$, and $g\in F[x]$ is irreducible and has a root in $K$, prove that $g$ splits over $K$. My proof (which I don't think is correct) is as follows: Let ...
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1answer
43 views

root of $f$ is smaller than that of $g$.

For a fixed natural no. $n\ge4$, consider $$f(x)=x^3-(n+2)x^2+2nx-2,$$ $$g(x)=x^3-(n+3)x^2+2(n+1)x-2,$$ It seems that smallest root of $f$ is smaller than that of $g$. Can someone show how to prove ...
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0answers
18 views

Finding the roots of this multivariable polynomial?

My polynomial is this ten term monster $P(x,y,z) = 6561 x^3+486 x^2+12 x+6561 y^3+1944 y^2+192 y+6561 z^3+6318 z^2+2028 z+223$ It's simplest form is ${1 \over 81} \left( (81x+2)^3 + (81y+8)^3 ...
2
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1answer
55 views

Norms on $\mathcal{P}_N$ Vector Space of Polynomials up to Order N

$\|p\|_\infty :=\sup_{x\in [0,1]}|p(x)|$ and $\|p\|_{L^1}:=\int_0^1 |p(x)| dx$. As the space of real-valued polynomials on $[0,1]$ up to order $N$ is a $N+1$ dimensional vector space and ...
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5answers
78 views

Given that the sum of two of its roots is zero,solve the equation: $6x^4-3x^3+8x^2-x+2=0$

Solution:-let $\alpha,\beta,\gamma,\delta$ be the roots of equation. It is given that,$\alpha+\beta=0$ and,$\alpha+\beta+\gamma+\delta=3/6=1/2$,this implies $\gamma+\delta=1/2$ and ...
0
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0answers
7 views

Hyperplane and cubic curves and their intersections.

Solve the following: $$a^4-a^2+A_{1}E u=0;$$ $$b^4-b^2+A_{2}Eu=0;$$ $$c^4-c^2+A_{3}Eu=0;$$ $$d^4-d^2+A_{4}Eu=0;$$ and $$a^2+b^2+c^2+d^2=E,$$ for $a, b, c, d,$ and $u$, when $A_{1}, A_{2}, A_{3}, ...
0
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1answer
33 views

Cubic polynomial with 1 real root and 2 complex conjugated roots (real coefficients)

I am stuck on this problem about cubic polynomials. I rely on the Wikipedia page on the topic. Using wikipedia notations (chapter "General formula for roots") : For the case where $\Delta > 0$, ...
1
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1answer
19 views

Efficiently check if two large factorized multivariate polynomials are the same without expansion

Given two large multivariate polynomials $f$ and $g$ in $m$ variables of degree $n$, which are factorized and have i.e. more than a 1000 terms, how can we check efficiently if they are the same? To ...
2
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1answer
17 views

Continuous dependence of matrix elements

I've stumbled upon several solution of linear algebra problems which use notion of "continuous dependence" of matrix polynomials on matrix elements. For instance (translated, so any inaccuracies are ...
0
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1answer
34 views

$p$ and $q$ be the roots of the polynomial $mx^2 + x(2-m) + 3$. [on hold]

Let $p$ and $q$ be the roots of the polynomial $mx^2 + x(2-m) + 3$. Let $m_1$ ,$m_2$ be two values of m satisfying $\frac{p}{q} + \frac{q}{p}= \frac{2}{3}$. determine numerical value of ...
0
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0answers
17 views

How to prove a self-recirpocal polynomials $P(z)$ to have all its zeros on the unit circle $|z|=1$?

Let $m(n)=10(n+1)^3$ and $$c_j(n)=\frac{2 (2j+1)}{\Gamma(j)}\sum_{k=1}^{n}(\pi k^2)^{j}\tag{1}$$, $$P(z)=\sum_{j=1}^{m(n)}(-1)^jc_j(n)\left(z^{4j+1}+z^{-(4j+1)}\right)\tag{2}$$ ...
1
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1answer
13 views

minimal polynomial of $\alpha - \beta$ from the minimal polynomial of $\alpha$

If $f \in k[x]$ is the minimal polynomial of $\alpha \in K$, show how to write down the minimal polynomial of $\alpha - \beta$ for $\beta \in k$. I've been playing around with the minimal polynomial ...
0
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1answer
33 views

What are prime and primitive polynomials?

Please, I am not a mathematician so highly mathematical textbook language will not make sense, that is why I am forced to post this question here. I am reading about Checksum and CRC data integrity ...
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0answers
14 views

A good version of truncated real radical ideal?

Suppose $\mathbb{R}[X]$ is the normal multivariate polynomial ring where $X = x_1,...x_n$. $\mathbb{R}[X]_t$ is the truncated set such that $\mathbb{R}[X]_t =\left\{f: f \in \mathbb{R}[X], \deg(f) ...
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1answer
29 views

Can a polynomial $p(x)$ generate only primes and 2-almost primes $\forall x \ge 0 \in \Bbb N$ or there is also a restriction for this to happen?

There is a simple demonstration to show that a polynomial of any degree can not generate only primes. Basically, if $p(x)=a_nx^n+...+a_1x^1+a_0$ is prime for every $x \in \Bbb N$ ($\Bbb Z$ would be ...
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1answer
188 views
+50

How many partial derivatives does a multivariate polynomial have?

My motivation for this question is from the following toy example; define the (nondeterministic) finite state machine generated by the polynomial $f(x_0 , \dots , x_n) \in \mathbb{Z} [x_0 , x_1 , ...
0
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0answers
30 views

Use Galois theory to find all complex roots of $T^4-2T^2-\sqrt{6}T+\frac{3}{4}$

I am currently studying Galois theory and a question that often comes up is "find all complex numbers which are roots of the polynomial $T^4+aT^2+bT+c$" where the coefficients are of the form ...
3
votes
2answers
28 views

Eigenvalue of a matrix and a polynomial of that matrix

Let $A$ be a $n \times n$ matrix over $F$, and let $c_1, ... c_n$ be its eigenvalues. Show that for every polynomial $g(x) \in F[x]$, the eigenvalues of $g(A)$ are $g(c_1), ... , g(c_n)$. I think by ...
0
votes
2answers
20 views

Conjugate Zeros Theorem for polynomials with real and complex coefficients.

Question: (a) Show that $2i$ and $1 - i$ are both solutions of the equation $x^2 - (1 + i)x + (2 + 2i) = 0$ but that their complex conjugates $-2i$ and $1 + i$ are not. (b) Explain why the result of ...
3
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0answers
78 views

On a family of polynomials related to the expansion of $(1+\epsilon)^{x/\epsilon}$ as a series in $\epsilon$

Consider the sequence of polynomials $(P_n)_{n\geqslant0}$ uniquely defined by the recursion $$(P_n)'=\sum_{k=0}^{n-1}\frac{P_k}{n-k+1},$$ valid for every $n\geqslant0$, with the initial conditions ...
2
votes
2answers
47 views

Given two polynomials $f$ and $g \in \mathbb{Q}[X]$, prove that $(f) + (g) = (h)$ and $(f)\bigcap(g) = (k)$

Given two polynomials $f(X) = 3X^2 + 7X - 6$ and $g(X) = 2X^2 + 5X - 3 \in \mathbb{Q}[X]$, prove that there exist $(h)$ and $(k) \in \mathbb{Q}[X]$ such that $(f) + (g) = (h)$ and $(f)\bigcap(g) = ...
1
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0answers
15 views

Most general polynomial of 2 or 3 variables and degree 4 with at least 4 minima

I am searching for the most general (if any) real polynomial that has the following features: -Degree 4 -$m+n$ variables, with $m \ge 2$ and $n \ge 0$ (focusing on $m=2, n=1$) -It is even in the ...
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2answers
33 views

Finding value of $m$ such that such that the polynomial is factorized

A polynomial $2x^2+mxy+3y^2-5y-2$ Find the value of $m$ much that $p(xy)$ can be factorized into two linear factors
4
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1answer
29 views

Is counting roots with multiplicites at all a geometric concept?

It is well known that a polynomial of degree $n$ admits $n$ roots when the field is algebraically closed. However, this comes with a caveat, in particular that the roots be counted with multiplicity. ...
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1answer
31 views

Irreduciblity of polynomial over field of prime element [on hold]

Let $f(x)=x^4-x^3+14x^2+5x+16$ be a polynomial and let $F_p$ denote the field with $p$ elements, where $p$ is prime number. Then which of the following are always true? Considering $f$ as a ...
2
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1answer
47 views

Show that $H = \mathbb{Z}_5[x]/\langle x^4+3x^3+x+4\rangle$ is not a field.

So I am looking over old exams in abstract algebra and I came across this question which seems to be a mistake. (Neither the original teacher who wrote it, nor my own teacher are available to answer) ...
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2answers
37 views

Stuck on a simple factoring problem

The answer to this question is probably very obvious but I can't figure it out for some reason: I simply want to factorise: $x^2+5x-2$ I solve $x^2+5x-2 = 0$ i find $x_1 = \dfrac{-5-\sqrt{33}}{2}$ ...
0
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1answer
21 views

Factor polynomial into linear factors with complex coefficients.

Question: A polynomial is given. $(a)$ Factor it into linear and irreducible quadratic factors with real coefficients. $(b)$ Factor it completely into linear factors with complex coefficients. $x^3 - ...
0
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1answer
41 views

How to factor cubic polynomials?

My polynomial: $2x^3 +7x^2+12x+9$. Now, I've tried both of the techniques given in this Wikihow page, but neither of them worked for this problem. Synthetic division is something which I think would ...
2
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2answers
102 views

What does a norm of a polynomial space mean?

When talking about polynomial vector space, the following example was provided. A polynomial of degree $n$ in two variables is $$p(X)=\sum_{0\leq k+j \leq n} a_{j,k}x_1^jx_2^k$$ where $k+j=n$ and ...
5
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2answers
400 views

A polynomial of degree 3 that has three real zeros, only one of which is rational.

Find a polynomial of degree 3 that has three real zeros, only one of which is rational. My answer: $(x - \sqrt{2})(x - 3)(x - \pi)$. Is this correct? It does have two irrational zeros, but I'm not ...
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2answers
40 views

Let $f(x)=x^3+ax^2+bx+c$ Prove that for $a^2\lt 3b$ there exists only one $x_0$ such that $f(x_0)=0$

Let $f(x)=x^3+ax^2+bx+c$ Prove that for $a^2\lt 3b$ there exists only one $x_0$ such that $f(x_0)=0$ Now we know this $x_0$exists because of the IVT. Also graphing different values of a and b, I've ...
0
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1answer
25 views

Difference of Squares Oddity

I was practicing factoring polynomials and ran across a problem I'd never seen before. $$x^{2m} -36y^2$$ I know this is a difference of squares but I'm not sure how to handle the '$m$' and the ...
3
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0answers
39 views

Bijective mapping between face polytopes of permutohedra and partitions of integers

The OEIS entries A019538, A049019, and A133314, relate a refinement of the face polynomials of the permutohedra (A049019) to partition polynomials (A133314) defined by multiplicative inversion of an ...
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3answers
37 views

Given that an expression $2x^3+px^2-8x+q$ is exactly divisible by $2x^2-7x+6$, determine the value of $p$ and $q$. [on hold]

Given that an expression $2x^3+px^2-8x+q$ is exactly divisible by $2x^2-7x+6$, determine the value of $p$ and $q$.
2
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0answers
36 views

Polynomial division for identifying an expression in terms of complex numbers.

This question is blatantly copied from here, for the sake of learning more I specify it a bit more: $$f(z)= (3x^2 + 2x - 3y^2 - 1) + i(6xy + 2y)$$ $$z = x+yi$$ I want to write $f(z)$ in terms of ...
1
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1answer
22 views

Find the multiplicity of root $x=a$ of polynomial $Q(x)= \frac{1}{2}$*$(x-a)(p'(x)+p'(a))-p(x)+p(a)$

This is the problem I have: We have polynomial $p(x)$, degree $n$, $n \in \Bbb N$. Find the multiplicity of root $x=a$ of polynomial $Q(x)= \frac{1}{2}$$(x-a)(p'(x)+p'(a))-p(x)+p(a)$. What have I ...
4
votes
3answers
250 views

Polynomial roots problems.

$$ X^5-55X+21$$ Prove that the given polynomial has 2 roots which satisfy the condition: $$X_1X_2=1$$ and find them. I have tried to make use of Viette's relations ,but couldnt get to a satisfying ...
3
votes
2answers
35 views

Adding Polynomials with exponents. Can't get same answer as answer key.

This question is from my final practice exam. simplify and express each answer using positive exponents only. $$(2x^3y^{-2}z^0)^2+8x^{-3}y^2 $$ After working out the problem this is the answer that ...
1
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2answers
34 views

Finding parameters for $f(x) = x^2 + 2(m − a)x + 3am −2 = 0$ that satisfy a condition.

1.find a for $f(x) = x^2 + 2(m − a)x + 3am −2 = 0$ such that for every m real, f has real roots 2.find m such that for every a real, f has real roots My ideea is to demonstrate that $\delta=4(m-a)^2 ...
2
votes
2answers
699 views

Getting K heads out of N biased coins problem (formula generation ).

Problem- Given a set of coins $n$ with each coin $i$ having $P_i$ probability to give heads. Find the probability of getting $k$ heads, when all coins are tossed together. Hi I have solved this ...
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votes
2answers
43 views

$p,q$ coprime polynomials - are $p^n,q^m$ coprime? [on hold]

Suppose that $p,q \in K[x]$ are coprime (there is no polynomial that divides both) and let $n,m \in\mathbb{N}$. Are $p^n$ and $q^n$ coprime and, if so, how to prove it?
3
votes
1answer
95 views

A variation of Buchberger algorithm

Let $I$ be an ideal of a polynomial ring $R$. Fix a monomial order. Denote the $S$-polynomial of $f, g\in R$ by $S(f, g)$ and denote the gcd of their leading terms by $T(f, g)$. Consider the ...
1
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1answer
32 views

Characterization of Groebner Bases in terms of uniqueness of remainders

Let $I$ be an ideal of a polynomial ring $R=k[x_1,\ldots,x_n]$ over a field $k$. A Groebner basis of $I$ is a finite generating set $\{g_1,\ldots,g_m\}$ such that every leading monomial (according to ...
1
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1answer
45 views

Conics and conics of the form $ax^2+by^2+c=0$

The problem of finding rational points on conics is usually discussed (for example in the book of Silverman and Tate) for conics of the form $ax^2+by^2+c=0$. I assume that those conics are in ...