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.

learn more… | top users | synonyms

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
49 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
56 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
39 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
66 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
13 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: ...
0
votes
1answer
18 views

let $P$ $\in$ $\mathbb{F}_{p}[x,y]$, if $P$ does not have multiple factors then $P-1$ is irreducible. [closed]

I need to show the following: Let $P$ $\in \mathbb{F}_{p}[x,y]$. If $P$ does not have multiple factors, then $P-1$ is irreducible." Please help.
0
votes
1answer
48 views

Is it correct this way to compute that radical ideal?

Is it correct to compute that radical ideal in this way? $$\sqrt{(x^2,xz^2-x,y-z)}=\sqrt{(x^2,xz^2-x,y-z,x)}=\sqrt{(y-z,x)}=(x,y-z)$$ In particular, I added $x$ to generators inside the 'root' ...
0
votes
1answer
57 views

Proof that $a^{n}+b^{n}$ is irreducible over $\mathbb Q$

The sum of fourth powers cannot be factored over $\mathbb Q$, since $ a^4+b^4 = (a^2+\sqrt{2}ab+b^2)(a^2-\sqrt{2}ab+b^2)$ And these quadratic factors does not have any real rational factors. How ...
3
votes
4answers
82 views

Find a recursive formula to the given closed formula

I'm asked to find a recursive formula to this closed formula: $$f(n) = 2n + 3^nn$$ I tried to transform this formula to a formula that I might get using the Characteristic polynomial method. As I ...
0
votes
1answer
20 views

If roots of a polynomial are complex number how to visualize Geometrically

I have one basic doubt in complex numbers. We know that if a polynomial equation $P(x)=0$ cuts the $X$ axis or touches the $X$ axis, then they represent Real roots of the polynomial or real roots with ...
0
votes
0answers
5 views

polynomial as product of distinct irreducible and separability

F field. $f \in F[x]$. I know if $f$ as product of irreducible is squarefree, then $f$ mayn't be separable if field is not perfect. But what is problem in following proof: Let f as product of ...
0
votes
0answers
34 views

Please help understand the polynomial transformation

I found this transformation in the explanation section of a text book (the simplest problems). It says that this expression: $$\frac{(b+c-a)^2}{2bc}\cdot\frac{abc}{a-b-c}$$ can be simplified to this: $...
2
votes
3answers
97 views

Is the solution from wolfram alpha for $x^4+bx^3+f=0$ only wrong for $b=0$?

edit, tldr: it seems like the question is interpreted as "for which combinations of a and f is there a solution to the expression"? I'm only interested in "for which combinations of a and f is the ...
0
votes
2answers
58 views

Understanding divison by monic polynomial in $R[x]$ where $R$ is an arbitrary ring

I read "Algebra: Chapter 0" by P.Aluffi. I encountered a topic where it says you can divide any polynomial in $R[x]$($R$ is any ring) by a monic polynomial(that is, a polynomial of the form $x^d + \...
1
vote
1answer
45 views

Simplify a polynomial in $e^{i\omega}$ without using trigonometry

Given $z=e^{\omega i}$, the following polynomial can be reduced to a quadratic using trigonometry. $$ P_0 (z^3 - z^{-3}) + P_1 (z^2 - z^{-2}) + P_2 (z^1 - z^{-1}) $$ One method is to exploit the ...
1
vote
2answers
47 views

Proof of Cyclic Redundancy Check validity

I'm looking to understand the use of a Cyclic Redundancy Check, in combination with the mathematics behind it. So far I have 1) For any message $$M(x)\cdot x^n = Q(x)G(x) + R(x)$$ Where $Q(x)$ is ...
0
votes
0answers
11 views

Is there an algebraic solution to this discount factor problem

I have a series of future cash flows ($CF$) to be discounted. Each time period has a known, unique, discount rate and I can solve for the present value ($PV$) of the cash flows. I would like to find ...
1
vote
3answers
31 views

Help with sum and product of roots.

I'm having trouble with a question from my textbook relating to roots of an equation. This is it: Let a and be roots of the equation: $x^2-x-5=0$ Find the value of $(a^2+4b-1)(b^2+4a-1)$, without ...
5
votes
1answer
88 views

Prove that $f(x) = x^4+ax^3+bx^2+cx+d$ does not has all rational roots

The quartic polynomial $f(x) = x^4 + a x^3 + b x^2 + c x + d$ is such that $ad$ is odd and $bc$ is even. Prove that $f(x)$ does not has all rational roots. My attempt: Clearly, f(x) will have ...
2
votes
2answers
305 views

How To Prove That The Rijndael Polynomial Is Irreducible?

I am learning about the AES algorithm which uses the finite field ${\mathbb{Z}_2[x]}\over{(p(x))}$, where $p(x)=x^8+x^4+x^3+x+1$. How do you prove that this polynomial is irreducible? I know that for ...
0
votes
2answers
53 views

Polynomial of degree $n$ with $n+1$ zeros

Prove that if $P $ is a polynomial of degree $n$ with $n+1$ zeros $P$ must be zero This can be proven easily by the fundamental Theorem of Algebra. However, how would one prove the statement ...
-2
votes
2answers
80 views

Prove or disprove $x^2-x$ is divisible by $x$ [closed]

Can someone prove or disprove this statement: Given a positive integer $x>1$, is it true that $x^2-x$ gives a number that is divisible by $x$?
0
votes
0answers
11 views

cusps of bivariate polynomial equation

I plotted an implicit curve: $$f(x,y)=\sum_i^N a_i x^i y^{N-i}=0 \;\;\;a_i\in R$$ and found that the curve has sharp corners (cusps). (area where f>0 are painted red, otherwise green) If this is ...
1
vote
0answers
30 views

Is there a simplified cubic formula?

I'm trying to make a calculator of sorts for entering the values from a cubic function. I found this website: http://www.math.vanderbilt.edu/~schectex/courses/cubic/ where it says that "$$p = -b/(...
0
votes
0answers
32 views

If $P(n)$ divides $P(P(n)-2015)$, prove that $P(-2015)=0$

Q. Let $P(x)$ be a non-constant polynomial whose coefficients are positive integers. If $P(n)$ divides $P(P(n)-2015)$ for every natural number $n$, prove that $P(-2015)=0$. In one of the sources,...
3
votes
1answer
63 views

Rationals that aren't in the image of polynomials

Consider multiple polynomials with coefficients in $\mathbb Z$ and of degree at least 2 (thanks Moos): $g_1,g_2..g_i$. How can I go about showing that there are an infinite amount of rationals number $...
2
votes
1answer
30 views

Does cancellation impact vertical asymptotes?

Question: Let $r(x) = \frac{(x^2 + x)}{(x + 1)(2x - 4)}$. Does the graph has $x = 1$ as one of its asymptotes? Answer: No. My reasoning: $\frac{(x^2 + x)}{(x + 1)(2x - 4)} = \frac{x(x + 1)}{(x + 1)(...
2
votes
1answer
35 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 $-(1^2+\cdots+n^2)=-\dfrac{n(n+1)(2n+1)...
2
votes
1answer
27 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 ...
3
votes
0answers
43 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 ...
1
vote
1answer
31 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 $a\in ...