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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0
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1answer
14 views

Counting monomials with $k$ variables

Say we expand $\left(\sum_{i=1}^n x_i\right)^k$ into monomials. If $k=3$ there are $3n(n-1)$ monomials with two variables: $3x_1x_2^2 + 3x_1x_3^2 +\dots + 3x_1^2x_2 + \dots$. Is there a closed form ...
4
votes
2answers
82 views

Express $1/(x-1)$ in the form $ax^2+bx+c$

Let $x$ be a root of $f=t^3-t^2+t+2 \in \mathbb{Q}[t]$ and $K=\mathbb{Q}(x)$. Express $\frac{1}{x-1}$ in the form $ax^2+bx+c$, where $a,b,c\in \mathbb{Q}$. I have proved that $f$ is the minimal ...
1
vote
3answers
97 views

How to solve $x^3 = 1$?

My intuitive side tells me to take the cube root of both the sides and get the answer $1$. However, I realize that it might be a problem for I'll lose solutions as given here: Is it the case that ...
3
votes
2answers
27 views

Find the cubic polynomial given linear reminders after division by quadratic polynomials?

A cubic polynomial gives remainders $(13x-2)$ and $(-1-7x)$ when divide by $x^2-x-3$ and $x^2-2x+5$ respectively. Find the polynomial. I have written this as: $P(x)=(x^2-x-3)Q(x)+(13x-2)$ ...
2
votes
1answer
53 views

Asymptotic behavior of integrals of Legendre polynomials

By definition $\int_{-1}^1 |P_n(x)|^2 dx = O(n^{-1})$. What about the other powers? Do we know how $\int_{-1}^1 |P_n(x)|^k dx$ behaves for any $k$? Maybe $O(n^{-k/2})$?
0
votes
2answers
33 views

Brute force roots of a univariate polynomial

I am given a polynomial, where (a, b, c) are integers (positive and negative). $$ ax^2+bx+c $$ I need to create a simple brute force method to find the roots of this univariate polynomial. Is ...
1
vote
2answers
794 views

Can you help me reverse the Minimum Curvature Method?

The minimum curvature method is used in oil drilling to calculate positional data from directional data. A survey is a reading at a certain depth down the borehole that contains measured depth, ...
-1
votes
0answers
19 views

maximization by vector

I would like to maximize the below equation $C_i = log_2 (1+ \frac{1}{x} h_i g_i g_i^H h_i^H )$ h and g both of them are vectors. constraints $\sum_{i=1}^N g_i ≤ N , N=1$ i am trying to to find ...
31
votes
9answers
2k views

why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$

why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$? I know this is well-known. But how to prove it rigorously? Even mathematical induction does not seem so straight-forward. ...
0
votes
0answers
14 views

Change of variable in biharmonic equation

I'm currently studying how to derive Michelle's Solution for plane elasticity in the cylindrical coordinate system. I have stumbled for days to understand how the following equation: ...
0
votes
1answer
52 views

Showing that $\mathrm{in}_<(f^m) = \mathrm{in}_<(f)^m$

I am currently in the following scenario: Let $f\in K[x_1, ..., x_n]$, and $<$ be a fixed term order. I want to show that $\mathrm{in}_<(f^m) = \mathrm{in}_<(f)^m$ (for some $m>0$). ...
0
votes
3answers
79 views

Dimension of $K[x]/x^{2}$ as a vector space

Let $K$ be a field, and $R=K[x]$ the polynomial ring over K. Let $J$ be the ideal generated by $X^{2}$ Show that $R/J$ is a K-space. What is its dimension? I know that the torsion submodule of $R/J$ ...
4
votes
8answers
149 views

Find the cubic equation of $x=\sqrt[3]{2-\sqrt{3}}+\sqrt[3]{2+\sqrt{3}}$

Find the cubic equation which has a root $$x=\sqrt[3]{2-\sqrt{3}}+\sqrt[3]{2+\sqrt{3}}$$ My attempt is ...
11
votes
4answers
1k views

How do you find the turning points of a polynomial without using calculus?

I have a polynomial $P(x) = -x^3+12x+3$, and I am asked to find the turning points of it, and hence state how many zeroes it has. Since this chapter is separate from calculus, we are expected to solve ...
1
vote
1answer
34 views

Noether's normalization lemma in practice (example)

I would like to know how to use the Noether's normalization lemma in practice. Noether's normalization lemma Let $k$ an infinite field, and $k[a_1,\dots ,a_n]$ be a finite $k$-algebra. There ...
0
votes
1answer
17 views

What is the theory of finding roots of a polynomial equation by looking at the factors of the $a_n$ and $a_0$ term called?

This is commonly taught in high schools in the context of factoring polynomials. I remember this method even has its own wikipedia page (with a proof) but I forget what was the theory called. Could ...
3
votes
1answer
43 views

Prime ideal in R[x] is either principal or $\mathfrak p = (q,f)$

$R$ is a PID and $\mathfrak p$ a prime ideal of $R[x]$. Show that $\mathfrak p$ is principal or $\mathfrak p = (q,f)$ for some $q\in R$ prime and $f \in R[x]$ monic. I can't figure out this ...
1
vote
1answer
20 views

If $f(c_i)=g(c_i)$ for $i=0,1,…,n$, prove that $f(x)=g(x)$ in $F[x]$.

Here is a problem I'm trying to solve: Let $F$ be a field. Let $f(x),g(x)\in F[x]$ have degree $\leq n$ and let each $c_i$ be a distinct element of $F$. If $f(c_i)=g(c_i)$ for $i=0,1,...,n$, ...
1
vote
2answers
130 views

Has the polynomial distinct roots? How can I prove it?

I want to prove that the polynomial $$ f_p(x) = x^{2p+2} - cx^{2p} - dx^p - 1 $$ ,where $c>0$ and $d>0$ are real numbers, has distinct roots. Also $p>0$ is an even integer. How can I prove ...
0
votes
0answers
12 views

Upper bound on the remainder of a polynomial (not taylor)

There are many ways of approximating a function with a polynomial, $\widehat{f}(x)\approx f(x)$. One way is the taylor polynomial. A nice property that goes along with the taylor polynomial is an ...
1
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0answers
38 views

Basis of the space of homogeneous polynomials clarification

I want to prove the following proposition. Let $H(n,m)$ denote the vector space of homogeneous polynomials of degree $m$ in $n$ variables over $\mathbb{C}$. Then here exist a finite number of ...
2
votes
3answers
60 views

Number of generators of a given ideal.

Let $I=\langle 3x+y, 4x+y \rangle \subset \Bbb{R}[x,y]$. Can $I$ be generated by a single polynomial? My approach: If $I$ can be generated by a single polynomial, then the two "apparent" ...
0
votes
1answer
46 views

$I=(f_1, \ldots, f_n)\subset k[x_1, \ldots, x_n]$ with $f_i\in k[x_i]$ irreducible polynomials

Let $A=k[x_1,\ldots, x_n]$ and $I=(f_1, \ldots, f_n)\subset A$ with $f_i\in k[x_i]$ irreducible polynomials. Is it true that $I$ is a maximal ideal in $A$? $I$ is a maximal ideal $\iff$ $1\in ...
4
votes
2answers
105 views

A complex polynomial satisfying $p(z)=p(\bar z) \forall z$ on the unit circle is constant

This question was part of a test on Complex Analysis. Let $p(z) \in \Bbb{C}$ be a complex polynomial such that $p(z)=p(\bar z) \forall z$ on the unit circle ($∣z∣=1$). Show that $p(z)$ is ...
-1
votes
2answers
49 views

How to perform long division on polynomials? [closed]

$$\frac{x^6 - 3x^5 + x^4 - 2x^3 - 3x^2 + x - 3}{x^2 + 1}$$ I got the answer in the book, but I can't figure out how it comes up with the x for $$\ { -2x^3 + 3x^3 }$$
0
votes
1answer
19 views

2nd order polynomial - finding the $x$ value of the top

I have a 2nd order polynome ($y = ax^2+bx+c$) from with I know $2.5$ points. $(450,5), (600,40)$ and $(q,0)$. I know also that $(q,0)$ is the top and $q<450$. How do I solve $q$?
2
votes
0answers
42 views

Find how many solutions of the equation $z^6+6z+10=0$ are in each quadrant. [duplicate]

Find how many solutions of the equation $z^6+6z+10=0$ are in each quadrant. This polynomial has six solutions by TFTA. I just don't know how to show what they are and where they lie. Any solutions or ...
1
vote
2answers
66 views

How do I integrate this expression involving exponential and polynomial

I tried a few ways (integral by parts, expanding), but I'm unable to compute this integral. $$\int_0^\infty\frac{\lambda^{n}e^{-\lambda}}{(\lambda + b)^2}\, \text{d}\lambda$$ n >= 0, b >= 0
2
votes
3answers
96 views

Show that $X^3+X^2+1$ has only one real root [duplicate]

Consider the polynomial $X^3+X^2+1 \in \mathbb R [X]$. Since it is of odd degree, it has at least one real root. How can I show that it's the only one?
2
votes
2answers
46 views

Prove that $\max_{|z| = 1} |P(z)| \ge 1$

I got stuck on this problem: Given a polynomial on complex plane $P(z) = z^n + a_{n-1}z^{n-1} + ... + a_1 z + a_0$ for $z \in \mathbb{C}$. Prove that $\max_{|z| = 1} |P(z)| \ge 1$ What I tried ...
2
votes
2answers
36 views

Roots of a Quartic (Vieta's Formulas)

Question: The quartic polynomial $x^4 −8x^3 + 19x^2 +kx+ 2$ has four distinct real roots denoted $a, b, c,d$ in order from smallest to largest. If $a + d = b + c$ then (a) Show ...
2
votes
1answer
28 views

how do i show that $x^2+1$ divides $x^{p-1}-1$ in $\mathbb{Z}_p[x]$?

$p$ prime integer. If we are given that $p \equiv 1$ modulo $4$ then $x^2+1$ divides $x^{p-1}-1$ in $\mathbb{Z}_p[x]$. I can show this by considering the cyclic subgroup $\langle g \rangle$of ...
3
votes
1answer
120 views

For which polynomials $f$ is the subset {$f(x):x∈ℤ$} of $ℤ$ closed under multiplication?

You surely know about the Brahmagupta–Fibonacci identity, $$(a_1^2 + b_1^2)(a_2^2 + b_2^2) = (a_1a_2 \pm b_1b_2)^2 + (a_1b_2 \mp a_2b_1)^2$$ which tells us that the product of two numbers, each of ...
1
vote
3answers
76 views

Prove that $(x^2+1)\mathbb Z[x]$ is a prime ideal of $\mathbb Z[x]$, but not maximal

Prove that $(x^2+1)\mathbb Z[x]$ is a prime ideal of $\mathbb Z[x]$, but not maximal. I'm supposed to show this for my homework. My first thought is to show that $\mathbb Z[x]/(x^2+1)\mathbb ...
0
votes
1answer
18 views

Undefined derivative of a third degree polynomial inverse?

I have the following equation: $$r_t=kr_s^{3}+r_s$$ I need to get $r_s'$, i.e. derivative of $r_s$ w.r.t. $k$, provided that value of $r_t'$ is already known. Taking derivatives of both sides w.r.t. ...
2
votes
2answers
462 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 ...
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. ...
2
votes
1answer
29 views

A question on roots of a polynomial of degree $n$

Under what conditions on cofficients of a polynomial $p(x)$, the roots of $p(x)$ are real and positive?
2
votes
3answers
34 views

Polynomials equations

I am trying to find the common solutions of this equations: $$ 2x^4+x^3-5x^2+2x=0 $$ $$ 2(2x-1)^{1998}+(4x-1)^{1997}=4x+1 $$. My idea is to solve the first one. I find the solutions:$ x=1$ , $x=0$ , ...
2
votes
0answers
20 views

Why polynomial $\psi^\top(t) A^{-1} \psi(t)$ attains maximum on $[-1, 1]$ at $\pm 1$, where $\psi_k(t) = t^k$?

Problem. Let $\psi(t) = (1, t, t^2, \ldots, t^{p-1})$ - a polynomial basis. Suppose there is a matrix $$ A = \int_{-1}^1 \psi(t) \psi^\top(t) dt, \ \text{i.e. } \ A_{ij} = [2 \, | \, i+j] \cdot ...
14
votes
3answers
3k views

Irreducible polynomial which is reducible modulo every prime

How to show that $x^4+1$ is irreducible in $\mathbb Z[x]$ but it is reducible modulo every prime $p$? For example I know that $x^4+1=(x+1)^4\bmod 2$. Also $\bmod 3$ we have that $0,1,2$ are not ...
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 ...
6
votes
0answers
98 views

Product of two random polynomials

Let $\alpha,\beta$ be two polynomials of the form $$\alpha(X)=\sum_{i=0}^{n}\alpha_iX^i,\quad \quad \beta(X)=\sum_{j=0}^n\beta_jX^j$$ where each coefficient is $1$ with a probability of $p$ and $0$ ...
16
votes
5answers
2k views

Do we really need polynomials (In contrast to polynomial functions)?

In the following I'm going to call a polynomial expression an element of a suitable algebraic structure (for example a ring, since it has an addition and a multiplication) that has the form ...
2
votes
1answer
45 views

Integer coefficients in a polynomial function

The problem: Find the polynomial $r(x)$ with the smallest degree and integer coefficients s.t. it has zeroes -1, 2/3 and -2 w/ resp multiplicities 2, 8 and 4 and $r(1) = -1$. What I tried: ...
1
vote
0answers
21 views

Finance Algeabra: Converting a Discount Polynomial Function to an Interest Rate Polynomial Function

I have a finance problem that is 99% mathematical. In finance, the price of a bond could be modelled as the discounted value of its future cash flows, so something like: ...
0
votes
0answers
23 views

Bounding the integral of the reciprocal of a complex polynomial

So, I would like to bound $\int_{C_R} \frac{1}{P(z)} dz$ where $C_R$ is the circle radius R centred at the origin, and $P(z)$ is a polynomial of degree $N=0,1,...,n$ i also want to deduce for what ...
0
votes
0answers
21 views

Using Euclid's algorithm, how do I find a polynomial $f_p(x)$ such that $f \cdot f_p \equiv 1 \pmod p$?

Suppose that we have a polynomial $f(x)$ with coefficients in $\mathbb{Z_3}$ and maximum degree $N-1$, where $N$ is prime. (In fact, we consider $f$ as a class of ...
-1
votes
1answer
36 views

$A\subset B$ if $A\cdot R[X] \subset B \cdot R[X]$? [closed]

Can we conclude $A\subset B$ if $A\cdot R[X] \subset B \cdot R[X]$ for ideals $A,B$ in $R$, where R is a commutative ring with unity and $A \cdot R[X]$ the ideal generated by the products $af$, ...
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 ...