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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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0answers
18 views

How do you create the polynomial basis function given data?

According to Wikipedia: However, i saw that basis functions applied before linear regression looks like: Is this saying that, Each observation x has D dimensions. then applying the wikipedia ...
4
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1answer
54 views

How to show that two trigonometric polynomials of degree $n$ combined have at most $2n$ zeros?

I am already aware of this question: Prove the following trigonometric polynomial has $2n$ zeros But it's not the same. Let be $P(x) = \sum_{k=0}^{n} a_k \cos (kx)$ and $\tilde{P}(x) = \sum_{k=0}^n ...
1
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2answers
43 views

Find Taylor Polynomial order 5 of $f(x) = \frac{1}{(1-x)}$, at $a = 0$

Find Taylor Polynomial order 5 of $f(x) = \frac{1}{(1-x)}$ , at $a = 0$ So I start of with: $f(0) = \frac{1}{(1-0)}=1$ $f'(0) = \frac{1}{(1-0)^2 }= 1$ $f''(0) = \frac{2}{(1-0)^3 }= 2$ $f'''(0) = ...
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1answer
32 views

Complex analysis: show for non constant polynomial that p(C)=C

From Remmert's Theory of complex functions chapter 9, page 269 Let p(z)$\in\mathbb{C}$[z] be a non constant polynomial, using the growth lemma and open set mapping theorem (but not the fundamental ...
0
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1answer
34 views

Factoring bivariate quadratic

I would like to factorise $$ax_1^2 + bx_2^2 + 2cx_1x_2$$ into two expressions, each involving only one variable. I'm trying to use a transform like $x_1 = x_+ + x_-$ and $x_2 = x_+ - x_-$ to hopefully ...
1
vote
1answer
38 views

Why does the long division method not work when trying to find GCD of 2 polynomials that have an actual GCD of 1? [duplicate]

I have written a program that calculates the GCD of 2 polynomials using long division (Euclid's algorithm/theorem). I am trying to find the GCD of x^2 - 3 and x + 5, which I already know is 1. ...
-1
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0answers
46 views

Factoring a polynomial with one variable [on hold]

Any good ideas to solve equation $x^3- 9x^2 + 24x - 20 = 0$ ?
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0answers
74 views

About $\sum\limits_{r=1}^{k}\exp\left[2i\pi\sum\limits_{k=1}^{n}\frac{r^{k}}k\right]$

Context: I recently saw user @David's profile picture and description: "My icon is the graph of the exponential sum $$\sum_{n=1}^{10620}e^{2\pi if(n)}$$ for $$f(n)=\frac{n}{20}+\frac{n^2}{9}+\frac{...
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0answers
43 views

What are the solutions to this equation $4x^4-3x^2+4x-3=0$? [on hold]

What is the solution to this equation $4x^4-3x^2+4x-3=0$. I tried solving it but I couldn't get to the answer.
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0answers
63 views

Find zero divisors for polynomials in several variables

I don't know how to find all zero divisors for polynomials in several variables. For example: $\mathbb{Z}_2[X,Y]/(X^2,XY,Y^2)\quad $ or $\quad \mathbb{Z}_4[X,Y]/(X^2,Y^2-XY)$ Can we to proceed ...
0
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1answer
34 views

Rook polynomial - how to understand the division principle of chessboard in this example?

I have this example from a discrete mathematics book and I can not understand the principle by which the chessboard in the example is being divided to smaller parts. The image is attached.By reading ...
0
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2answers
38 views

Show some polynomial satisfies Eisenstein's Criterion

Consider a polynomial $$f(X) = X^{(p-1)p^{n-1}} + X^{(p-2)p^{n-1}} + \cdots + X^{p^{n-1}} + 1$$ Now I need to show $f(X+1)$ safeties Eisenstein's criterion. My argument is that $$f(X+1) = (X+1)^{...
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0answers
19 views

Solutions to cubic and higher degree functions

If $u = ax^2+a_2x+a_3$ is a quadratic polynomial then there exists a solution to $w=v^2$ where $w=cu+c_2$ and $v = bx+b_2$, which can be easily shown to be true. For instance when $u=x^2+x+1$, $w=4u-...
0
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1answer
22 views

Resolving a second degree inequation using its factored form

I'm trying to solve an exercise and I'm having troubles to get it right. We have two functions $f(x)$ and $g(x)$ defined as follow : $f(x) = 2x-3$ $g(x) = -x²+x-3$ Question : We have to demonstrate ...
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2answers
37 views

Show that $c^2 + a^2d=abc$ for a monic quartic polynomial

I apologize in advance for asking a homework question, but I have genuinely no idea on how to approach part b). The question is as follows: Consider the polynomial equation $\rm{P}(x)=x^4 + ax^3 + ...
2
votes
1answer
33 views

Determining the number of real roots of a certain function

I apologize in advance for asking a homework question, but I have genuinely no idea on how to approach part b). The question is as follows: (a) Show that the polynomial expression $x^4 -x^2 + x +\...
2
votes
2answers
52 views

Non-Separable Polynomials and their Derivatives

We say that a polynomial $f(x) ∈ F[x]$ is inseparable if it has a repeated root in some field extension. Otherwise we say that $f(x)$ is separable. Prove that $f(x)$ is separable $\iff\gcd(f, Df) = 1$....
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1answer
34 views

Factorization of $x^p-1$ modulo $p^n$

What are the (monic) divisors of the polynomial $x^p-1$ in the ring $(\mathbb{Z}/p^n\mathbb{Z})[x]$? For $n = 1$, the ring $(\mathbb{Z}/p\mathbb{Z})[x]$ is a UFD, and we have $x^p - 1 = (x-1)^p$. ...
4
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1answer
72 views

Find a remainder when $(x^5+1)^{100} + (x^5-1)^{100}$ is divided by $x^4+x^2+1$

The question is: Find a remainder when $f(x)=(x^5+1)^{100} + (x^5-1)^{100}$ is divided by $x^4+x^2+1$ I first began by decomposing $$x^4+x^2+1=(x^2+x+1)(x^2-x+1)$$ and using $$x^3-1=(x-1)(x^2+x+1)\...
1
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1answer
70 views

6th degree equation [duplicate]

The number of real roots of $$ \frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19} = x^2 -11x -4 $$ How to solve without actually finding the roots or is it the only way? I know that if the ...
0
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1answer
63 views

Why is $x = 25n + 9$ in the equation: $3=(3388997632 x^{23}) \text{ mod } 25$?

Apparently the general solution of this: $3=(3388997632\cdot x^{23}) \text{ mod }25$ is $x = 25n + 9$, where $n$ is any natural number, it seems? I get how there is connection with $25$ as modulo, ...
0
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1answer
40 views

How to take this variable out of this equation?

I'm working to develop a function in my embedded code for i2c initialization. I came up to this equation which is to calculate the speed of the i2c clock. But if I want to develop a function that ...
-2
votes
0answers
31 views

Symmetry in the roots of a quadratic [on hold]

Given $m$ is a root of $x^2+ax+b=0$ find all the possible values of $(a,b)$ such that $m^2-2$ is also a root.
0
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1answer
21 views

Median of a continuous r.v.

Let X be a continuous r.v., with pdf $f_X(x) = kx(1-x), 0 < x <1$ I evaluated k = 6 and found the cdf $F_x(x) = 3x^2 - 2x^3$, but then I am asked to find the median. The equation $F_x(x) = 1/2$ ...
2
votes
2answers
66 views

Factorization of a polynomial in $\Bbb F_7$

I need to reduce as much as possible the polynomial: $x^6+3x^5+2x^4+6x^3+4x^2+5x+2$ in the finite field $\Bbb F_7$. It has no roots over the field, and I don't think that it is necessary to check ...
1
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4answers
59 views

Finding the eigenvalues of $A=\left(\begin{smallmatrix} a & 1 & 1 \\ 1 & a & 1 \\ 1 & 1 & a \\ \end{smallmatrix}\right)$

I would like to calculate the eigenvalues of the following matrix $A$, but the factorization of the characteristic polynomial does not seem to be easy to compute. $A=\pmatrix{ a & 1 & 1 \\ 1 &...
0
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1answer
32 views

Solving $z^2+\frac{9z^2}{(3+z)^2}=-5$

Solve the equation $$z^2+\frac{9z^2}{(3+z)^2}=-5$$ PS.: The expanded form of a 4 degree polynomial is $$z^4+6z^3+23z^2+30z+45=0$$
4
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1answer
51 views

Low-degree polynomial $T\in\mathbb F[x,y]$ with $T(P(z),Q(z))=0$

Given polynomials $P,Q\in\mathbb F[z]$ over a finite field $\mathbb F$, one can find a non-zero polynomial $T\in\mathbb F[x,y]$ such that $T(P(z),Q(z))=0$ for any $z\in\mathbb F$. Is there a way to ...
0
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0answers
34 views

polynomial equation with non-integer powers

If for every $t$ $$\sum_{i=0}^{k_1}\left[\left(a_i^Tx-b_i\right)t^i\right]=0$$ where $a_i \in \mathbb{R}^{n \times 1}$, $x \in \mathbb{R}^{n \times 1}, \forall i \in \{0,\dots, k_1\}$, and $b_i \in \...
3
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3answers
61 views

Solution to $\sqrt{\sqrt{x + 5} + 5} = x$

There are natural numbers $a$, $b$, and $c$ such that the solution to the equation \begin{equation*} \sqrt{\sqrt{x + 5} + 5} = x \end{equation*} is $\displaystyle{\frac{a + \sqrt{b}}{c}}$. Evaluate $a ...
0
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3answers
40 views

Check if true : Atleast one of the integers, a, b, c must be even

Suppose a, b, c are integers such that the equation $ ax^2 + bx + c =. 0 $ has a rational root. Check if true : Atleast one of the integers, a, b, c must be even. I know for rational roots $ b^2 - ...
0
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2answers
29 views

The number of quadratic equations which are unchanged by cubing their roots is

I want to find the number of quadratic equations which are unchanged by cubing their roots. Let $ax^2 + bx + c $ be a quadratic whose roots are $ \alpha$ and $\beta $. I know that quadratic whose ...
2
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1answer
11 views

Projection map for polynomial rings

Let $K$ be a field and Consider the projection map $\pi_{i,j} : K[X]/(X^i) \to K[X]/(X^j)$, for $j \leq i$. This is well-defined since $(X^i) \subseteq (X^j)$. I'm wondering what it looks like, is it ...
0
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1answer
42 views

For how many real numbers 'b', does $x^2 + bx + 6b = 0$ have one integral root?

The question states. For how many real numbers 'b', does $ f(x) =x^2 + bx + 6b = 0$ have one integral root ? My line of thinking : Let $\alpha , \beta$ be the roots of of $f(x)$. $\alpha + \beta = -b....
0
votes
0answers
19 views

f(x) is invertible polynomial function of degree ‘n’ {n≥3} then f"(x) = 0 has exactly ‘n – 2’ distinct real roots if

$f(x)$ is invertible polynomial function of degree $n\geq 3$ then $f''(x) = 0$ has exactly $n - 2$ distinct real roots if A)$f′(x)=0$ has $n−1/2$ distinct real roots B)$f′(x)=0$ has $n−1$ ...
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0answers
14 views

Is this proof of a irreducibility criterion in an integral domain correct?

This is an exercise from Grillet's "Abstract Algebra" (page $145$, proposition $10.10$). Let $R$ be an integral domain, let $I$ be an ideal of $R$, and let $\pi\colon R\to R/I$ be a canonical ...
7
votes
5answers
212 views

Remainder theorem for polynomials (JUEE 1990)

Suppose the polynomial $P(x)$ with integer coefficients satisfies the following conditions: (A) If $P(x)$ is divided by $x^2 − 4x + 3$, the remainder is $65x − 68$. (B) If $P(x)$ is divided by $x^...
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0answers
44 views

How to solve mixed power and exponential equation?

The exercise says that I should find "real zeros" of the following function: $$f(x)=7-2x^2+2^{-x}$$ I have tried several ways, but I keep getting stuck as I take log-s of both sides. Can I solve ...
3
votes
1answer
134 views

Find the coefficient of $x^{32}$ in $(x^3 +x^4 +x^5 +x^6 +x^7)^7$

I don't understand the explanation in the book and why the final answer looks the way it does. I know I am supposed to factor and it will equal to $(x^3(1+x+\cdots+x^4))^7$. But after that, I am ...
0
votes
1answer
35 views

How can I convert this equation

How can I convert this equation: $$321^2 - 196^2 = 64625$$ to be in this form: $$X^2 - Y^2 + X = 64625$$ Whereas $X$ and $Y$ are Odds and $X > \sqrt{64625}$ I tried to find $X$ value by testing ...
0
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1answer
52 views

What determines how many times a polynomial can be differentiated before 0 is reached?

Also, does it relate to the degree of the polynomial in any way? I am struggling to get a high-level understanding of the characteristics of different degrees of polynomials - for example, their shape ...
0
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2answers
73 views

Inverse of $y = 0.595x^2 + 0.387x^3$

[Edit] Simplify to: $$y = x^2 + x^3$$ Steps for solving an inverse would be to trade x and y (or f(x)) and then solve for y... can you do that here with x and y both still present in the equation? ...
0
votes
1answer
28 views

If $P \in V$ is a polynomial with maximal support $\Sigma$, then $|\Sigma| \geq dim(V)$

I was wondering if anyone could help me understand a statement in the proof of Theorem 4 of this paper: https://www.jstor.org/stable/pdf/24906443.pdf. I'm aiming to prove the theorem for $q=3$ and $\...
0
votes
2answers
32 views

the value of a determinant

Let be a polynomial function $P\in \mathbb{R}[X]$.If I divide $P$ by $(x-1)(x-2)(x-3)(x-4)$ I get a remainder without "free term" ( like $ax^{3}+bx^{2}+cx$ ) I have to calculate the determinant: $$\...
0
votes
2answers
44 views

Find the Taylor Polynomial $T_{3}$ for the Function $f(x) = \frac{5x}{2+4x}$

Find the Taylor Polynomial $T_{3}$ for the Function $f(x) = \frac{5x}{2+4x}$ So I have this problem and I'm struggling, but below is what I am attempting to do: Plan: Attempt to translate series ...
0
votes
2answers
32 views

Conversion to polynomial. [closed]

It is given that $x^2–x–1=0$. Convert $(x^3+x+1)/x^5$ into a polynomial form for that particular set of value(s) of x. Thus problem is troubling me and its part of my maths homework. Please help me ...
2
votes
0answers
76 views

Proving that a prime ideal is principal

Suppose $Q_1, Q_2\in \mathbb{C}[X_0,\dots,X_n]$ are irreducible homogeneous quadratic polynomials such that $V(Q_1, Q_2)$ is an irreducible projective variety of degree two and codimension two in $\...
4
votes
2answers
106 views

Number of real solutions to $x^7 + 2x^5 + 3x^3 + 4x = 2018$

Find the number of real solutions of $x^7 + 2x^5 + 3x^3 + 4x = 2018$? What is the general approach to solving this kind of questions? I am interested in the thought process. Few of my thoughts ...
4
votes
6answers
111 views

Writing $x^2+xy+y^2$ as a product [closed]

How can this polynomial be written as a product of two complex factors? I know it has something to to with the n th root of 1 but i got stuck.
0
votes
0answers
23 views

The minimum degree of interpolating polynomial that fits table data points.

What is the minimum degree that an interpolating polynomial that fits all five data points exactly can have? The following table data points are given for (x,f(x)): (-0.5,5),(0,15),(0.5,9),(1,3),(1.5,...