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
votes
3answers
118 views

Finding the range of a $y=-x^2(x+5)(x-3)$ without calculus?

I was helping a precalculus student with this question. The graph wasn't given. My only idea was to find the inverse and try to find its domain. When trying to find the inverse, I arrived at ...
12
votes
3answers
136 views

Why $|x|$ is not rational expression?

I'm 9th grade student, and my teacher said that $|x|$ is not rational expression ( expression like $\frac{p(x)}{q(x)}$ s.t $p(x)$ and $q(x)\neq 0$ are polynomial) but he didn't have convincing reason. ...
0
votes
1answer
28 views

Characterization of the elements of a quotient ring

I'm in trouble with the following exercise: Consider the ideal $ I = (X^2-Y^3,Y^2-Z^3) $ in the polynomial ring $ k[X,Y,Z] $, where $k$ is any algebraically closed field. Show that every element of $ ...
0
votes
1answer
10 views

Factors/divisibility of monotonically-increasing integer polynomial

For positive integers $n$ and $x$, let $f_n(x)$ be a polynomial in $x$ of degree $n-1$, such that $f_n(x)$ is monotonically increasing for increasing $x \ge 1$. Now assume that there exist positive ...
0
votes
0answers
45 views

Why $f_1,f_2,f_3$ don't have a common factor?

Let $p(x,y)$ and $q(x,y)$ are polynomials in $x,y\in \mathbb{R}$. ($p,q\ne0$) $p$ and $q$ are coprime to each other ,($\frac{{\partial p}}{{\partial x}}=p_x$,$\frac{{\partial p}}{{\partial y}}=p_y$) ...
1
vote
1answer
25 views

How to draw cubic plane curve?

In Python, using MatPlotLib, given [vector] parameters $a$ and $b$ and [scalar] parameter $c$, I want to draw a general cubic plane curve in 2-dimensional space (regular plane with $x$ and $y$ axes): ...
1
vote
1answer
28 views

What is the family of parabolas with the following characteristics? [closed]

How does one describe the family of parabolas that contain the point $(0,0)$ and are tangent to the parabola $x = y^2 + 1$?
0
votes
0answers
38 views

Exhibit a reducible polynomial of the form $x^p -x-c$ having no roots in a field of characteristic 0

Is it possible for a polynomial, $x^p -x-c$ where $p$ is prime, to be reducible in a field of characteristic $0$, yet have roots in that field? I know for a fact that the general form is true, ...
1
vote
1answer
15 views

Do the integer roots of a polynomial $P(x) \in \Bbb Z[x]$ have to divide the constant coefficient?

By Gauss Lemma, the roots of a polynomial $P(x) = a_nx^n + \cdots + a_1x + a_0 \in \Bbb Z[x]$ are either integer, irrational or complex. Vietà's formulas imply that the product of all roots equals ...
1
vote
1answer
71 views

Logarithm as limiting case of $n$th root

Let $f_n(x) = x^{1/n}$ where $n \in \mathbb N$, and let $g(x) = \log(x)$. We can compute $f_n'(x) = \frac{1}{n}x^{-1 + \frac{1}{n}}$ and $g'(x) = x^{-1}$. Let's define $f_\infty(x) = \lim_{n ...
2
votes
1answer
40 views

Denominator is product of irreducibles with cyclic Galois group

Short version of the question: Guess the next terms in the sequence : $D_{17},D_{19},D_{23}$ etc where $$ \begin{array}{lcl} D_3 &=& (a\pm 1) \\ D_5 &=& (a\pm 1) (a^2-1 \pm 11a) \\ ...
0
votes
1answer
33 views

solution of system of polynomials

I have 3 equations as following: $$ \left\{ \begin{array}{c} (\Delta_{11}*y^2 + \Delta_{12}*y + \Delta_{13})x^2 + (\Delta_{21}*y^2 + \Delta_{22}*y + \Delta_{23})x + \Delta_{31}*y^2 + \Delta_{32}*y + ...
0
votes
1answer
16 views

Jacobian of a system of equations

I'm asked to compute the Jacobian of a system of equations $x_1^4+x_2^4-1=0$ $x_2-\sin(5x_1)=0$ $x_1-x_3^2=0$ What's the Jacobian of a system of equations? Do I perhaps need to infer the individual ...
1
vote
2answers
36 views

Find a polynomial f(x) of degree 5 such that 2 properties hold.

I have been trying to find a polynomial $f(x)$ such that these $2$ properties hold: $f(x)-1$ is divisible by $(x-1)^3$ $f(x)$ is divisible by $x^3$ To start, I set $f(x) =ax^5 + bx^4 + cx^3 + dx^2 ...
2
votes
1answer
27 views

Show that this sum of polynomials has no zeros with positive real part

Let $0 < \lambda_1 \leq \ldots \leq \lambda_n $ and $k_1, \ldots, k_n> 0$. Let further $$ \begin{align} P(x)&:=\prod_{i=1}^n (x+\lambda_i) = (x+\lambda_1)\cdot \ldots \cdot (x+\lambda_n) ...
2
votes
3answers
56 views

Prove that every symmetric $2\times2$ matrix is diagonalizable over the reals

Prove that every symmetric $2\times2$ matrix is diagonalizable over the reals. Every symmetric matrix looks like this: $$A = \begin{pmatrix} a_{11} & a_{12} \\ a_{12} & ...
0
votes
0answers
9 views

How to get the coefficients between different orthogonal polynomial basis

There are two different coordinate systems. One is $ (x_1,x_2) $ and the other is $ (y_1,y_2) $ such that $ x_1 x_2=y_1, $ and $ x_2^2=y_1^2+y_2^2. $ With the following basis in Legendre $P_n$, ...
5
votes
3answers
527 views

How to prove that my polynomial has distinct roots?

I want to prove that the polynomial $$ f_p(x) = x^{2p+2} - abx^{2p} - 2x^{p+1} +1 $$ has distinct roots. Here $a$, $b$ are positive real numbers and $p>0$ is an odd integer. How can I prove that ...
1
vote
1answer
164 views

Reducibility of $x^2+1$ in $\mathbb{Z}_n[x]$ [closed]

I want to prove or disprove the statement: $x^2+1$ is reducible in $\mathbb{Z}_n[x]$ $\iff$ there exists $a$ such that $a^2=-1$ in $\mathbb{Z}_n$. How can I prove or disprove the proposition?
0
votes
2answers
47 views

Bézout's Identity of polynomials?

Let $P=X^3−7X+6$, $Q = 2X^2+ 5X − 3$ and $R = X^2 − 9 ∈\mathbb Q[X]$. What are $S$ and $T ∈\mathbb Q[X]$ such that $PS + QT = R$? I have calculate the greatest common divisor of $P,Q,R$ are ...
0
votes
1answer
26 views

What does the first x represent in {x, (x+1), (x-3)}?

The question is: "Part of the graph of a polynomial function is shown. Which of the following sets contains only elements that are factors of the polynomial?" The two answer choices left are B. ...
1
vote
3answers
90 views

Showing reducibility of a polynomial in a Discrete Valuation Ring

Let $R$ be a complete discrete valuation ring with uniformiser $\pi$. I would like to show that a polynomial $f$ in $R[X]$ is reducible. Does it suffice to show that $f$ is reducible in ...
0
votes
2answers
16 views

Get $M$ and all the roots of ecutions to be real

Get $M$ and all the roots of ecutions to be real $x^4 - (2m-1)x^2 +4m -5 =0$ If I set $x^2=t$ I get some delta but idk how to solve that
0
votes
2answers
44 views

What is the Order (Big O) of this polynomial?

$$\frac{2n^{14} + 7 n^8 - 3}{3n^8 - n^4 + 3}$$ If this division is $p(n)$, I have to write $p(n) = O(n^k)$ I guess the answer is $n^6$, but how can i solve it step by step?
0
votes
0answers
9 views

Which polynomial factorization method leads directly to $(1-\alpha_0 z)(1-\alpha_1 z)$

I know how to factor a polynomial $p(z)$ so that it looks like $a_n(z- z_0)\cdots(z-z_n)$, where $z_k$ are its zeros. Now I could squeeze this form into the wanted $(1-\alpha_0 z)\cdots(1-\alpha_n ...
0
votes
2answers
20 views

Is $x^p-ax-b$ with $a,b\neq 0$ irreducible in a field with characteristic a prime p?

It's a part of a bigger problem I'm facing. Not only I don't know how to prove it, I don't know if it's true or false at all (so I have no idea what to try to prove and so I don't know where to ...
2
votes
1answer
36 views

A non-constant polynomial with odd-integer co-efficients and of even degree , has no rational root?

Let $f(x)$ be a non-constant polynomial in $\mathbb Z[x]$ with odd-integer co-efficients and even degree ; then is it true that $f$ has no rational root ?
1
vote
1answer
19 views

$F$ be a finite field , then are there infinitely many polynomials $f(x) \in F[x]$ such that $f(a)=0 , \forall a \in F$ ?

Let $F$ be a finite field , then is it true that there are infinitely many polynomials $f(x) \in F[x]$ such that $f(a)=0 , \forall a \in F$ ?
0
votes
3answers
32 views

Is there a non-exponential function whose limit at infinity is a real, irrational number?

$e$, for example, can be calculated through a non-polynomial function $(1+1/x)^x$, but I cant think of an example for a non-exponential function (or rational function) where the limit to infinity ...
0
votes
1answer
47 views

Can every polynomial generate an ideal?

Suppose an arbitrary polynomial $f$ in a polynomial ring $R$. Is $\langle f\rangle$ always an ideal? Helper parts Consider a finite polynomial ring. Let $R=R[x_1,\ldots,x_n]$. Is the answer ...
3
votes
3answers
74 views

Is $\frac{x^2+x}{x+1}$ a polynomial?

Is $\frac{x^2+x}{x+1}$ a polynomial? Fist question can be: on which field/ring or etc? In basic, let's take over $\mathbb R$. Actually, it is $x$ if $x \ne -1$. Can we say again it is a polynomial, ...
0
votes
0answers
30 views

How is “Binomial” defined in Algebraic Geometry?

I am learning ideal arithmetics and I was flabbergasted that $\langle x\rangle$ is binomial ideal, as observed with Macaulay2 here. $x$ is clearly not a polynomial with two terms. Then I read paper ...
0
votes
1answer
51 views

How to show that $a^3+b^3+c^3+d^3\geq abc+abd+acd+bcd$ if $a,b,c,d>0$

How can I prove that if $a,b,c,d>0$ then $$a^3+b^3+c^3+d^3\geq abc+abd+acd+bcd?$$ I think there is some simple proof but I can't remember... is this a special case of some general inequality? ...
0
votes
2answers
34 views

Factor trinomials dividing by the common GCF

I have a doubt with the following problem I found in a book. You have to simplify a polynomial using the GCF. Now, this is the problem I am not able to grasp: $$6x^2-19x-7$$ According to the book, ...
0
votes
1answer
16 views

Find the limit points of this sequence of polynomials: $p_n(z)=z^2+\frac z{n(n-1)}-\frac1{n(n-1)}$ for different topologies

A) Let $A=\{a_n|p_n(a_n)=0\}$ be a subset of complex numbers regarded as a sequence in the set of complex numbers with the metric topology. Determine all limit points in A. B) Let ...
0
votes
2answers
63 views

Is $x^4+x^2+1$ an irreducible polynomial over $\Bbb Z/2\Bbb Z$?

Is $x^4+x^2+1$ an irreducible polynomial over $\Bbb Z/2\Bbb Z$? According to this site the answer is no. But I can't find the factors. Can you?
4
votes
3answers
39 views

How to find $W^{\perp}$ in the following polynomial inner product space?

Consider $P_3(\Bbb{R})$ with inner product $\langle p(x),q(x)\rangle=\int^1_{-1} p(x)q(x)dx$ and let $W=\{ p(x)\in P_3(\Bbb{R})|p(0)=p'(0)=p''(0)=0\}$. How to find $W^{\perp}$? Let's set ...
-1
votes
0answers
101 views

Is it true that; $\frac{{\partial f}}{{\partial x}}$ and $\frac{{\partial f}}{{\partial y}}$ don't have a common factor?

Let $f(x,y) = ({x^2} + {y^2})p{(x,y)^2} - q{(x,y)^2}$ and where $p(x, y)$ and $q(x, y)$ are real polynomials. Is it true that; $\frac{{\partial f}}{{\partial x}}$ and $\frac{{\partial f}}{{\partial ...
0
votes
1answer
26 views

$f,g \in \mathbb Q[x]$ , $f(a)=g(b)=0$ ; $f$ is irreducible in $\mathbb Q(b)[x]$ iff $g$ is irreducible in $\mathbb Q(a)[x]$?

Let $a,b$ be complex roots of irreducible polynomials $f(x),g(x) \in \mathbb Q[x]$ . Let $F:=\mathbb Q(a) , K:=\mathbb Q(b)$ ; then is it true that $f(x)$ is irreducible in $K[x]$ if and only if ...
0
votes
2answers
45 views

Discriminant of Quintic (Galois theory)?

I am working on the irreducible polynomial $x^5-Npx+p=0$ where $p$ is a prime and $N\in\mathbb Z_+$. I need to calculate the discriminant of this to determine its Galois group, background here by ...
2
votes
1answer
16 views

A polynomial $P(x)$ of degree $5$ with lead coefficient one,increases in the $(-\infty,1)$ and $(3,\infty)$ and decreases in the interval $(1,3)$

A polynomial function $P(x)$ of degree $5$ with leading coefficient one,increases in the interval $(-\infty,1)$ and $(3,\infty)$ and decreases in the interval $(1,3)$. Given that $P(0)=4$ and ...
0
votes
0answers
14 views

GCD of monic polynomial and regular polynomials

Requesting your assistance in proving the following: Let $p$ be a monic polynomial and $f, g$ some two other polynomials. prove that $GCD(pf,pg) = p * gcd(f,g)$ Thanks
0
votes
1answer
51 views

A question on polynomials

Let $f(x,y) = ({x^2} + {y^2})p{(x,y)^2} - q{(x,y)^2}$ and $p$ and $q$ are two polynomials. Is it true that; $\frac{{\partial f}}{{\partial x}}$ and $\frac{{\partial f}}{{\partial y}}$ don't have a ...
0
votes
2answers
46 views

Can we say that; $\frac{{\partial f}}{{\partial x}}$ and $\frac{{\partial f}}{{\partial y}}$ don't have a common factor?

Let $f(x,y)$ be a polynomial. Can we say that; $\frac{{\partial f}}{{\partial x}}$ and $\frac{{\partial f}}{{\partial y}}$ don't have a common factor?
0
votes
0answers
37 views

Showing that the definite integral of a polynomial is unequal to zero.

Let $n$ be odd and let $\sigma_k=a+\frac{k}{n}(b-a)-\frac{a+b}{2}$ for $k=0,\ldots,n$ and some $a<b$. I need to show that $\int^{\frac{b-a}{2}}_0 \prod^n_{k=0}(s-\sigma_k)d s\neq 0$ but my attempts ...
0
votes
1answer
27 views

Homogeneous polynomial identity

Suppose we have a homogeneous polynomial $P$ of degree $k$ in $n$ complex variables. The following formula seems to be true (can be proven by straightforward computations for $2$ ora $3$): ...
0
votes
0answers
33 views

Given n points, show there is a unique polynomial through all of them

if x0, x1,..., xn are distinct points in [a,b] and $A$ = { a0, a1,..., an } ∈ Rn+1, show there is a unique polynomial $p$A of degree at most "n" such that $p$A(xj) = aj for each "j". Then, show ...
1
vote
1answer
33 views

Bessel function $J_{3\over 2}(x)$

How can i find $J_{3\over 2}(x)$ and $J_{5\over 2}(x)$ by use this formula : $$J_{p+{\frac{1}{2}}}(x)=\left(\frac{2}{\pi}\right)^{\frac{1}{2}} \cdot(-1)^p x^{p+{\frac{1}{2}}} ...
2
votes
3answers
110 views

$t^4+t^3+t^2+t+1 $ has no linear factors

I am trying to understand why the given polynomial has no linear factors over $\mathbb{Q}$. I am trying to do it using elementary methods (no eisenstein criterion etc to show it's irreducible). I ...
0
votes
3answers
30 views

How do I show that something only has one stationary point?

I have this question in my homework where I can't quite grasp the solution. I have been asked to show that a curve with equation $y=(1+2x)^4+(1-2x)^4$ has one singular stationary point and what its ...