Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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
votes
3answers
25 views

Factorise of polynomial

Factorize $$x^4 - 5x^3 - 5x^2 - 5x - 6$$ I have tried different methods to solve but could no be able to do so. Please can somebody help. Your individual contributions would be greatly appreciated. ...
0
votes
1answer
33 views

$x_i=\prod_{j \neq i}x_j$ for $i=1,2,..,6$

Find all real $x_i^s$ satisfying the system of equations $x_i=\prod_{j \neq i}x_j$ for all $i=1,2,..,6$ It is obvious that $(x_1,x_2,..x_6)=(0,0,..0),(1,1,...,1),(-1,-1,-1,...-1)$ are obvious ...
1
vote
0answers
16 views

Chebyshev Polynomials: Properties of Derivatives

Show that: $T_n'(x)$=$2n\sum_{k=0\\k+n~~odd}^{n-1}\frac{1}{c_k}T_k(x)$ $T_n''(x)$=$\sum_{k=0\\k+n even}^{n-2}\frac{1}{c_k}n(n^2-k^2)T_k(x)$ where $c_0=2$ and $c_n=1$ for $n\geq1$ I tried using the ...
0
votes
0answers
30 views

factoring a polynomial which cannot be done with rational root test.

I try to factor the polynomial t$^3$ $-$ 5t$^2$ $-$ 5t $-$ 1 = 0. But it cannot be done with rational root test obviously, plug in 1 and $-$1 does not work. Any other way to solve it quickly?
1
vote
2answers
51 views

Polynomial Roots with no complex roots

Let $(a_1,b_1),$ $(a_2,b_2),$ $\dots,$ $(a_n,b_n)$ be the ordered pairs $(a,b)$ of real numbers such that the polynomial $$p(x) = (x^2 + ax + b)^2 +a(x^2 + ax + b) - b$$has exactly one real root and ...
0
votes
0answers
19 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
votes
1answer
58 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
vote
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) = ...
0
votes
1answer
33 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 ...
1
vote
1answer
43 views

Decomposing bivariate quadratic form into sum of two squares

I would like to decompose $$ax_1^2 + bx_2^2 + 2cx_1x_2$$ 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 ...
1
vote
1answer
40 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
votes
0answers
46 views

Factoring a polynomial with one variable [on hold]

Any good ideas to solve equation $x^3- 9x^2 + 24x - 20 = 0$ ?
10
votes
0answers
93 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{...
-2
votes
0answers
44 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.
0
votes
0answers
65 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
votes
1answer
35 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
votes
2answers
40 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)^{...
0
votes
0answers
20 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
votes
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 ...
1
vote
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 $\iff\gcd(f, Df) = 1$....
0
votes
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
votes
1answer
73 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
vote
1answer
71 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
votes
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
votes
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 [closed]

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
votes
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
vote
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
votes
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
votes
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
votes
0answers
35 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
votes
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
votes
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
votes
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
votes
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
votes
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 $n−1$ ...
0
votes
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
213 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^...
0
votes
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
votes
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
votes
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 ...