Polynomials are expressions like $15x^3 - 14x^2 + 8$. Questions tagged with this concern common operations on polynomials, like adding, multiplying, polynomial long division, factoring and solving for roots.

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Polynomial in 2 variables

Let $\mathbb{P}_k(T)$ the set of polynomials of degree less than or equal to $k$ defined on $T\subset\mathbb{R}^2$ and $\tilde{\mathbb{P}}_k(T)\subset \mathbb{P}_k(T)$ defined by: $p\in ...
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1answer
38 views

Three positive real numbers [closed]

Let $a$,$b$,$c$ be positive real numbers such that $abc$ is not equal to $1$ and $$ [(ab)^2]=[(bc)^4]=[(ca)^k]=abc $$ Then $k$ is equal to
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0answers
21 views

Finding the roots of fourth degree polynomial [duplicate]

$$ax^4 + bx^2 +cx + d = 0$$ How do I find just the real roots not even complex roots ?
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0answers
19 views

ratio based question

If p,q and r are three distinct real numbers such that $(pq+1):(qr+1):(rp+1)$ is $q:r:p$,then prove that $|pqr|=1$.
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1answer
22 views

maximal injective neighborhoods centered at the zero of a polynomial

I was working on a particular problem involving the injectivity of a certain polynomial, $p(z) = z^5 + z -1$, $z \in \mathbb{C}$, in which I needed to find a neighborhood around it's real root so that ...
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2answers
36 views

$P(x+2)=2x^3-4x^2+2x+3$. Find the remainder of $\dfrac{P(x)}{(x-3)}$

$P(x+2)=2x^3-4x^2+2x+3$. Find the remainder of $\dfrac{P(x)}{x-3}$ I've tried this: $P(x)=2(x-2)^3-4(x-2)^2+2(x-2)+3$. What should I do next?
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1answer
35 views

How do we divide $P(x)$ with $ax+b$?

How do we divide $P(x)$ with $ax+b$? I've tried this: $x=-\frac ba$ $p(x)=(x+\frac ba)Q(x)+v=a(x+\frac ba)\cdot \frac 1a \cdot Q(x)+v$. Is this correct?
2
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2answers
33 views

Minimum Number of Values to Guess a Polynomial with Non-Negative Coefficients

My math teacher claimed that he could guess any polynomial with non-negative coefficients given two values that he asked for. For example, he asked me to write down a function of which I wrote down ...
3
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4answers
70 views

How many zeros does $f(x)= 3x^4 + x + 2 $ have?

How many zeros does this function have? $$f(x)= 3x^4 + x + 2 $$
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1answer
46 views

Integral extension and field

I came a cross a question that I don't know how to solve Problem: $A,B$ are commutative domains and $A\subseteq B$. Show if that $B$ is a field and every element of $B$ is the root of a non-trivial ...
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2answers
34 views

Factorize this polynomial $ax^2+bx+c$ into factors of the first exponent in the cases when D>0, D=0

The previous request was to prove the identity $ax^2+bx+c=a[(x+(b/2a)^2-(D/4a^2)]$, where $D=b^2-4ac$ And I proved it from the left to the right, which means I managed to express $ax^2+bx+c$ as ...
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0answers
26 views

Is $r_1 \cdot f_1 + r_2 \cdot f_2 $ uniformly distributed?

Consider $f_1$ and $f_2$ are fixed polynomials, $r_1$ is a random linear polynomial, $r_2$ is a random polynomials, degree($r_2$)=degree($f_i$)=$d$. We define $f_i$ and $r_i$ over $R[x]$ where $R$ can ...
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6answers
97 views

Finding the roots of $x^2+(3+5i)x+(7+11i)=0$

how can I solve following equation analytically $$x^2+(3+5i)x+(7+11i)=0$$ I need the roots as follow $x=a+bi$
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1answer
22 views

Factoring $x^{2n} +2 x^n \cos{na\pi}+1$ polynomial

I have a very strange polynomial to factorize and can't even get started with it, so if anyone could give me a hint on how to get started, not the exact solution. $$x^{2n} +2 x^n \cos{na\pi}+1$$ ...
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0answers
38 views

Linear combination of m polynomials equals variable times another linear combination of those polynomials

Let $p_1,p_2, \dots , p_m \in \mathbb C [x,x_1,x_2,\dots , x_n ]$ . Assume that these polynomials have the property that for every $c_1,c_2, \dots , c_m \in \mathbb R$ , there exists $d_1,d_2, \dots ...
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1answer
20 views

Polynom as sum/product of symmetric polynoms

I have a polynom $(x_1^2x_3 + x_2^2x_1 + x_3^2x_2)(x_1^2x_2 + x_2^2x_3 + x_3^2x_1)$ and I need to express as sum/product of elemental symmetric polynoms $s_1,s_2,s_3$. I know there is an algoritm for ...
2
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0answers
29 views

Eigenvalues of 5x5 matrix given equation involving matrix

I have been given the matrix $A$ and we are told it is a $5\times 5$ matrix s.t. $A^4=A^2\neq A$. I want to find the eigenvalues so I tried $A^2(A-I)(A+I)=0$ so the eigenvalues are $0, 1, -1$ but I ...
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0answers
42 views

To prove given $ r \cdot f_1+f_2\cdot (s+1)$ one who knows $f_2$ cannot find out what $f_1$ is

We define the polynomials $r,f_1,f_2,s\in R[x]$. Where $r$ is a random degree 1 polynomial and $s$ is a random polynomial such that: $\deg(s)=\deg(f_1)=\deg(f_2)$. Let $R$ be $\mathbb {Z}_q$ where $q$ ...
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1answer
35 views

Newton Raphson interval choice for multiple roots

I am using Newton Raphson to find the roots of multiple polynomials as part of a homework exercise. However, for polynomials that have complex roots I don't understand how to select the intervals to ...
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0answers
1 views

Factorization of quasi-homogeneous polynomials over C

Let $f(x,y) \in C[x,y]$ be a quasi-homogeneous polynomial, with $f(t^{w_1}x,t^{w_1}y)=t^df(x,y)$ Why we can always write it as: $f(x,y) = \underset{i=1}{\overset{d}{\prod}}(y^p-a_{i}x^q)^{c_i}$, ...
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1answer
20 views

An $\Bbb{R}\to\Bbb{R}$ function with two plateaus of different heights and a valley

I am looking for a $\Bbb{R}\to\Bbb{R}$ function $f$ with two plateaus of different heights and a valley. The function has a minimum for $x=a$ and $f(a)=b$. The first (the one for smaller $x$) ...
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1answer
35 views

Build a polynomial

I have $f=x^3 + ax^2 +bx +c \in \mathbb C[x], \alpha_1,\alpha_2,\alpha_3 \in \mathbb C$ are roots of $f$. $\beta_1 = {\alpha_1 \over \alpha_2} + {\alpha_2 \over \alpha_3} + {\alpha_3 \over \alpha_1}, ...
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1answer
34 views

Polynominal odd function [closed]

If $f(x$) is an odd function and $x-y$ is a factor. show that $x^2-y^2$ is a factor as well I'm having trouble to solve this
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0answers
34 views

Prove that $p(x)=(x-1)(x-2) \cdots (x-n) + 1$ is irreducible over $\mathbb{Z}$ for all $n \geq 1$, $n \neq 4$. [duplicate]

Prove that $p(x)=(x-1)(x-2) \cdots (x-n) + 1$ is irreducible over $\mathbb{Z}$ for all $n \geq 1$, $n \neq 4$. I do not clearly see how to solve this problem and what is so special about the integer ...
4
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0answers
22 views

How to find out if a polynomial equation has real solutions?

I have a polynomial equation of $N$th order. The coefficients of the equation are parametrized by two variables, let's call them $a$ and $b$, both of which are real and positive. For general $N$, I ...
2
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1answer
12 views

Differential Equation involving Polynomial Discriminants

So this is a homework question in my algebra class that I'm getting really stuck on... it should be straightforward, but I'm not sure how to interpret the differential equation. Any hints (solutions ...
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1answer
38 views

Fourth degree polynomial with rational coefficients and a real root

If a quartic has rational coefficients and one real root, how would one go about showing that the real root is rational? I understand that the condition is equivalent to showing that having a ...
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1answer
39 views

Writing a particular polynomial as product of irreducibles in various rings.

I want to factor the polynomial $x^3-10x+4$ into a product of irreducibles over each of the fields $\mathbb{Z}[i]$,$\mathbb{Q}[\sqrt{2}]$, $\mathbb{Q}[\sqrt{2},\sqrt[3]{2}]$, $\mathbb{Z}/ 11 ...
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2answers
30 views

Using Chebisev polynomials to express sin(nx) & cos(nx) as polinomials of sin(x) and cos(x)

$Sin(nx)$ and $cos(nx)$ can be expressed as polynomials of sin(x) and cos(x). I am interested in the way of this expression and a proof (preferably at secondary-school level) as well.
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1answer
39 views

gcd of polynomials over Z_7

I want the gcd of the two polynomials: $$f=x^5+3x^4+5x^3+x^2+x+3$$ $$g=2x^3+4x^2+x$$ in $Z_7[x]$. My approach: I use the euclidean algorithm and continue until I get no remainder. ...
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1answer
45 views

Is Rufinni's rule the quickest hand-method to find roots in high order polynomials?

I'm wondering if there's another method where I do not have to "trial-error" with every guess neither using numerical methods. When the searched root is 3*π/5 with Ruffini's rule I cannot find it ...
8
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1answer
195 views

Existence of a Polynomial

Does their exist a non-linear polynomial $P(x)$ such that for every rational number $y$ there exists a rational number $x$ such that $y=P(x)$?
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1answer
63 views

Find exact value of $\cos (\frac{2\pi}{5})$ using complex numbers.

Factorise $z^5-1$ over the real field. Show that $\cos \frac{2\pi}{5}$ is a root of the equation $4x^2+2x-1=0$ and hence find its exact value. I have worked out that $$ ...
3
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1answer
56 views

How to show that the polynomilal $\sum_{k=0}^n \dfrac 1{3^{k^2}}x^k$ has $n$ distinct real roots for any positive integer $n$ ?

From this Rational roots of polynomials ; How might we show that the polynomilal $\sum_{k=0}^n \dfrac 1{3^{k^2}}x^k$ has $n$ distinct real roots $\forall n \in \mathbb N$ ?
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2answers
277 views

Can we make a sequence of real numbers such that polynomial of any degree with co-efficients of the sequence has all its roots real and distinct ?

Does there exist a sequence of real numbers $(a_n)$ such that $\forall n \in \mathbb N$ , the polynomial $a_nx^n+a_{n-1}x^{n-1}+...+a_o$ has all $n$ real roots ? Can we make a sequence so that all ...
7
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2answers
108 views

Real roots of a polynomial of real co-efficients , with the co-efficients of $x^2 , x$ and the constant term all $1$

Can all the roots of the polynomial equation (with real co-efficients) $a_nx^n+...+a_3x^3+x^2+x+1=0$ be real ? I tried using Vieta's formulae $\prod \alpha=\dfrac {(-1)^n}{a_n}$ , $(\prod \alpha ...
11
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6answers
210 views

If the number $x$ is algebraic, then $x^2$ is also algebraic

Prove that if the number $x$ is algebraic, then $x^2$ is also algebraic. I understand that an algebraic number can be written as a polynomial that is equal to $0$. However, I'm baffled when showing ...
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1answer
13 views

GCD of polynomials by using Euclid's algorithm

Let $g = x^2 +6x -7$ and $f = x^4 - 1$. Find the GCD of $f$ and $g$. So I started by evaluating $f/g$ and the result is $q = x^2-6x+43, r = -300x+300$. I tried to follow the algorithm one step ...
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1answer
45 views

Find a, b, c in the polynom P(x)=ax^2+bx+c if P(x+1)+P(-1)=8x^2+6x+10

I've tried to put if P(u)=8(x-1)^2-6(x-1)+10+8(u+1)^2-6(u+1)+10. How should I solve it?
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0answers
13 views

Localization of polynomial ring as differentiable functions

Let $a \in \mathbb{R}$ be a point and $S=\mathbb{R}[x]_{(x-a)}$ the localization of the polynomial ring $\mathbb{R}[x]$ with maximal ideal $(x-a)$. i) Describe the elements of $S$ as differentiable ...
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1answer
8 views

The number of polynomial in a polynomial ring

If we define a poylnomail ring $R[x]$, I need to know the number of polynomial (in this ring) of a particular degree, $d$, please.Let $R=Z_P$, where $p$ is a prime number.
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1answer
151 views

How to solve: $x^4+x^2=1$

I solved $x^4+x^2+1=0$. But, the above one is hard. The equation is too hard for me to understand. Can anyone solve it? Please help.
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0answers
36 views

Integral points of proper rational functions

Let $f \in \mathbb{Q}(X_1,\dots,X_n)$ be an arbitrary rational function which is not a polynomial, and let $D = \{ x \in \mathbb{Z}^n : f(x) \in \mathbb{Z} \}$ be the set of integral points of $f$. ...
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0answers
20 views

Is the number 5 saying hello to the composed Bernoulli polynomials?

If you compose the Bernoulli polynomials umbrally with themselves, you get polynomials defined by the following e.g.f. $$(\frac{t}{e^t-1})^2e^{x t} = e^{p.(x)t} .$$ The first few are (I think--my ...
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0answers
55 views

question on ideals in rings

Let $S=K[x_1,\dots,x_n]/J$ be a ring where $K$ is a field of characteristic $0$ and $J$ is an ideal with $Z(J)$ being the zero set of the ideal. For every $\tilde{q}\in S$, let $q$ be the lowest ...
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2answers
34 views

It is div:$\mathbb{P}_k\rightarrow \mathbb{P}_{k-1}$ surjective?

My question is: It is the operator $\text{div}:\;(\mathbb{P}_k){\color{red}{^3}}\rightarrow (\mathbb{P}_{k-1})$ surjective ? Here $\mathbb{P}_k$ denotes, as usual, the set of polynomial with degree ...
2
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0answers
43 views

Does the equation $\tan(x)=y$ have any non-zero rational solution?

Trivially $\tan(0)=0$ but it seems this is the "unique" solution of the equation $\tan(x)=y$ on rational numbers. In fact if we try to make $y$ rational we usually use irrational (transcendental) ...
0
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2answers
39 views

Second Degree Polynomial Interpolation, error related

We want to create a table of the exponential integral function $$E_{1}(x)=\int_{x}^{\infty}\frac{e^{-t}}{t}dt, x>0$$ over the interval $x \in [1,10]$ with stepsize $h$. How large can $h$ be if a ...
0
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1answer
102 views

Derivation of Cubic Formula

How do we derive the so called cubic formula without using Cardano's method or substitution? I would like to see a step by step proof of where wolfram alpha derives this answer. And also explain where ...
0
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1answer
17 views

synthetic division/long division divisor sign

I know that if you are dividing by $x-3$ with long, then if you do it with synthetic division it's going to be positive 3 that gets used, the value used in synth. division is opposite. So if your ...