Questions tagged [polynomials]

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 and checking for irreducibility.

17,839 questions
40 views

Finding minimal polynomial given one root and two values

I'm trying the following exercise: Find a minimum degree polynomial $P(x) \in \Bbb C[x] : P(1)=2 \land P(i-1)=i \land P(-\pi i)=0$. The root lead to $P(x) = (x + \pi i)Q(x)$ now using one ...
67 views

Find B(x) such that $A(x) = P(x) \cdot B(x)$

A(x) is enumerator (generating function) of partitions of number such that contain exactly $1$ (but maybe multi times) of $2,3,5$. P(x) is enumerator of all partitions. Find compact pattern for $B(x)$ ...
44 views

Is multiplicity of root of polynomial meaningful in any way? [closed]

Is multiplicity of root of polynomial meaningful in any way? I encounter this problem, when I find roots of some polynomial and there are fewer roots found than the order of the polynomial. Which ...
36 views

Discrepancy in finding the coefficient of a polynomial

After asking a question here let us consider on the following case which is simpler, this might help me to make some progress: suppose we have the following polynomial: $$f(x)=(x^0+x^1+x^2+x^3)^3$$ ...
31 views

Limit at infinity / Show inequality

I am looking at the following: Show that if $p,q$ are real polynomial of the same degree, then $p(x)\approx q(x)$ as $x\rightarrow \pm \infty$. Show that the funcions $n^{-x}, n\in \mathbb{N}$ ...
227 views

82 views

How to find $P(-2)$ here?

Where $$P(n)=\sum_{i=1}^n i^{10}$$ And $n=1,2...$, $P:R\to R$. Maybe there is a easier way of solving this without actually calculating the sum, which is not that pleasing.
34 views

Lost in algebraic expressions: is there any available visual /chart / Venn-diagram representation of the typology?

The presentation of the classification of algebraic expressions is not always as clear as one may wish ( to me at least). One can sometimes read things like : " a polynomial is a sum of monomials......
36 views

searching for a matrix, which brings $f(x)$ to $f(x-1)$ for polynomials up to degree of $2$ and considering the basis $(1,x,x^2)$

I am trying to understand how to produce such a matrix. Do I have to write down a system of linear equations in order to see that? I do not see , how to solve it yet. Thanks
68 views

How to approach this

$2^{33x-2} + 2^{11x+2} = 2^{22x+1} +1$ The question is to find the sum of all the solutions for $x$. How does one approach these type of problems? I tried to take $2^{11x}$ as $n$, but failed to ...
66 views

Prove that there is no $a\in \mathbb R$ so that $f(x)= e^x(x^2+a)$ has only one extremum

As to my understanding in order to calculate the behavior of the extrema of a function, a good approach is to examine the behavior of the first derivative of the given function. So by calculating the ...
45 views

How do you check if a polynomial system has solutions

Given the polynomials $f(x,y)=4x^{12}+7y^{18}-1=0$ and $g(x,y)=5x^{10}+9y^{14}-1=0$ I have to figure out: a) if the real solutions set of the above system is empty or not b) if the set of real ...
17 views

Example of basis for d-variate polynomials for numerical application

I want to construct a polynomial basis for the space of $d-$variate polynomials of degree $Q$ for numerical purposes. Let's pick for example $Q = 2$ and $d = 3$. Now, if $d = 1$ it would be easy: I ...
16 views

How to judge that a Boolean polynomial cannot be divisible by another?

Let $f(x),g(x)\in\mathbb{F}_2[x]$, and assume that $f(x)\nmid g(x)$, $\deg (f) \ll \deg(g)$ So how to confirm this fact $(f(x)\nmid g(x))$ as soon as possible? Further more, is there an algorithm ...
53 views

At most two of the three polynomials $p(x)$, $p(x) - 1$, $p(x) - c$ are squares in $k[x]$

I am trying to show the following statement: Let $k$ be a field (no assumptions about algebraic closure or similar). Let $c \in k$ be a constant, which is not $1$ or $0$. Let $p(x) \in k[x]$ be a ...
57 views

Find roots of primitive polynomial $x^2+4x+2$ in $\mathbb{Z}_{11}/(x^2+x+8)$

I need to find the roots of primitive polynomial $x^2+4x+2$ in $\mathbb{Z}_{11}$ over field $\mathbb{Z}_{11}/(x^2+x+8)$ ($x^2+x+8$ is also primitive). As far as I'm concerned the answer is the ...
9 views

Concerning a certain differential equation in $k[x,x^{-1},y]$ multiplied by a positive power of $x$

Let $f,g \in \mathbb{C}[x,x^{-1},y]$. Truly, I wish to solve $f_x+g_y+f_xg_y-f_yg_x=0$, see this question. Assume that this equation involves negative powers of $x$, so if we multiply it by an ...
51 views

Polynomial division over an extension [duplicate]

Suppose $F \subseteq K$ are fields. I'm trying to show that if $P,Q \in F[X]$ and $P \mid Q$ over $K$, then $P \mid Q$ over $F$ as well. Denote the field of fractions of $F$ with $F(X)$. Then I have ...
34 views

Nilpotent Operator Minimal Polynomial

Given a Nilpotent Operator $F: V\rightarrow V$ with index $k$, that being the smallest $k$ such that $F^k = 0$, show the minimal polynomial is $M(t) = t^k$. It's obvious that $M(F) = 0$, but how do I ...
40 views

How likely is it for the polynomial obtained from a base-$b$ expansion of $n$ to be reducible?

Let $n$ and $b$ be integers larger than $1$, and write the base-$b$ expansion of $n$ as $n = \sum\limits_{i=0}^\infty a_i b^i$ (only finitely many of the $a_i$ are nonzero). We can define the ...
33 views

A polynomial equation on the integers

A polynomial $p(x)$ is called self-centered if it has integer coefficients and $p(100) = 100.$ If $p(x)$ is a self-centered polynomial, what is the maximum number of integer solutions $k$ to the ...
51 views

Fourth degree equation with parameter

For what values of the parameter a equation $$x^4-a*x^3+(3-2a)*x^2+a*x+1=0$$ has both positive and negative roots?
58 views

Can we reduce finding matrix roots to finding roots of Jordan blocks?

I just found some interesting question about matrix square roots and I came to think of one way to find them, or at least reduce them to a set of simpler problems. Assume we have a matrix $\bf A$ and ...
104 views

Holder unsuccessful here!

For $x,\,y,\,z> 0$, prove that $$\frac{y}{\sqrt{2\,z(\,x+ y\,)}}+ \frac{z}{\sqrt{2\,x(\,y+ z\,)}}+ \frac{x}{\sqrt{2\,y(\,z+ x\,)}}\geqq \frac{3}{2}$$ I tried $\lceil$ HOLDER!inequality $\rfloor$ ...
18 views

Commutative Property Matrices that Differ by a Constant

Let $A$ be a matrix, $P$ a Polynomial in its linear factors $$P(x):=\prod_{i=1}(\lambda_i-x)$$ How does one know that the multiplication of Matrices when we calculate $P(A)$ is commutative, as the ...
44 views

Determine polynomial function of degree 4.

The graph of a polynomial function $f(x)$ of degree $4$ with real coefficients has a local maximum at $(-3|3)$ and a local minimum at $(1|0)$ and no other local extremal points. Determine the function....
16 views

Order Polynomial of Matrix Commutative Property

Given a matrix $A$ and a polynomial $P$ as its linear factors $$\prod_{i=1}^{n}(\lambda_i-x)$$ For $P(A)$, does it matter in which order the linear factors are multiplied, seeing as matrix ...
23 views

Kernel of endomorphism for polynomials from $f(x)$ to $f(2x+1)$

I am looking for a kernel of a map, producing $f(2x+1)$ out of $f(x)$. where $f(x)$ is an arbitrary polynomial of degree $n$. I thought of trying to write this transformation as a matrix and then to ...
I have been given the definition of a Sturm sequence ($p_1,\dots, p_m$) as follows: $$p_0 = a \in \mathbb{Q}[x]\\ p_1 = a'\\ \forall 1 < i < m: p_i(t) = 0 \implies \text{sig}(p_{i-1}(t))=-\text{... 1answer 78 views what is the root of this polynomial? Let f_n(x)=\prod\limits_{i=1}^n (x+i)-n!=(x+1)(x+2)\cdots(x+n)-n! n is a positive integer. What are the roots of the polynomial for a given n except 0? Or determine the real part of the ... 0answers 54 views Prove that \mathbb{Z}[x]/(x^2 - 3) \cong \mathbb{Z}[\sqrt{3}]. Here is the statement of the problem: Let (x^2 - 3) be the ideal of \mathbb{Z}[x] generated by x^2 - 3 and let \mathbb{Z}[\sqrt{3}] = \{a + b\sqrt{3} : a,b \in > \mathbb{Z}\}. Prove ... 2answers 61 views I am stuck on this one question where I have to find the coefficient of x^2. It is non-calculator. [closed] What is the coefficient of x^2 in$$ \left(4-x^2\right)\left[\left(1+2x+3x^2\right)^6-\left(1+4x^3\right)^5\right] $$A calculator cannot be used. 1answer 68 views Can we let n go to \infty after applying the fundamental theorem of algebra to a polynomial of degree n? Recently I came across a proof of the infinite product for \sin z (https://www.sciencedirect.com/science/article/pii/0022247X77902347). It applies the fundamental theorem of algebra to$$p_{n}(z)=\...
Prove that the equation$$x^4+(a-2)x^3+(a^2-2a+4)x^2-x+1=0$$ does not admit $$x=-2$$ as a triple root.