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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6
votes
4answers
78 views

Showing that for $f \in K[x]$, we have $f(x) \mid f(x + f(x))$

Let $K$ be a field an $f \in K[x]$. I now want to show that $f(x) \mid f(x + f(x))$ (in $K[x]$). I know that I need to find a polynomial $g \in K[x]$ so that $f(x) g(x) = f(x + f(x))$. So I thought ...
2
votes
5answers
57 views

Solving $2x^4+x^3-11x^2+x+2 = 0$ [duplicate]

I am having no idea how I can solve this problem. I need help! Here's the problem $2x^4+x^3-11x^2+x+2 = 0$ I am learning Quadratic Expressions and this is what I need to solve, and I can't ...
1
vote
3answers
144 views

Does $K = \mathbb Q[X]/(X^4 - 2)$ contain the imaginary unit $i$?

Let $P(X) = X^4 - 2 \in \mathbb Q[X]$. a) Prove that $P(X)$ is irreducible. b) Prove that the field $K = \mathbb Q[X]/(P(X))$ is an algebraic extension of $\mathbb Q$ and find a generator of it....
0
votes
1answer
19 views

square of polynomial still harmonic? [on hold]

Let $P(z)=\sum_{i=0}^n a_i z^i$ be a polynomials on $\mathbb{C}[z]$ such that $a_i$ are real numbers. $|P(z)|^2$ is a harmonic function ?
0
votes
0answers
21 views

Techniques to turn expressions involving integer roots into polynomials by substitution?

Inspired by this question involving an Equation for a Torus How to find a parametrization for a torus? I started wondering if there is some systematic approach to do substitutions to make equations ...
1
vote
1answer
63 views
1
vote
5answers
94 views

How to prove that $(x-1)^2$ is a factor of $x^4 - ax^2 + (2a-4)x + (3-a)$ for $a\in\mathbb R$?

Let $a \in R$. Verify that $(x − 1)^2$ is a factor of $$p(x) = x^4 − ax^2 + (2a − 4)x + (3 − a)$$ How can I solve this question?
0
votes
2answers
23 views

Find $a+b$ for $a, b$ such that $(x+1)^{n}(x^{2}+ax+b) \equiv 2^{n}(x-1) \mod (x-1)^{2}$

Since $2^{n} = \sum_{0}^{n}\binom{n}{k},$ we have from the given congruence the congruence $$\sum_{0}^{n}\binom{n}{k}(x^{k+2} + ax^{k+1} + bx^{k} - x +1) \equiv 0 \mod (x-1)^{2}.$$ The given answer ...
3
votes
2answers
69 views

Image of polynomial is $\mathbb{Q}$

I know that the polynomial $f(x)=\frac{1}{2} x^2 + \frac{1}{2}x \in \mathbb{Q}[x] $ has that $f(\mathbb{Z})\subset\mathbb{Z}$. My question is a bit different: Does there exist a polynomial $f \in \...
0
votes
0answers
41 views

A problem with infinitely many eigenvalues on a finite dimensional vector space

I want to develop some theory before posing the problem. Kindly stay with me. Consider $ Aut (k[x_1,...,x_n])$ where $k$ is an algebraically closed field, you can take $k=\Bbb C$. $\alpha \in Aut(k[...
4
votes
1answer
97 views

The other $47$ roots of the minimal polynomial for $\cos 1 ^\circ$

The minimal polynomial for $x=\cos 1 ^\circ=\cos \frac{\pi}{180}$ is: $$281474976710656 x^{48}-3377699720527872 x^{46}+18999560927969280 x^{44}- \\ -66568831992070144 x^{42}+162828875980603392 x^{40}-...
29
votes
2answers
8k views

Number of monic irreducible polynomials of degree $p$ over finite fields

Suppose $F$ is a field s.t $\left|F\right|=q$. Take $p$ to be some prime. How many monic irreducible polynomials of degree $p$ do exist over $F$? Thanks!
0
votes
0answers
31 views

How to get a feel for rigor/form used in mathematics?

I'm an engineer, and while you get introduced to many concepts of mathematics, but only with a subset of the vocabulary, and none of the rigor and proofs. So while trying to read a mathematical book, ...
5
votes
2answers
108 views

Roots of $y=x^3+x^2-6x-7$

I'm wondering if there is a mathematical way of finding the roots of $y=x^3+x^2-6x-7$? Supposedly, the roots are $2\cos\left(\frac {4\pi}{19}\right)+2\cos\left(\frac {6\pi}{19}\right)+2\cos\left(\...
0
votes
1answer
34 views

Proof Check: Prove Relation Between Invariants Is the Only Relation

Consider the finite matrix group $C_{4} \subset$ GL$(2,\mathbb{C})$ generated by $$A = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} \in \text{GL}(2,\mathbb{C}).$$ (a)Prove that $C_{4}$ is ...
0
votes
0answers
18 views

What is the difference between common components of $f(x,y)$ and common factor of $f(x,y)$?

Let $f(x,y)$ be a polynomial.What is the difference between common components of $f(x,y)$ and common factor of $f(x,y)$?
4
votes
0answers
30 views

Sum of divisors in $\mathbb{Z}_2[x]$

Let $A$ denote a polynomial on $\mathbb{Z}_2[x]$ and $\sigma(A)$ denote the sum of divisors on $A$. Let$$A = x^h(x + 1)^k P^l Q^{2n - 1},$$where $P$, $Q$ are irreducible polynomials with degree at ...
12
votes
1answer
106 views

Does there exist a polynomial $p(x) \in \mathbb C[x]$ such that $p(x) \notin \mathbb R[x]$ and $p(x)p(-x)=p(x^2)$?

Does there exist a polynomial $p(x) \in \mathbb C[x]$ such that $p(x) \notin \mathbb R[x]$ and $p(x)p(-x)=p(x^2)$ ? I have noticed that if $a_n$ is the leading co-efficient of $p(x)$ then $a_n=(-1)^n ...
1
vote
5answers
60 views

Is $F[x,y]$ a Euclidean Domain?

I was wondering if this is just common knowledge. So far for a field $F$ and transcendental $x$ and $y$, I know one can define the degree by $1) \deg c =0$, for any $c \in F-\{0\}$ $2) \deg x^{n_1}...
1
vote
1answer
45 views

$k[x,y,z]/(y-x^2,z-x^3)\cong k[x]$, where $k$ is a field

This is generalizing from a previous question, which asks to prove that $k[x,y]/(y-x^2)\cong k[x]$. The way I proved that was by using the homomorphism $\phi:k[x,y]/(y-x^2)\to k[x]$, $\phi(\overline{f(...
3
votes
2answers
39 views

Is it possible to convert the polar equation $\ r = k \cos (\theta n) + 2$ into cartesian form?

Is it possible to convert the polaer equation $$\ r = k \cos (\theta n) + 2$$ into cartesian form? Here, $k$ is some constant and $n$ is any positive whole number greater than $2$. The ...
4
votes
1answer
126 views

Minimal polynomial for $x=\tan \left( \frac{2}{5} \arctan p \right)+\tan \left( \frac{3}{5} \arctan p \right)$

I found numerically that the minimal polynomial for: $$x=\tan \left( \frac{2}{5} \arctan p \right)+\tan \left( \frac{3}{5} \arctan p \right)$$ has the following form: $$(3p^2-1)(p^2-1)\color{blue}...
-1
votes
1answer
36 views

Finding the limit of these functions

Do you mind explaining me how to find the limit of these functions? $\lim_{n\rightarrow \infty}\frac{7n^5-2}{(n+4)^5n}$ $\lim_{n\rightarrow \infty}\frac{(n^3+1)n^3}{((n+1)^3+1)(n+1)^3}$ Thank you ...
0
votes
1answer
24 views

In an $\Bbb{N}$-graded domain $A$, units are homogeneous

Let $A$ be a graded domain, with additive subgroups $A_n,\,\forall\,n\geq 0$, s.t. ${A_n\cdot A_m}\subseteq A_{n+m}\,\forall\,n,m\geq 0$, and $A=\bigoplus_{n=0}^\infty\, A_n$ as abelian groups. I wish ...
0
votes
1answer
27 views

Fast way of suming up polynoms

You probably will think that this question is more about algorithms, but actually i'am searching for right formula. What i want to do is to compute $(A^n + A^{n-1} + .. + A + 1))$ mod $(10^9 + 7)$ ...
0
votes
1answer
34 views

Radius of Convergence for Polynomial

I have to find the radius of convergence for this one, but I haven’t found a solution for this type of a term. $\mathbb f (x)$ = $1 + 7x^3 + 5x^4 + x^{13} + x^{2015}$ Could you help me out?
4
votes
0answers
86 views

Injectivity of $R \to R[t]/(f)$ for non-constant $f\in R[t]$

Question: Let $R$ be a (unital commutative) ring and $f = a_0 t^n + \cdots + a_n \in R[t]$ a non-constant polynomial. What are (necessary and sufficient) conditions on the coefficients $a_0,\ldots,a_n ...
3
votes
0answers
42 views

How are varieties related polynomials?

My teacher says that varieties and ideals are related to each other while I tend to mix polynomials and varieties in my terminology. Could some explain how varieties are related to polynomials? And ...
0
votes
1answer
37 views

Find the numbers at which the polynomial is irreducible over $\mathbb{Q}$

How can I find the integers $a$ at which the polynomial $f(x)$ is irreducible over the field $\mathbb{Q}$? Thank you! $$f(x) = 5x^4 - 6x^3 - ax^2 - 4x + 2$$
3
votes
1answer
67 views

Find real parts of the complex roots of this $9^{th}$ order polynomial in explicit form

I have a following polynomial. (See WolframAlpha ): $$x^9-6x^8+14x^7-16x^6+36x^5-56x^4+ 24x^3-320x+\frac{640}{9}=0 \tag{1}$$ Wolfram says that $(1)$ has three real roots and three pairs of complex ...
0
votes
0answers
17 views

Cubic's Irreducible case.

I know how to solve a cubic equation using Cardano's method, but some equation when solved through Cardano's method give complex numbers under a cube root sign. For Example: $x^3-15x=4$. I want to ...
2
votes
3answers
57 views

Show that polynomial is irreducible over $\mathbb{Q}$

How I can prove that polynomial $f(x)$, where$$f(x) = x^4 + 3x^3 + 3x^2 - 5$$ is irreducible over $\mathbb{Q}$? Thank you
0
votes
2answers
84 views

Polynomial with infinite roots

I started Ring theory recently and I came across this statement while reading polynomial rings.. If $F$ is an infinite field and let $f(x)\in F[x]$ . If $f(a)=0$ for infinitely many elements $a$ ...
2
votes
3answers
613 views

Solve $x^4+3x+20=0$ by Ferrari's method

Comparing the equation $$x^4+3x+20=0$$ With the equation $$(x^2+\lambda)^2-(mx+n)^2=0$$ we get $m^2=2\lambda,$ $-2mn=3,$ $n^2=\lambda^2-20$ Now, $4m^2n^2=9\Rightarrow 4(2\lambda)(\lambda^2-...
0
votes
0answers
53 views

Sum of the divisors of polynomials in $\mathbb{Z}_2[x]$ [on hold]

Let $A$ be a polynomial in $\mathbb{Z}_2[x]$ and $\sigma(A)$ denote the sum of divisors of $A$. Let $A=x^h(x+1)^kP^lQ^{2n-1}$, where $P,Q$ are irreducible polynomials with degree $\geq 2$, $l\neq 2^r-...
6
votes
2answers
252 views

The largest root of a recursively defined polynomial

Suppose that for all $x \in \mathbb{R}$, $f_1(x)=x^2$ and for all $k \in \mathbb{N}$, $$ f_{k+1}(x) = f_k(x) - f_k'(x) x (1-x). $$ Let $\underline{x}_k$ denote the largest root of $f_k(x)=0$. I ...
2
votes
2answers
489 views

Symbolic polynomial interpolation

I'm trying to create polynomials from some symbolic points to discretize derivations. This means I'm having data like $(a, \phi(a)),\ (b, \phi(b) ) $and $(c, \phi(c))$ and want to fit a second order ...
-1
votes
1answer
70 views

How to solve this problem on abstract algebra? [closed]

I feel hard in Abstract Algebra in application. Help me... with this problem.. Let $P(z),Q(z)$ be two complex nonconstant polynomials of degree $m,n$ respectively. The number of roots of $P(z)Q(z)$...
2
votes
1answer
40 views

Values of Polynomial in $\mathbb{F_{2^n}}$

$ \phi~:~ \mathbb{F_{2^n}} \rightarrow D, ~\phi(X)= X+X^2+...+X^{2^{(n-1)}}$ Show that $D=\{0,1\}$ for any n and $\phi(X)=0$ exactly for half of the $X \in \mathbb{F_{2^n}}$. Hi, got a bit rusty ...
0
votes
0answers
16 views

Simplify the expression $\varphi(x)=\sum_{i=1}^{n+1}\alpha_i Q(x+\lambda_i)$

Let the polynomial $𝑄 (𝑥)$ $𝑛-$th degree with real coefficients and set real numbers $\lambda_1 <\lambda_2 <... <\lambda_{𝑛+1}.$ Simplify the expression $$\varphi(x)=\sum_{i=1}^{n+1}\...
-4
votes
3answers
58 views

THEORY OF EQUATIONS [closed]

I would like to solve this problem: If $a$ is a special root of the equation $x^8-1=0$, then prove that $$1+3a+5a^2+7a^3+....+15a^7=16/(a-1)$$
11
votes
4answers
1k views

Why can a quartic polynomial never have three real and one complex root?

It seems that a quartic polynomial, (degree 4) either can have 0 real, 1 real, 2 real, or 4 real roots, and the rest is complex roots. Why can it not have 3 real roots and 1 complex?
0
votes
2answers
42 views

Prove that $q(x)$ does not divide $p(x)$?

"Let $F$ be a field and suppose that $p(x),q(x) \in F[x]$ are the two polynomials $p(x) = x^5 - x^4 + x^3 - x^2 + x - 1$ and $q(x) = x^2-1$ (i) Prove that $q(x)$ does not divide $p(x)$ when $F = \...
2
votes
1answer
44 views

Explicit delta for polynomial limit

I'm looking for an explicit formulation for $\delta$ in the $\epsilon-\delta$ formulation of the limit for a polynomial $p(x) = \sum_{n=0}^N a_nx^n$. For example, in the the specific linear case $p(...
1
vote
1answer
40 views

How many polynomials are squarefree?

Of course, this depends on the field, and how we measure "how many," but it seems I cannot find an answer to this except over finite fields. My question specifically is If we have a field $F = \...
0
votes
1answer
29 views

Decompose $f = x^3 - x^2 \in Z_2$ in a product of irreducible polynomials

$$x^3 - x^2 = x^2(x-1)$$ So, after reaching this point, $x-1$ is surely an irreducible polynomial because $gr(x-1) = 1$, but how to continue?
1
vote
0answers
55 views

How to expand a fraction of polynomials into a series?

I am trying to expand into a series (sorry, I'm not sure of proper terminology here but hopefully it is clear) the ratio of a polynomial in $x^2$ at two consecutive values: $$\frac{a_0 (x+1)^n + a_2 (...
1
vote
0answers
31 views

Is there a nonzero polynomial in $\mathbb{H}[x]$ which vanishes in all $\mathbb{H}$?

I know that over any infinite field $F$ there are no nonzero polynomials in $F[x_1,\cdots,x_n]$ which vanish in all $F^n$ (Proof is by induction with basis step given by the fact that polynomials in $...
1
vote
2answers
32 views

Determining a basis for a space of polynomials.

Let $V = \mathbb R[x]_{\le 3}$ I have the space of polynomials $U_2 = \{ p = a_0 + a_1x + a_2x^2 + a_3x^3 \in V \mid a_1 - a_2 + a_3 = 0, a_0 = a_1 \}$ I am asked to find a basis, so I proceed by ...
0
votes
1answer
48 views

Alternate proof for Vieta's formula (formula for the summing the roots of a polynomial)

I just saw Vieta's formula for the first time, where it was stated that given some polynomial $$p(x)=a_nx^n+\cdots+a_0,$$ let $x_1,\ldots,x_n$ denote its roots. Then $$\sum_{i=1}^n x_i=-\frac{a_n}{...