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.

0
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1answer
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 ...
0
votes
1answer
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 $B(x)$ ...
0
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3answers
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 ...
0
votes
1answer
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: $$f(x)=(x^0+x^1+x^2+x^3)^3$$ ...
0
votes
0answers
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 $n^{-x}, n\in \mathbb{N}$ ...
10
votes
5answers
227 views

Determining the coefficients of $(1 + x + x^2 +\cdots+x^n)^{n-1}$

Suppose we have the following polynomials: $$f_1(x)=(1 + x + x^2)$$ $$f_2(x)=(1 + x + x^2 + x^3)^2$$ $$f_3(x)=(1 + x + x^2 + x^3 + x^4)^3$$ $$f_4(x)=(1 + x + x^2 + x^3 + x^4 + x^5)^4$$ $$\vdots$$ $...
2
votes
2answers
89 views

For which $\lambda \in \mathbb{N}$ the equation $x^3 - \lambda x - 2 =0$ has only rational roots?

For which $\lambda \in \mathbb{N}$ the equation $x^3 - \lambda x - 2 =0$ has only rational roots? My attempt: I start to plug values for $\lambda$ but I only find that $\lambda = 3$ works, but is ...
0
votes
0answers
17 views

Concerning $(x,y) \mapsto (\lambda x^2y+A,\mu x^{-1}+B)$

Let $f : (x,y) \mapsto (p,q)$ be a map from $\mathbb{C}[x,y]$ to $\mathbb{C}[x,x^{-1},y]$ satisfying the following two conditions: (i) $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in \mathbb{C}-\{0\}$. (...
2
votes
0answers
68 views

Analytical coefficients of a polynomial

Suppose I have the following polynomial, $$f(x)=(1+x)^2(1+x+x^2+x^3)^2$$ expanding this gives: $$f(x)=1+4x+8x^2+12x^3+14x^4+12x^5+8x^6+4x^7+x^8$$ now suppose I want to extend this as follow: $$f(x)=...
1
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1answer
35 views

Prove reciprocal polynomial is not primitive(field).

Let $f$ be a polynomial with a non-zero constant term and define its reciprocal polynomial $f^*=x^nf(\frac{1}{x})$. (Here $\deg {f}=n$. I see several other names like "reverse/inverse polynomial", and ...
0
votes
1answer
37 views

Is there any polynomial that increases and decreases for every integer change in variable?

I was looking at method of differences to solve a next in sequence problem. The method is applicable to sequences that can be expressed as a polynomial. I started thinking about sequences whose sign ...
-3
votes
1answer
20 views

Which of the following answer choices is a factor of P(x)? [closed]

A polynomial P(x) such that P(3/4) =0. Which of the following answer choices is a factor of P(x)? a. 4x-3 b. 4x+3 c. 3x-4 d. 3x+4 Thank you
2
votes
1answer
40 views

Geometry of roots of a polynomial with real coefficients

Consider the polynomial $p(x) = x^n(1-x) - r$, where $r$ is a real number. For sufficiently small $r$, there are two positive zeros of $x^n(1 - x) -r = 0$. (I came to this conclusion by graphing $x^n(...
0
votes
1answer
70 views

Show that there is no polynomial P(x) of degree 998 with real coefficients satisfying the equation $P(x)^2 - 1 = P(x^2 + 1)$ for all real numbers x.

A friend of mine explained this solution to me but there are a few parts which I still don't understand. Her solution is as follows: By substituting in $x$ and $-x$ into the equation given, it can be ...
6
votes
1answer
47 views

Enumerating polynomials over finite fields without multiple roots

Suppose that $p > n$. We are interested in the number of monic polynomials of degree $n$ defined over $\mathbb F_p$ without multiple roots. I hope someone could provide a proof (hopefully to be ...
0
votes
2answers
44 views

Ideals and order of a polynomial ring

Consider the ideal $I=(X^3+\hat2X+\hat1)$ of polynomial ring $R=\mathbb Z_3[X]$. Is $R/I$ an integral domain? How many elements does $R/I$ have? Find the inverse of $X^3+\hat1$ in $R/I$. (1) $(X^3+\...
0
votes
1answer
44 views

$n\times n$ Matrix with Fewer than n Eigenvalues

I have a problem which asks for all JCF of a 7x7 matrix with characteristic polynomial $t^3(t-1)^2$. Shouldn't the algebraic multiplicity of all Eigenvalues add up to 7? Why not? Are there any ...
1
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0answers
15 views

General form of Tutte polynomial for a family of graphs

I'm looking at a family of graphs that look like this: $n = 1$ $n = 2$ $n = 3$ So for every $n$ you add another "shell" and connect it with the previous outer shell. What i'm interested in is if ...
0
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1answer
30 views

What are those $p = X_{1}$ and $p = X_{1}^{2}$ and $p = X_{1}^{2} X_{2} X_{3}^{3}$ in $R[X_{1}, X_{2}, X_{3}]$?

In my textbook Analysis I by by Amann/Escher, there are definitions as follows Let $R$ be a nontrivial (not necessarily commutative) ring with unity. The formal power series ring over $R$ is the ...
1
vote
1answer
51 views

Find the degree of a field extension

Suppose $F$ is the minimal subfield in $\mathbb{C}$ containing all the roots of polynomial $x^4-x^2+1$. Find the degree of a field extension $[F:\mathbb{Q}]$. I understand that I need to find the ...
0
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2answers
26 views

Congruence in F[x], proof involving infinity

The statement to be proven is: "Show that there are infinitely many distinct congruence classes modulo $x^2 -2$ in $\mathbb{Q}[x]$." My question is if the following solution suffices. My apologies if ...
2
votes
4answers
141 views

Completing the square to find if quadratic form is positive definite.

I have the quadratic form $$g=x_1^2+6x_2^2+8x_3^2-4x_1x_2-6x_1x_3-x_2x_3$$ I have problems completing the square. I tried to rewrite the expression as follows $$g=x_1^2-4x_1x_2+6x_2^2-x_2x_3+8x_3^2$...
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votes
6answers
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.
1
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0answers
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......
0
votes
2answers
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
1
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2answers
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 ...
2
votes
1answer
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 ...
0
votes
2answers
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 ...
0
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0answers
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 ...
0
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0answers
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 ...
3
votes
1answer
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 ...
1
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1answer
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 ...
0
votes
0answers
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 ...
2
votes
3answers
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 ...
0
votes
1answer
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 ...
4
votes
0answers
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 ...
0
votes
2answers
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 ...
0
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3answers
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?
4
votes
1answer
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 ...
0
votes
1answer
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 $\rfloor$ ...
0
votes
1answer
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 ...
2
votes
3answers
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....
1
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1answer
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 ...
1
vote
1answer
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 ...
1
vote
0answers
19 views

Why is the sign of the last term in a Sturm sequence always constant?

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{...
1
vote
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 ...
3
votes
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 ...
-1
votes
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.
3
votes
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)=\...
0
votes
2answers
42 views

Prove that polynomial doesn't admit a particular solution

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.