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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2answers
26 views

Splitting field of $x^2 +1 \;$ over $\mathbb Z_7 $

I need to find the splitting field of $\; x^2+1 \in \mathbb Z_7 [x] \;$ over $\mathbb Z_7 $. The roots of the polynomial are $-i \;$ and $i$. Therefore I would conclude that the splitting field is ...
0
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2answers
32 views

Integral domains, polynomials and division

We know that : If $K$ is a field and $P\in K[X]$ is a polynomial with coefficients in $K,$ then $\alpha\in K$ is a root of $P$ if and only if $(X-\alpha)$ divides $P$. My question is the ...
2
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1answer
26 views

A circulant matrix and its transpose

It is well-known that a circulant matrix $A$ of size $n \times n$ is isomorphic to a polynomial $$p(x) \bmod x^n - 1.$$ If we consider the transpose $A^T$, what is the corresponding polynomial ...
2
votes
0answers
30 views

Finding the roots of a polynomial with limited information

Let $\ f(x) \in \mathbb{R}[x]$ be a 7-th degree polynomial, such that. $$ f(0)=0 \land f(i)=-3i $$ $$ f'(0)=0 \land f'(i)=-21 $$ Find all the complex roots of $ f(x)-3x^7$. Find all possible ...
3
votes
1answer
35 views

Let $a,b,c,d$ be distinct integers such that the equation $(x-a)(x-b)(x-c)(x-d)-9=0$ has an integer root $r$,then find the value of $a+b+c+d-4r.$

Let $a,b,c,d$ be distinct integers such that the equation $(x-a)(x-b)(x-c)(x-d)-9=0$ has an integer root $r$,then find the value of $a+b+c+d-4r.$ As $r$ is the integer root of the equation ...
2
votes
1answer
57 views

How do to find all the roots of $x^4-2yx^3+3y^3x-2y^4=0$?

The main question is how do we factorize $x^4-2yx^3+3y^3x-2y^4=0$ where $y$ is a parameter. I thought we could use Vietta's formula and solve the following system: $$\begin{cases} \begin{split} ...
1
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2answers
63 views

Find all complex roots of $T^4-{1/2}T^2-\sqrt{15}T+{69/16}$

I want to find all complex roots of $T^4-{1/2}T^2-\sqrt{15}T+{69/16}$. The only way I can think to do it is to find 1 complex root, $\alpha$, by inspection, so we can rearrange the polynomial to ...
1
vote
1answer
95 views

Statistical problem: how many books of different widths fit it into a self of a limited certain width?

Let's say I have N sets of books, being the size of the books in a set the same. The cardinality of the every set is different: I might have 3 books of width 5 units (first set), 6 books of width 10 ...
5
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1answer
183 views

Polynomials with some roots whose product is 1

Consider the complex coefficient polynomial equation \begin{eqnarray} ...
8
votes
1answer
120 views

Functional Equation for $f(x-y)+f(y-z)+f(z-x)=2f(x+y+z)$

The following functional equation proved quite difficult. $1.$ $f(x)$ is a polynominal with real coeffecients. $2.$ $f(1)=2,f(2)=20$. $3.$ When for real $x,y,z$ satisfies the condition ...
3
votes
1answer
37 views

Are all rationally parametrized plane curves algebraic? How does one find their degree?

Suppose a plane curve is given parametrically by $x=p(t),y=q(t)$, where $p,q$ are rational functions. I originally assumed that this means that the parametrized curve is algebraic, i.e. that it is the ...
3
votes
1answer
65 views

How do computer programs find roots of high-degree polynomials?

My question is motivated by curiosity about the optimization of high-degree polynomial functions. Let's say your experiment data are modeled by a non-trivial 15th degree polynomial. Taking the ...
2
votes
1answer
27 views

Solution to Sextic Polynomial with Two Real Roots

I have the polynomial $$f(x;a)=3ax^6+6x^5-9ax^4-4x^3+9ax^2+6x-3a$$ where the variable $a$ is a random variable from the uniform distribution in the range $[0,1)$. When I analyze this function using ...
0
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0answers
15 views

How to solve for the constants of a non-linear equation?

I don't know the correct method to solve for the constants in equations like these (when I am trying to find the solution to a trial non-homogeneous recurrence): $$a\cdot n^2 + b\cdot n^3 + c\cdot ...
0
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0answers
3 views

Relation between polynomials and derivatives in Savitsky-Golay fitting

I am seeking a more thorough explanation of some of the properties of Savitsky-Golay (S-G) filters, that are maybe not intuitive to people who have worked with least squares in other contexts. To ...
0
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1answer
18 views

My nonhomogeneous recurrence trial solution fails

$$T_n = 6T_{n-1} - 13T_{n-2} + 12T_{n-3} - 4T_{n-4} + 5n^2 + 3n + 2 + 2^n + n2^n$$ The characteristic polynomial is $x^4 - 6x^3 + 13x^2 - 12x + 4 = 0$, or $(x-2)^2 (x-1)^2 = 0$. Therefore the ...
2
votes
1answer
34 views

How to factor a polynomial of degree 4 that is the product of two irreducible quadratic polynomials

It is easy to construct a polynomial of degree four with integer coefficients that doesn't have any real roots, if it has a real root then it can be factored by division by $x-a$ where $a$ is the root ...
0
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0answers
7 views

Solve $\sum_i^n p_i L^{n-i}s^i(1-x)^{n-i}x^i=\prod_i^{n/2} \left( a_ix^2+ b_ix+c_i\right)$

$p(x)$ is a polynomial with coefficients in terms of S, L. $p_i, S, L$ are rational numbers. $$ p(x)=\sum_i^n p_i L^{n-i}s^i(1-x)^{n-i}x^i=\prod_i^{n/2} \left( a_i(s,L)x^2+ b_i(s,L)x+c_i(s,L) \right) ...
0
votes
0answers
9 views

numerically obtaining relative condition number of a monic polynomial with respect to finding its roots

Is there a way to numerically find the relative condition number (and the associated perturbation vector) of a monic polynomial with respect to finding the roots of that polynomial? For example, if I ...
0
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0answers
17 views

General Hermite interpolation problem

Good evening, i wonder if someone could help me with the proof of the general hermite interpolation error.
3
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2answers
36 views

$(x+y)(x+1)(y+1) = 3$ and $x^3 + y^3 = \frac{45}{8}$

I've came across this problem : If $x$ and $y$ are real numbers such that $(x+y)(x+1)(y+1) = 3$ and $x^3 + y^3 = \frac{45}{8}$, find $xy$. This is what I've tried so far: $$x^3 + y^3 = ...
0
votes
1answer
11 views

Determining a homogeneous polynomial (with N indeterminates) from an integer

Imagine you have some computer program that requires an input of N values (say $a,b,c$), and calculates some homogeneous polynomial (with some small natural number coefficients) returning an integer ...
1
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2answers
37 views

Polynomials and common roots

When dividing $f(x)$ by $g(x)$: $f(x)=g(x)Q(x)+R(x)$. How to find the quotient $Q(x)$ and the remainder $R(x)$? For example: $f(x)=\ 2x^4+13x^3+18x^2+x-4 \ $ , $g(x)=\ x^2+5x+2 \ $ At first $g(x)= ...
3
votes
1answer
46 views

How do you find the value of $N$ given $P(N) = N+51$ and other information about the polynomial $P(x)$?

Problem: Let $P(x)$ be a polynomial with integer coefficients such that $P(21)=17$, $P(32)=-247$, $P(37)=33$. If $P(N) = N + 51$ for some positive integer $N$, then find $N$. I can't think of ...
1
vote
1answer
32 views

How to solve this non-homogeneous recurrence?

I've made a few threads recently asking how to solve non-homogeneous recurrences and I think I've gotten the hang of it, but now I want to try a complicated thing like this: $T(n) = 4T(n-1) + 2T(n-2) ...
0
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2answers
52 views

Finding angles of a quadrilateral

There is a quadrilateral. Length of all $4$ sides are known (lets say $a,b,c,d$). All $4$ angles are $\leq 180^{\circ}$, but their exact value is unknown. lets say the four angle names are ...
6
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2answers
119 views

Looking for a function $g(x)$ such that $g(2x+2) = g(x) + 2x+2$

So recently I got bored in maths class (I'm in tenth grade) and made up a little equation that looked something like this: $$g(f(x)) = g(x) + f(x) $$ My original goal was to find different $g(x)$ to ...
0
votes
1answer
24 views

How to solve non-homogeneous recurrences?

I am trying to find a way to solve non-homogeneous recurrences by solving the homogeneous and non-homogeneous parts separately. I can use generating functions for the whole thing, but I want to learn ...
4
votes
3answers
52 views

Monic irreducible polynomials of degree 6 in $F_{5}[X]$

Question A How many monic irreducible polynomials of degree 6 in $F_{5}[X]$ Question B Give an example of an irreducible polynomial of degree 6 in $F_{5}[X]$ Idea for a Such a polynomial would be ...
0
votes
0answers
13 views

Count the number of positive real solutions of a polynomial of arbitrary degree?

Let $$P(x) = a_0 + a_1 x + \dots + a_n x^n$$ be a generic polynomial of order $n$. I need to know the number of positive real solutions to the equation $P(x)=0$. Specifically, I need to determine if ...
0
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0answers
32 views

Modulus-amplifying polynomials

I'm trying to prove that the family of polynomials $\lbrace P_k \rbrace$ defined as follows \begin{equation*} P_k(x) = (-1)^{k+1}(x-1)^{k} \left( \sum_{j = 0}^{k-1} \binom{k+j-1}{j} x^j \right) + 1. ...
0
votes
2answers
35 views

Solving a third degree polynomial without calculator

I was just wondering if it is possible to find a solution to this without using a CAS/Calculator (with wolframalpha I get $x \approx 3.865$) $\dfrac{1}{x^3} = \dfrac{4}{(10-x)^{3}}, x\in\mathbb{R}$ ...
3
votes
1answer
73 views

An inequality on three constrained positive numbers

Assume $a,b,c$ are all positive numbers, and $2a^3b+2b^3c+2c^3a=a^2b^2+b^2c^2+c^2a^2$. Prove that: $$2ab(a-b)^2+2bc(b-c)^2+2ca(c-a)^2\ge(ab+bc+ca)^2$$
0
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1answer
44 views

How to factor polynomials by hand?

Is there a good approach for factoring polynomials by hand (e.g. if you're in an interview situation without access to a computer)? For example $1−4z+5z^2−2z^3$?
1
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0answers
36 views

approximation with polynomials without stone weierstrass

This is from a course in real analysis. Let $f:G\rightarrow \mathbb{R}$ be a continuous bounded function and $G\subset\mathbb{R}^n$ an open bounded set. Prove that for every compact set $K\subset G$ ...
1
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0answers
27 views

If $Q(P(x), x)$ is the zero polynomial, then $Q(x, y) = (y - P(x))A(x, y)$ for some $A$

Let $Q(x, y)$ be a bivariate polynomial over some field $\mathbb{F}$, and $P(x)$ a univariate polynomial over $\mathbb{F}$ such that $Q(P(x), x) = 0$ for every $x$. Show that then, $Q(x, y) = (y - ...
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0answers
24 views

Monic polynomial terminology

If the constant term of a monic polynomial is one or negative one, is there a name for that special kind of monic polynomial?
2
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0answers
133 views

Prove that $(a-b)^n\mid (a^n-b^n) \iff n=1$ under given conditions

Suppose that $a,b,(a-b)$ are pairwise co-prime (i.e. $a\perp b\perp (a-b)\perp a$), and that $\frac{a}{2}<b<a$, where $a$ and $b$ are both positive integers greater than $2$. Let $n$ be odd. ...
3
votes
2answers
93 views

Describe the rational points on $y^2 = 11 - 2x^3$

Describe the rational points on the title curve. My attempt: Consider the line $L$ with slope $m$ that passes through the point $(1, 3)$. To obtain rational solutions, we need to have $m$ rational. ...
0
votes
1answer
36 views

How to do polynomial long division with absolute value?

I wanted to know whether anyone can tell me how to solve i.e. $( 4x^5 + abs (x)^5 )$ : $( x^4 -4)$, because I want to find asymptotes. Shall I look at : $4x^5 + x^5$ and $4x^5 - x^5$ separately? ...
-1
votes
1answer
40 views

Is there a general formula the solutions of a polynomial equation of the form $Ax^n + Bx^{n-1} + C = 0$?

Is there a general formula for the solutions of a polynomial equation of the form $$Ax^n + Bx^{n-1} + C = 0,$$ where $A$, $B$, $C$, and $n$ are constants?
1
vote
5answers
96 views

Proving $x^4+2$ cannot be factored into $2$ degree polynomials

My book says that it can't be because if I try to write $x⁴+10x³+15x²+5x+12$ as: $$x^4+2$$ (which is $p(x) mod 5$) then $x⁴+2$ is irreducible because: $$x^4+2 = (ax²+bx+c)(a'x²+b'x+c)$$ is ...
-1
votes
0answers
33 views

Any ideal is an extended one

It is true for any commutative rings $S$ and $T$ with $1$ and any ring homomorphism $f:S\to T$ that the set $E$ of extended ideals in $T$ equals $\{J\mid J^{ce}=J\}$. In fact, if an ideal $J$ of $T$ ...
0
votes
0answers
9 views

$d(x)$ is an gcd $iff$ $a\cdot d(x)$ is a gcd ($a\neq 0)$

My definition for gcd is that: If $d(x)$ is a common divisor of $p_1, \cdots, p_n$, and for any other $d'(x)$ that is also a common divisor, this $d'(x)$ will divide $d(x)$, then $d(x)$ is the ...
0
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0answers
23 views

Significance of various lex sort on polynomials

I have just finished writing a monomial order sort package for Maxima CAS which supports variety of lexicographic orders. But I want to know what are the uses of these sorts on polynomials. One that ...
1
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0answers
44 views

Computing Krull dimension of $\mathbb{Z}[X_1,\ldots,X_n]/I$ [closed]

Let $I$ be an ideal of $\mathbb{Z}[X_1,\ldots,X_n]$. How does one compute the Krull dimension of $\mathbb{Z}[X_1,\ldots,X_n]/I$? Are there any general methods? Or methods which work in special cases?
4
votes
3answers
56 views

Find all irreducible polynomials of degree $2$ over $\mathbb{Z}_5$

Obviously if I write all the possible ones and try the roots I'd get a LOT of polynomials $(125)$ and I'd have to test $5$ roots for each of them, which would be a LOT. Is there any idea? I must also ...
0
votes
4answers
87 views

Irreducibility of $x^4-5$ over $\mathbb{Z}_{17}$

Obviously, it'd be hard to try all the $17$ elements to see if there is some root, and even if there is none, it'd be necessary to verify if it can't be factored into two irreducible 2 degree ...
0
votes
2answers
37 views

What does this notation mean: $f(x)\in\mathbb{Z}[x]$

I have come across this excercise: Find all $f(x)\in\mathbb{Z}[x]$ such that $x^6+x^3f''(x)=f(x^3)+x^3$. What is the meaning of this notation: $f(x)\in\mathbb{Z}[x]$?
1
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2answers
29 views

Which of the sets are ideals and maximal ideals?

The exercise asks me to prove which of the sets are ideals, and if they are, which of those are maximal. I have these 4 cases: $$ a) J = \{f(x)\in \mathbb{Q}[x]: f(1)=f(7)=0 \} \\b) J = \{f(x)\in ...