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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2
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1answer
34 views

General method: show subset of $\mathbb{C}$ is connected

Consider the two sets $$ A = \{z \in \mathbb{C} : |z^2 - 3| < 1\}, ~~~~ B = \{z \in \mathbb{C} : |z^2 - 1| < 3 \} $$ $B$ is connected, while $A$ is not. However, I have no idea how to prove this....
2
votes
2answers
64 views

The expansion of $(a+b+c+d)^{20}$ [closed]

Let us consider the expansion of $$(a+b+c+d)^{20}.$$ Find: The coefficients of $a^{11}b^6c^2d$ and $a^{11}b^9$, The total number of terms of this expansion, The sum of all the coefficients. Thank ...
2
votes
0answers
36 views

How to define hypergeometric function ${}_1 F_1(-n+1;-n+1;z)$ for $n$ positive integer

Consider a truncated Taylor series of the exponential function to approximate $e$: $$ E(n) = \sum_{k=0}^{n-1} \frac{1}{n!} $$ I thought of computing this using the hypergeometric finite series $_1 F ...
1
vote
0answers
38 views

fifth degree equation

The general form of the fifth degree equation to achieve Bring-jerrard form some coefficients are complex . Should they be considered only real part of the coefficients that the roots of the fifth ...
1
vote
3answers
170 views

On every other finite field at least one of −1, 2 and −2 is a square, because the product of two non squares is a square

[Except on field extensions of $\mathbb{F}_2$] On every other finite field at least one of $−1$, $2$ and $−2$ is a square, because the product of two non squares is a square. I don't see why this is ...
1
vote
2answers
63 views

Finding an interval in which all the real roots of a polynomial lie

I'm making a program which uses simple bisection method to find the roots of a polynomial. For me to implement this method, I need a rough interval where it can be said with absolute certainty that ...
0
votes
2answers
80 views

Solve $z^5=-32$ and draw its solutions in complex space, then describe their characteristic geometrical property.

I'm solving past exam questions in preparation for an Applied Mathematics course. I came to the following exercise, which poses some difficulty. If it's any indication of difficulty, the exercise is ...
1
vote
2answers
70 views

Limit of polynomial function

Good evening to everyone! How can I solve a limit of this type. I would try l'Hospital or factoring out but it won't work for sure. $$ \lim _{x\to 0}\left(\left|\frac{2x^7+x^2}{x^7+5x^4}\right|\right) ...
7
votes
5answers
131 views

Can all functions over $\mathbb{Z}/4\mathbb{Z}$ be “described” by a polynomial?

Consider the set $T$ of functions from $\mathbb{Z}/4\mathbb{Z}$ to $\mathbb{Z}/4\mathbb{Z}$. I'm now asked to prove or disprove the statement that all functions in $T$ can be described by a polynomial ...
0
votes
0answers
23 views

Solve system of polynomials

I have four polynomials with four unknowns $x_{1},x_{2},y_{1}$ and $y_{2}$ as following $$ \left\{ \begin{array}{c} m_{1} + m_{2}x_{1} + (m_{5} + m_{6}x_{2})y_{1} + (m_{3}+m_{4}x_{1})y_{2} ==0 \\ n_{...
0
votes
1answer
30 views

Comparing the growth of two polynomials?

For two polynomials, $P(x)$, $Q(x)$, $x \in \mathbb{N}$, both having all positive integer coefficients, and the degree of $P$ is greater than the degree of $Q$, is it true that $\frac{P(x)}{P(x-1)} &...
0
votes
0answers
30 views

Prime factors polynomials

I have proved a theorem which I will state: For $f(x)=x^n+\sum_{i=1}^nh_ix^{n-i}$ a polynomial of degree $n$ where $h_i=r_i+d_i$ with $r_i$ real and $d_i$ infinitesimal. Then if $x^n+\sum_{i=1}^nr_ix^{...
0
votes
0answers
52 views

A question on polynomials..

Let $f(x,y)$ be reduced polynomials with coefficients in $\mathbb{R}$. Can we say that $f$ and $\frac{{\partial f}}{{\partial x}}$ and $\frac{{\partial f}}{{\partial y}}$ don't have any common factor?(...
0
votes
0answers
20 views

Hermite Polynomials using gram-schimdt

How does one generate these polynomials using the gram-Schmidt algorithm? I know how it should work, but I get 0 as the value for the scalar product of (p1,q0) and q1 should be 2x not x. $$q1\left(x\...
1
vote
0answers
68 views

Polynomial taking irrationals to irrationals

Problem: Find all polynomials from $\mathbb{R}\to \mathbb{R}$ $f$ with integer coefficients taking irrationals to irrationals. My attempt: It is clear that the problem statement is equivalent to ...
14
votes
6answers
2k views

Can the product of two polynomials result in a single term?

Assume that the polynomials that we multiply consist of more than one term. I don't think we can get a result containing only a single term, but I don't know how to prove it.
5
votes
1answer
62 views

L-series through integrals of rational functions

Recently I stumbled upon this short proof here: $$L(1,\chi_2)=\sum_{j=0}^{+\infty}\left(\frac{1}{3j+1}-\frac{1}{3j+2}\right)=\int_{0}^{1}\frac{1-x}{1-x^3}\,dx=\int_{0}^{1}\frac{dx}{1+x+x^2}$$ so: $$\...
0
votes
0answers
45 views

Show that $f(x)=x^3+(2+i)x+(1+i)$ is irreducible in $\mathbb{Z}[i][x]$

Problem says: Show that $f(x)=x^3+(2+i)x+(1+i)$ is irreducible in $\mathbb{Z}[i][x]$. I think I have to use Eisenstein's criterion with substituting $x$ with something but for me to show that ...
5
votes
2answers
87 views

Functional Equation of iterations

Problem: Let $f : \mathbb{Q} \to \mathbb{Q}$ satisfy $$f(f(f(x)))+2f(f(x))+f(x)=4x$$ and $$f^{2009}(x)=x$$ ($f$ iterated $2009$ times). Prove that $f(x)=x$. This is a contest type problem ...
0
votes
0answers
32 views

Matrices over integer fields to solve complex polynomials.

Inspired by the fruitful answer to this question regarding numerically solving polynomial equations in terms of simpler fields (in that case representing real numbers as fractions of integers), I ...
0
votes
3answers
77 views

Least possible degree of polynomial with atleast one irrational root [closed]

Let $p(x) = a_0 + a_1x + a_2x^2 + ..... + a_nx^n$ be a nonzero polynomial with integer coefficients. If $p(\sqrt 2 +\sqrt 3 +\sqrt 6) = 0$, then the smallest possible value of $n$ is?
0
votes
0answers
54 views

About correct meaning of radical solution to polynomials

Suppose that we have the equation $(1)$ $x^2 = a$ Whose root is $ x = \mp \sqrt{a}$. This is the "radical solution" of the equation. Suppose that we have $\sqrt[3]{m +\sqrt{n}}+ \sqrt[3]{m -\sqrt{...
0
votes
0answers
42 views

Prove that if $R$ is real closed field, $f \in R[X]$, and $f(a)f(b) <0$ for some $a < b$ in $R$, then $f(r) = 0$ for some $a < r < b$

Prove that if $R$ is real closed field, $f \in R[X]$, and $f(a)f(b) <0$ for some $a < b$ in $R$, then $f(r) = 0$ for some $a < r < b$. Suppose that $f$ is a irreducible polynomial of ...
3
votes
1answer
32 views

Using the Factor Theorem

We are asked to use the above Theorem to show that: $f(x)=x^{91} + 3x^{73} - 2x^{37} - 2$ has ($x-1$) as a factor. It's easy to show that $f(1)=0$ by inserting $1$ for $x$, yielding: $f(1)=1^{91} + ...
1
vote
1answer
49 views

Iterations with matrices over simple fields approximating solutions for more complicated fields.

Inspired by this question I started wondering if there exist some systematic way to construct approximation to any number one can find using matrices over a preferrably simpler field. In the question ...
0
votes
0answers
26 views

Doesn't the recursive Fast Fourier Transform violate f(-x) =/= f(x) for odd functions?

When you recursively split into $Y_{even}$ and $Y_{odd}$, from the second recursion onwards don't these sets have their even-ness and odd-ness violated? I.e., assume you are running the FFT algorithm ...
2
votes
2answers
92 views

$p(x)$ is a polynomial in $R[x]$ such that $p(0)=1$ , $p(x) \ge p(1)$ and $\lim_{x \rightarrow \infty} p''(x)=4$ Find $p(2)$

$p(x)$ is a polynomial in $R[x]$ such that $p(0)=1$ , $p(x) \ge p(1)$ and $\lim_{x \rightarrow \infty} p''(x)=4$ Find $p(2)$ I assumed the degree of $P(x)$ to be smaller ones like $2,3$ and found $P(...
3
votes
0answers
47 views

How a complex root $\eta$ of $x^2 + x + A$ affects the ring $\mathbb{Z}[\eta]$

While reading a statement in P. Pollack's Not Always Buried Deep: A Second Course in Elementary Number Theory I came across a statement that seemed obvious and I am wondering if I am oversimplifying ...
3
votes
1answer
88 views

Seeking an “easy ” way to show that $p(x)=x^6+\cdots+x^2+x+1$ is irreducible over $\Bbb{Z_{17}}$

As the title suggests, we need to Show that $p(x)=x^6+\cdots+x^2+x+1$ is irreducible over $\Bbb{Z_{17}}$ We can immediately answer that it is indeed irreducible since it is the cyclotomic ...
1
vote
3answers
81 views

Why does degree determine the amount of zeros?

We just learned about complex numbers in my math class and I have a question. Why does the degree of a polynomial equal the amount of zeros it has? The degree of $f(x) = x^3 - x^2 + x - 1$ is $3$, ...
0
votes
0answers
25 views

The ability to solve the multivariate nonlinear equations

For m nonlinear polynomial equations with n variables and the highest degree 3, how is the current ability to solve such equations? In the webpage of IBM cplex, it says that: IBM ILOG CPLEX ...
0
votes
0answers
43 views

Proof that there are No Modulo Invertible Polynomials with f(1) = 0

I was reading an article from the NTRU Cryptosystem (probably the first one): NTRU: A Ring-Based Public Key Cryptosystem And I don't know how to prove the assertion he makes in parenthesis in ...
1
vote
1answer
55 views

Polynomial that is irreducible over $ \mathbb{Q} $ but reducible over every finite field [duplicate]

I want to prove that $ X^4 - 10X^2 + 1 $ is reducible in $ \mathbb{F}_p[X] $ for every prime number $ p $, but it is irreducible over $ \mathbb{Q} $. I am not sure how to approach this problem; any ...
0
votes
0answers
17 views

Vanishing polynomial in complex projective space

Assume we are working in $n$-dimensional complex projective space. Why does a (homogeneous) polynomial of degree less than or equal to $d - 1$ which equals $0$ on $d$ points on a line $L$ in ...
1
vote
1answer
65 views

Showing a polynomial of degree 7 is not solvable by radicals. [closed]

Show that the polynomial of $x^7-10x^5+15x+5$ is not solvable by radicals. For a polynomial of degree 5, we simply consider the derivative and determine the number of real and complex roots from ...
1
vote
1answer
30 views

Riesz Representation Thereom for Polynomials with real coefficients problem

Find a polynomial q(x) $\in$ P$_2$($\Bbb R$) Such that $ p ( 1/4 ) = $$\int_0^1 p(x)q(x) \,dx$$ $. I'm sorry to ask this question, but I've been working on it for some time. The inner product on P$_2$(...
2
votes
1answer
47 views

Galois Group Isomorphic to $S_3$.

Let $f \in \mathbb{Q}[x]$ be an irreducible polynomial of degree 3. Suppose $f$ has one real root, we want to show that $$\text{Gal}(L/\mathbb{Q}) \cong S_3,$$ where $L$ is the splitting field of $f$. ...
0
votes
1answer
20 views

proving an inequality involving a linear spline / piecewise polynomial

I have $n+1$ sample points $x_i = \left(\frac{i}{n}\right)^4$ and want to approximate the function $f(x)=\sqrt{x}$ by a linear spline $f_n \in S^{1,0}(\mathcal{T_n})$ on the interval $[0,1]$. I know ...
1
vote
1answer
35 views

A problem about ring of polynomials over a field [duplicate]

For $K$ is an infinite field and $f(x_1,x_2,\ldots,x_n)$ $\in K[x_1,x_2,\ldots,x_n]$ . Prove that If $f(a)=0 $ for any $a \in K^n$ then $f=0$. Can any one help me?
1
vote
2answers
33 views

A system of polynomial equations of degree $2$ in two variables

I need to find an explicit solution of this system of polynomial equations of degree $2$ in two variables $x,\,y$: $$\begin{cases} p_1x^2+q_1y^2+r_1xy+s_1x+t_1y+u_1=0\\ p_2x^2+q_2y^2+r_2xy+s_2x+t_2y+...
1
vote
0answers
34 views

Any way to characterize this family of polynomials?

I have a family of polynomials generated by the recurrence relation $P_{n+1}(w) = (1+w)P_n ^{\ \prime}(w) -(3n-1 +nw)P_n(w) \\ P_1(w) =1$ The family is related to the Lambert $W$-function by its ...
0
votes
1answer
33 views

Why this linear transformation has a basis of eigenvectors?

Consider $W$ one $n$-dimensional vector space over $\mathbb{C}$ and $T:W\to W$ one linear transformation such that $T^3 = \mathbf{1}$ where $\mathbf{1}$ is the identity. I've read that this implies ...
0
votes
1answer
31 views

Get a matrix of polynomial coefficients from the roots

I've got the polynomial $P(z) = \Phi_0 - \Phi_1z $ defined by the following matrices of coefficients: $$ \begin{eqnarray} \Phi_0 = \left[ \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0.2 & 1 &...
0
votes
0answers
37 views

Difference between rationalizing factor and conjugate surd

I have some confusion regarding rationalizing factor and conjugate surd. For binomial surds for example $2+\sqrt{3}$ is conjugate of $2-\sqrt{3}$ and it is also rationalizing factor of $2+\sqrt{3}$. ...
0
votes
1answer
56 views

If the $100$-th derivative of $f$ vanishes on $\Bbb R$, then $f$ is a polynomial.

I have the following statement: If $f^{100}(x) = 0$ for every real number $x$, then $f$ is a polynomial. I couldn't find a counter example so I would like to get some help for prove/disprove. ...
4
votes
1answer
65 views

Prove the polynomial $P_a=X^5 + a$ is reducible over a field

Let $(K, +, \cdot)$ a finite field so that the polynomial $P=X^2-5$ is irreducible. Prove that: a) $1+1 \ne 0$ b) The polynomial $P_a=X^5 + a$ is reducible $\forall a \in K$ a) ...
1
vote
1answer
44 views

I was going through Jordanization by Jonathan Nilsson.

I was going through Jordanization by Jonathan Nilsson. Here he describes the algorithm for Jordanizing any square complex matrix $A$. Here $T := (A - \lambda I )$. Now finding the sub spaces $Im(A -...
0
votes
0answers
12 views

How to factorize a cubic/biquadratic polynomial mentally (without long division method)?

This is essential for me to know because I am preparing for a competition which depends heavily on speed of solving problems (without calculator).
1
vote
2answers
53 views

Polynomial nth derivative

I was wondering how is this done? Let $a,b_0,...,b_n \in \mathbb{R}$. Show that there exists a polynomial $f(x)$ of degree at most $n$ such that $f(a) = b_0, f'(a) = b_1, f''(a) = b_2, ..., f^{(n)}(...
1
vote
1answer
26 views

How to find point in polynomial regresion

I have the following data set: ...