All Questions
23
questions
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Non-positively algebraic number
Let us say that an algebraic number $\alpha \in \mathbf{R}$ is non-positively algebraic if it is the root of a monic polynomial $p(x) = x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + \dots + a_1 x + a_0$ ...
- 193
0
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1
answer
134
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How to find modular roots of $x^{22}-2x^{11}-x+2$ (to show it has more than $22$ solutions by CRT).
Consider a polynomial $P$ defined by $P(x)=x^{22}-2x^{11}-x+2,$ how to show that there exists an integer $n\geq1$ such that the equation $P(x)\equiv0$ modulo has more than $22$ solutions modulo $n?$
*...
- 329
4
votes
3
answers
259
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For which $n$ does $p_n(x)=\sum\limits_{{k=1,(k,n)=1}}^n o(k) x^k $ have exactly two real roots?
Let $n\in\mathbb{N}$ be fixed and denote by $o(k)$ the multiplicative order of $k$ modulo $n$. Define $$p_n(x)=\sum_{\substack{k=1 \\ (k,n)=1}}^n o(k) x^k ;$$here the sum is taken over $k$ that are ...
- 8,369
2
votes
1
answer
64
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Finding roots of polynomial $X^2 - X + 19$ in $\mathbb Z/61 \mathbb Z$
For $p = 61$.
I was given the roots of $X^2 + 3$ in $\mathbb Z/p \mathbb Z$, which are $\pm 27 + p\mathbb Z$.
I then must find the roots of $X^2 - X + 19$ in $\mathbb Z/p\mathbb Z$, which I have found ...
- 35
1
vote
1
answer
107
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Elementary Number Theory and Congruences
Let $f(x) = x^m + a_1x^{m−1} + · · · + a_{m−1}x + a_m$, with $a_j \in \mathbb{Z}$, be a polynomial with integer coefficients and $m \geq 1$.
(i) Show that if $a$ and $b$ are integers with $a \equiv b ...
- 131
0
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0
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96
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Span of an equation
I use this term to denote a set of numbers of the form $\sum z_n x^n$, where $x$ is the solution of some polynomial $x^a=\sum z_ix^i$ for $i=0$ to $a-1$.
$a$, $i$, and the $z_j$ are all in $\mathbb{Z}...
- 6,914
-3
votes
2
answers
171
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Find an integer $a$ such that $(x-a)(x-10)+1=(x-b)(x-c)$ for some integers $b$ and $c$ [closed]
Can someone help with this Olympiad question?
Find an integer $a$ such that $$(x-a) (x-10) +1$$ can be factored as $$(x-b) (x-c)$$ with $b$ and $c$ integer.
3
votes
0
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210
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Apply Cardano's method to find the roots of $f(x)=x^3 - 6x + 4$
Apply Cardano's method to find the roots of $f(x) = x^3 − 6x + 4$
There was a part a) to this question with $f(x)=x^3 - 6x - 6$ which yielded a positive number under the square root using Cardano's ...
- 689
0
votes
3
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83
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Prove that if the roots of the polynomial $x^2+ ax+ b + 1 $ are positive integers, then a and b are integers.
Prove that if the roots of the polynomial $x^2+ ax+ b + 1 $ are positive integers, then a and b are integers.
I'm having problems with this exercise, I'd like to know how to start it.
Thank you.
0
votes
1
answer
155
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Polynomial p(a)=1, why does it have at most 2 integer roots?
Suppose is p(x) is a polynomial with integer coefficients. Show that if p(a)=1 for some integer a then p(x) has at most two integer roots.
I have read the answers given in an earlier post but don't ...
- 31
2
votes
3
answers
246
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Find $\frac {\alpha}{\beta} + \frac {\beta}{\alpha}$ if $\alpha^2+3 \alpha+1=\beta^2+3\beta+1=0$
The question:
Let $\alpha$ and $\beta$ be $2$ distinct real numbers which such that $\alpha^2+3 \alpha+1=\beta^2+3\beta+1=0$. Find the value of $\frac {\alpha}{\beta} + \frac {\beta}{\alpha}$.
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- 2,781
1
vote
2
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140
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What are the roots of this equation?
I have a quadratic equation $ ax^2 +bx+c =0 $, where $ a,b,c $ all are positives and are in Arithmetic Progression.
Also, the roots $\alpha$ and $\beta$ are integers.
I need to find out $ \alpha + \...
- 1,471
0
votes
0
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49
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Finding the roots of this multivariable polynomial?
My polynomial is this ten term monster
$P(x,y,z) = 6561 x^3+486 x^2+12 x+6561 y^3+1944 y^2+192 y+6561 z^3+6318 z^2+2028 z+223$
It's simplest form is ${1 \over 81} \left( (81x+2)^3 + (81y+8)^3 +(81z+...
- 7,252
0
votes
3
answers
212
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Proving there exists such a polynomial
I'm having trouble proving the following statement:
For all primes $p$, there exists a non-constant polynomial $f(x)\in \mathbb Z_p[x]$ such that f(x) does not have a root in $\mathbb Z_p$
What I ...
- 725
2
votes
0
answers
115
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Number of "possible" rational roots of a univariate polynomial
Is it possible to determine the number of possible rational roots of a single variable polynomial?
polynomial a0 + a1x + a2x^2 + ... + anx^n.
This is to find ...
- 21