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.

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1answer
13 views

Confusion regarding the proof of the Multinomial theorem

We saw the following theorem in class: If $n \in \mathbb{N}$ and $z_1, \dots, z_m \in \mathbb{C}$ we have: $ (z_1 + \dots + z_m)^n= \sum_{k_1+\dots+k_m=n} \binom{n}{k_1, \dots, k_m} z_1^{k_1} \dots ...
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1answer
32 views

How to factorize $P=(X^2-4X+1)^2+(3X-5)^2$ in $\mathbb R[X]$?

First of all, I searched for roots. I knew that $\exists x\in\mathbb R,\, P(x)=0 \iff \exists x\in\mathbb R,\, x^2-4x+1 = 0 \text{ and } 3x-5 = 0$. However, It is really easy to say that there is no ...
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0answers
14 views

Linearly independent of $f_1(x), f_2(x),…, f_m(x)$ and 1

Let $a_1, a_2, ..., a_m$ be points in $R^n$. Let $f_i(x)=||x-a_i||^2-\delta^2$. There is $f_i(a_j)=0$ when $i\neq j$, $f_i(a_j)=-\delta^2$ when $i=j$. Show that $f_1(x), f_2(x),..., f_m(x)$ and 1 are ...
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How to find undefined singularity point of a polynomial equation

I was given two equations and I had to find if they are equal or are they undefined and the condition given was: (https://latex.codecogs.com/gif.latex?x\neq&space;0) The equation : (https://latex....
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1answer
39 views

Factorize $8(p-2q)^{2}-2p+4q-1$

I got this problem in my mathematics exams and was not able to solve it even after trying a lot. The problem was to factor the polynomial: $$8(p-2q)^{2}-2p+4q-1$$
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1answer
56 views

Test to determine if a polynomial has only real roots?

Given a polynomial $p(x)=x^n + c_{n-1} x^{n-1} + \cdots + c_0$ with real coefficients $c_{n-1}, \ldots, c_0$, is there an efficient method to determine whether all roots to the polynomial are real and ...
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0answers
18 views

Frobenius map on certain elements

I would appreciate some help with understanding the following setup: Let $a\in \mathbb{F}_q$ is a primitive root of unity with $q=3^d$. Consider a rational function $f(x) \in \mathbb{F}_3(x)$, where $...
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1answer
9 views

Proof of the existence of generating polynomials of an ideal

I'm going through Shilov's Linear Algebra, and I've come to the chapter on canonical representations of linear operators, which uses polynomial algebras to develop the JCF. The topic of algebras of ...
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2answers
34 views

A cubic polynomial over $\mathbb{Q}$ with a degree three splitting field?

I'm wondering if it's possible to have a cubic polynomial $f(x)\in\mathbb{Q}$ with three distinct roots (in $\mathbb{C}$) that has a degree three splitting field? If $f(x)=(x-a)(x-b)(x-c)$, with $a,b,...
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1answer
75 views

Determining a polynomial $p$ from its image of rationals.

Let $f$ and $g$ be polynomials with rational coefficients, and let $F$ and $G$ denote the sets of values of $f$ and $g$ at rational numbers. Prove that $F = G$ holds if and only if $f(x) = g(ax + b)$ ...
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1answer
20 views

How do I know what the gain is of a second order system which is expressed in this format?

Can someone please explain how to know the gain of a second order system which is expressed as such $$\frac{number_{1}}{(s+ number_{2})(s+ number_{3})}$$ For example $$\frac{40}{(s+0.2)(s+20)}$$ Thank ...
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1answer
22 views

Composition of polynomials and Zariski-topology

I am learning algebraic geometry and have a rather vague question about the the correspondence between polynomial ideals and closed sets in the Zariski-topology in the context of composition of ...
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1answer
21 views

Lagrange polynomial interpolation maximum degree

I want to prove that no polynomial of degree $1$ that passes through $(0, \cos(0))$, $(0.6, \cos(0.6))$ and $(0.9, \cos(0.9))$. By the following theorem: Theorem 1. If $x_{0}, x_{1}, \ldots, x_{n}$ ...
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1answer
41 views

Finding all monic complex polynomials $P(x)$ such that $P(x)|P(x^2)$ [duplicate]

Find all monic complex polynomials $P(x)$ such that $P(x)|P(x^2)$. My progress so far is that I have find that for degree 1, $P(x)=x, x^2$ are the only ones. For degree 2, they are $P(x)=x^2+x+1, x^2,...
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Solving a system of generalized weighted power sums

I am interested in the generalization (weighted and higher dimensional) of recovering polynomial roots from their Newton coefficients (power sums): [Dimension 1 formulation:] Given the system $b_k = \...
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4answers
41 views

Determining the value of $k$ if $x+2$ is a factor of $3x^3-kx^2+kx-4$ [closed]

Determine the value of $k$ if $x+2$ is a factor of $3x^3-kx^2+kx-4$. I have absolutely no idea how to tackle this problem, I have tried trial and error but I think I am doing something wrong. Could ...
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3answers
28 views

Finding a polynomial $f(x)$ that when divided by $x+3$ yields quotient $2x^2-x+7$ and remainder $10$

I'm struggling to grasp this particular question: When a polynomial $f(x)$ is divided by $x+3$, the quotient is $2x^2-x+7$ and the remainder is $10$. What is $f(x)$? This is what I did: $$\begin{...
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2answers
54 views

Test whether a given function is polynomial

You have a black box function to which you can give real number inputs and from which you can receive real number outputs. How would you test whether it is likely to be a polynomial? One expensive ...
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1answer
47 views

Necessary and sufficient condition for a degree 4 polynomial to be a sum of 4th powers

This is a problem I encountered in Hall and Knight's Higher Algebra. Suppose we have a polynomial $p(x,y)=a_0x^4+4a_1x^3y+6a_2x^2y^2+4a_3xy^3+a_4y^4.$ I want to find a necessary and sufficient ...
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0answers
40 views

Roots of a certain sixth order polynomial

I am looking for the roots (or basically any information regarding them) of the sixth order polynomial $$p(x):=ax^6+(a+1)x^4+2bx^3-b^2$$ for positive, real constants $a,b$. Since $p(0)=-b^2<0$ and $...
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2answers
77 views
+50

Cube root of $-2+i$

Edit: My question comes from finding the solutions of this equation using Cardano's method(because our teacher said :D ): $$x^3-6x+4=0$$ And finally I got: $$x=(\sqrt[3]{2})\sqrt[3]{-2+\sqrt{-1}}⠀+(\...
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1answer
47 views

Unique solutions for $\frac{A^2}{a}+\frac{B^2}{b}+\frac{C^2}{c}=1,\quad\text{and}\quad a+b+c=1.$

Let $A,B,C$ be strictly positive real numbers satisfying $A+B+C=1$ and let $a, b, c $ be real variables. Suppose $a, b, c$ satisfy the following system of equations: $$\frac{A^2}{a}+\frac{B^2}{b}+\...
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1answer
17 views

The minimum root of determinant polynomial of PSD matrices are all non-negative?

Suppose $A\in \mathbb{R}^{n\times n}$ is a PSD matrix. Define a multi-affine polynomial $$p(z_1,\cdots,z_n) = \det[diag(z_1,\cdots,z_n)-A] .$$ Suppose $a = (a_1,\cdots,a_n)$ satisfying $diag(a)-A$ is ...
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2answers
95 views

Polynomial such that $e^{2i\pi P(n)} \rightarrow 1$

here is a problem i’ve been having quite a lot of trouble with . Let $P$ be a polynomial such that the sequence $e^{2i\pi P(n)}$ converges to 1 $(i^2=-1).$ Show that $\forall n ,P(n)$ is an integer. ...
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0answers
14 views

Simplify the sum of the cube of a polynomial having fourth degree.

Today I learn about polynomial again, because I want to improve my knowledge. Thank you for your support and time for sharing information and experience. From question : If $a, b, c$ and $d$ are the ...
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1answer
25 views

Best polynomial approximation of a partially constant function

Suppose $\alpha \in [0;1]$. Let's define $f_\alpha$ as the function $[0;1] \to [0;1]$ with the following formula: $$f_\alpha(x) = \begin{cases} 0 & \quad x < \alpha \\ 1 & \quad x \geq \...
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3answers
59 views

If $(x+y-7)[z(x+y)+24]=(y+z-7)[x(y+z)+24]=(z+x-7)[y(z+x)+24]$, find $x^2+y^2+z^2$

Let x, y, z be pairwise distinct real numbers, if $$(x+y-7)[z(x+y)+24]=(y+z-7)[x(y+z)+24] $$ $$=(z+x-7)[y(z+x)+24]$$, find $x^2+y^2+z^2$ I've tried many ways but couldn't find a working way to solve ...
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1answer
52 views

Morphisms from quotient ring

Let $f(x)$ be a polynomial in $\mathbb{Z}[x], \left<f(x)\right>$ be the ideal it generates. Let $R$ be a ring. Prove that giving a ring homomorphsim $\mathbb{Z}[x]/\left<f(x)\right> \...
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0answers
12 views

How to prove that a spline is a polynomial

Let $\Delta$ be a subdivision of $[a,b]$: $a=t_1 < t_2 <\ldots < t_N =b$. Then $\mathbb{S}_{m}^{k}(\Delta )=\{s\in \mathcal{C}^{k}([a,b]) : s_{\mid_{[t_i , t_{i+1}]}}\in\mathbb{P}_{m}, i=1,\...
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1answer
38 views

Invert “matrix of polynomials” for a change of presentation of an Ideal

If I start with a polynomial ideal generated by a set of polynomials $f_i$, which is not a standard basis for the ideal, and then I obtain a standard basis for the ideal to be a set of polynomials $...
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0answers
76 views

General solution for a cascading recurrence relation

My main question: Starting from a simple recurrence relation: \begin{equation} Y_i = aY_{i-1}+(1-a)X_i\tag{1} \end{equation} I can easily find that the general solution, if $Y_0=0$, is: \begin{...
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1answer
55 views

Given that $x^3=x^2+x+1$ find $x^4$ in terms of $x^2$ and a constant.

I don't know is this can be done, but I'm just interested in it, and I'm hoping that it is possible. Here's the question: Given that $x^3=x^2+x+1$ find $x^4$ in terms of $x^2$ and a constant. I ...
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1answer
41 views

What is the function for the following plot?

I have the following plot and have huge difficulties finding the corresponding function $X=1,2,3,6,12,40,100$ $Y=-2,-1,0,1,2,3$ $R=$ the results $\mathbf{-2}$ $\mathbf{-1}$ $\mathbf{0}$ $\mathbf{1}$ $...
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1answer
89 views

Asymptotic expression of root of $x^n+\frac{1-x^n}{x-1}=0$ [closed]

How does one prove that given $$f_n(x) = x^n+\frac{1-x^n}{x-1}$$ Then for $x_n=2+\dfrac{1}{1-2^n}$ we have $\lim\limits_{n\to\infty}f_n(x_n)=0$ That is, how does one prove that such value $x_n$ is an ...
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1answer
18 views

Local minima of the modulus of a square-free complex polynomial is always zero?

I came across the following questions in my research: Let P(z) be a complex polynomials which has only simple roots. Consdier the two real-variable function h(x,y)=|P(x+yi)|. Then is it true that h ...
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0answers
13 views

Finding dimensions of a square based prism [closed]

How do I go about solving this? Seem’s so simpl but I just can’t grasp the proper formula. A package sent by a shipping company has the shape of a square base prism with a side length of x centimetres....
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2answers
44 views

Help with polynomials give zeros

My teacher hasn’t covered this type of material yet and I’m really struggling. All I know is that it’s a third degree polynomial and the roots are $x=2$ and $x=-3i$. Also given is that the polynomial ...
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0answers
50 views

Given $f(x)=ax^6+3x^4+cx^3+dx^2+ex+8$, determine the values of $a,c,d,e,f$ and $k$.

I’ve tried a combination of different things but still haven’t found a viable path. Any help would be appreciated. Here is the full question (title does not allow that many characters) Given $f(x)=ax^...
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3answers
64 views

Finding every $t\in \Bbb R$ such that $f(x)=x^3-tx^2+4$ Determined all values of t, and has two equal roots

Given the function $f(x)=x^3-tx^2+4$, determine all values of t, where $t\in \Bbb R$ such that $f(x)$ has two equal roots. Anybody have any ideas? I am currently attempting it but have no ideas how ...
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1answer
33 views

I need help finding the roots of a characteristic polynomial

I am trying to find the roots of characteristic polynomials. This can be difficult to do by hand, especially when I have a characteristic polynomial of the 3rd or 4th degree. I always try to factorise ...
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0answers
24 views

Artin Algebra, 10.3.7

$R$ will denote a commutative unitary ring. (3.4) Proposition. Substitution Principle: Let $\varphi: R \to R'$ be a ring homomoprhism. Then given elements $\alpha_1,$...$,\alpha_n \in R'$ there is a ...
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0answers
32 views

Rings of polynomials with very high amount of variables

Let $R$ a commutative ring. We can create the polynomial ring with uncountably many variables $R[\mathbb{N}^\mathbb{R}]$. However, it seems like anything we do on this ring could be done on the ...
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0answers
32 views

Artin Algebra 10.2.13 Existence and uniqueness of commutative ring structure on set of polynomials of a ring

The polynomials over a commutative ring $R$ are defined as usual: $R[x] = \{p \in R^\mathbb{N}: \#\mathrm{supp}(p) < \#\mathbb{N} \}$ where $\mathrm{supp}$ is the set of non-zero points and $\#$ is ...
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3answers
47 views

Factoring out $4x^3+2x^2y-2xy^2-y^3$

This can be factored as follows: $$4x^3+2x^2y-2xy^2-y^3 = (2x^2-y^2)(2x+y)$$ What is a systematic way for finding this factorization?
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0answers
85 views

Help solving a cubic polynomial function

I’m tutoring a high school pre-calc student who has to find the roots of a 3rd degree polynomial equation for homework, and I’m struggling to figure out what they are without using an online ...
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1answer
64 views

How can we prove that f(a)*g(a)=(f*g)(a) for any two polynomials f,g?

I have an idea how to prove it, but I'm not sure if it's right. So a verification would be nice, and if it's wrong, a nudge in the right direction would be appreciated. This is my solution: Let $deg(f)...
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3answers
41 views

Solving two simultaneous equations in x and y

During solving a complex numbers question, after comparing the real and imaginary parts, it finally came down to these 2 equations- $$x^3-3xy^2 = y \quad \text{ and }\quad y^3-3yx^2= -x.$$ I am ...
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0answers
156 views
+50

Prove that there exists four polynomials $p_1,p_2,p_3,p_4$ in $x,y,z$ so that $(x^2+y^2+z^2)^3-8(z^3x^3+x^3y^3+y^3z^3)=p_1^2+p_2^2+p_3^2+p_4^2$

Prove that there exists four polynomials $p_{1}, p_{2}, p_{3}, p_{4}$ in $x, y, z$ so that $$\left ( x^{2}+ y^{2}+ z^{2} \right )^{3}- 8\left ( z^{3}x^{3}+ x^{3}y^{3}+ y^{3}z^{3} \right )= p_{1}^{2}+ ...
0
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1answer
48 views

Polynomial $f$ over $\mathbb{C}$.

I came across a question which asks to find all polynomials $f$ over $\mathbb{C}$. Does this mean the coefficients of the polynomial can be complex or the polynomial can take complex values?
2
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1answer
31 views

How to find the Image of the linear transformation?

I tried to find the image of the linear transformation: $T:\mathbb{R}^3\to\mathbb{P}_3$ defined: $T\left[\begin{array}{c}a\\ b\\ c\end{array}\right] = (a+b)x^3+(-b+c)x^2+(a+c)x+(2a+b)$ I define a ...

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