Tagged Questions

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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1
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2answers
45 views

Factorize x^3+3

Task: Factorize $x^3+3$ in $\mathbb{R}$ and $GF(7)$ I think the solution is $x^3+3$ in both cases, so the polynomial already is irreductible. Is my assuption is right, how do i show that?
0
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1answer
13 views

Disprove that this subset of P3 is not a subspace by using a counterexample

The set of all polynomials with degree 3 plus the zero polynomial. A hint would be appreciated to get me going :)
0
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1answer
39 views

What does “two polynomials have no zeros in common” mean?

The question is Given two constant-coefficient operators $A$ and $B$ whose characteristic polynomials have no zeros in common. Let $C=AB$... What does that mean by "no zeros in common"?
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1answer
21 views

Converting expressions to polynomial form

My question is from Apostol's Vol. 1 One-variable calculus with introduction to linear algebra textbook. Page 57. Exercise 12. Show that the following are polynomials by converting them to the ...
0
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0answers
10 views

Finding set of integer pairs for which two integer polynomials intersect

I am wondering if there is a theorem in number theory that addresses the following issue: Suppose we have two polynomials, f and g, with integer coefficients. Is there a general way to find elements ...
3
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1answer
25 views

Polynomial curve fit

Well I have a 2 (or 3) data points - and some extra limits - and a polynom needs to be fitted through those points (exactly). The polynom needs to be of the smallest order, and not a least square, it ...
2
votes
2answers
28 views

Prove that one of x,y,z is smaller than 3 and one is bigger than 5 if…

If $x+y+z=12$ and $x^2+y^2+z^2=54$ then prove that one has to be smaller or equal to 3 and one has to be bigger or equal than 5. So I got that $xy+yz+zx=45$ and with that I had a function with x,y,z ...
2
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1answer
15 views

Is the Frobenius automorphism a polynomial?

Let $p$ be a prime and $\Bbb{Z}/\Bbb{Z}_p$ the field of integers mod $p$, and since $\Bbb{Z}/\Bbb{Z}_p$ is a field we have the ring of polynomials in $X$ with coefficients in $\Bbb{Z}/\Bbb{Z}_p$ ...
0
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0answers
15 views

Proof Explanation: Vector Space of Polynomials with Average Value 0 around a circle

The question is from Putnam 2009 B4. Problem: Say that a polynomial with real coefficients in two variable, $x,y$, is balanced if the average value of the polynomial on each circle centered at the ...
3
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0answers
63 views

On polynomials over finite fields?

Pick prime $q\in\Bbb Z$ such that $q>B^{3t}>((n+1)B)^{2t}$. Suppose I have a multilinear polynomial $g(x)\in \Bbb Z[x_1,\dots,x_n]$ of degree $t$ whose coefficients are bound by $|B|>2$, I ...
1
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2answers
19 views

How to determine the smallest interpolation degree required?

Given a set of $n$ points $(x_k, y_k)\ (k\in\{1,...,n\})$, of course a polynomial of degree $n$ can fit all points. However, in some cases the coefficient of the higher degrees actually vanish and one ...
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1answer
28 views

$\frac{-1}{2}$ Zero of odd powers sum polynomials?

Consider the polynomial $S_k(x) \in \mathbb{Q}[x]$ such that $S_k(n)=\sum_{i=1}^{n}i^k, \forall n \in \mathbb{N}$. Now if i recall correctly the definition it should be that ...
1
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2answers
41 views

Number of roots of $x^3+2x-1$ in $\mathbb Q$

How many roots does $x^3+2x-1$ have in $\mathbb Q$? I know that it has one real and two complex conjugate roots because the determinant is $-59$.
2
votes
1answer
50 views

What is the name for a polynomial with all coefficients equal to 1?

I am looking for a good google search word for polynomials that have all coefficients equal to 1. An example of a such polynomial is: $$1+x^{23}+x^{57}+x^{101}$$ One such polynomial could also be ...
1
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1answer
37 views

$P_n(x):=1+ \sum_{m=1}^n\dfrac{x^m}{m!}$ has no real root for even $n$ and exactly one real root for odd $n$

Is it true that $P_n(x):=1+ \sum_{m=1}^n\dfrac{x^m}{m!}$ has no real root for even $n$ and exactly one real root for odd $n$ ? I can only prove that the polynomial cannot have any multiple roots . Am ...
3
votes
2answers
47 views

Solving $(1-x)^3 = -1$ over the complex field

What are the solutions of: $(1-x)^3 = -1$ over $\mathbb{C}$? We have one real solution which is $2$ so there are two complex solutions.
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0answers
18 views

Sensitivity of polynomial global minimizers with respect to perturbations in the coefficients.

I'm trying to find the value of a global minimizers of a multivariate polynomial (4 variables) of high order numerically. The numerical values of the coefficients are coming from noisy measurements ...
1
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1answer
24 views

synthetic division with $i$ in divisor

I divided $x^3-4x^2+4x-16$ by $-2i$ using synthetic division and got a remainder of $-8i-8$. Is that right? I'm not sure I'm doing this right.
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0answers
11 views

find $f(x) \in \Bbb Z[x]$ s.t $f(x)-\frac {p(x)}{x^k} \in P=\{\frac {(x^2+1)q(x)}{x^k}| k\geq 0, q(x) \in \Bbb Z[x]\}$ where $p(x)\in \Bbb Z[x]$.

Given $\displaystyle\frac {p(x)}{x^k}$, find $f(x) \in \Bbb Z[x]$ s.t. $f(x)-\displaystyle\frac {p(x)}{x^k} \in P=\{\frac {(x^2+1)q(x)}{x^k}\mid k\geq 0, q(x) \in \Bbb Z[x]\}$ where $p(x)\in \Bbb ...
0
votes
1answer
18 views

Polynomial has right inverse implies invertible?

If $p:\Bbb R\rightarrow \Bbb R$ is a real polynomial such that $p$ has a right inverse $q$, does it follow that $p$ is invertible? That is, must $q$ also be a left inverse of $p$? The question ...
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0answers
18 views

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 ...
0
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1answer
33 views

Three positive real numbers [on hold]

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
19 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 ?
3
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0answers
16 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 $|pqr|=1$.
3
votes
1answer
19 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 ...
1
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2answers
30 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?
0
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1answer
28 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
votes
2answers
31 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
votes
4answers
66 views

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

How many zeros does this function have? $$f(x)= 3x^4 + x + 2 $$
2
votes
1answer
40 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 ...
0
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2answers
25 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 ...
0
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0answers
24 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 ...
0
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6answers
91 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 $x=a+bi$
0
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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. $$x^{2n} +2 x^n \cos{na\pi}+1$$ ...
0
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0answers
35 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 ...
1
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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
votes
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 ...
1
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0answers
40 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 $q$ ...
2
votes
1answer
34 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 ...
0
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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: $f(x,y) = \underset{i=1}{\overset{d}{\prod}}(y^p-a_{i}x^q)^{c_i}$, ...
0
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1answer
18 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 $x$) ...
0
votes
1answer
33 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}, ...
1
vote
1answer
34 views

Polynominal odd function [on hold]

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
2
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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
votes
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
votes
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 ...
1
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1answer
37 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 ...
1
vote
1answer
38 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 ...
1
vote
2answers
27 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.
0
votes
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. ...