All Questions
17
questions
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Extended Euclidian algorithm for polynomials [duplicate]
this question follows one from yesterday that got deleted because it was a duplicate.
The problem was about solving this equation (in $ℚ[x]$) :
$$f(x)(2x^3 + 3x^2 + 7x + 1) + g(x)(5x^4 + x + 1) = x + ...
- 93
4
votes
2
answers
217
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prove that $x^{20}+(1-x)^{20}-20$ is square free
Prove that $x^{20}+(1-x)^{20}-20$ is square free (i.e. has no repeated roots).
Note that the claim holds iff $p'(x)$ is coprime to $p(x)$, where $p(x)=x^{20}+(1-x)^{20}-20$. $p'(x)= 20x^{19}-20(1-x)^{...
- 2,031
1
vote
2
answers
275
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Find all polynomials $P(x)$ so that $P(x)(x+1)=(x-10)P(x+1)$
Find all polynomials $P(x)$ so that $P(x)(x+1)=(x-10)P(x+1)$.
I'm looking for a general solution to the above problem. For instance, say I was trying to find all polynomials $P$ satisfying $(x+1) P(x)...
- 2,031
1
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0
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Find the least common multiple of three polynomials [duplicate]
I am recently writing a math module about polynomials in python. And I encountered this question when it comes to compute the least common multiple of several polynomials.
Let [] denote lcm and let () ...
- 605
1
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2
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194
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Find all prime numbers that divide 2 polynomials [duplicate]
I am trying to pass some time during the COVID-19 era. I was going through my mails and found a problem. A friend of mine said her daughter had this problem in some math contest about 2-3 years ago ...
-1
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2
answers
1k
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GCD of polynomials over GF(2)
I have two polynomials:
$$f(x)=x^5+x^3+x+1\\ g(x)=x^4+x^3+x+1.$$
I have to find out $\gcd(f,g)$ over $\operatorname{GF}(2).$
I think the gcd is: $x+1.$ But I am not sure, because here is the ...
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3
votes
2
answers
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What are the steps involved in finding the Greatest Common Divisor of two polynomials?
Ultimately I'm trying to define all the steps necessary to go from this toy quartic polynomial modulus:
$$x^4 + 21x^3 + 5x^2 + 7x + 1 \equiv 0 \mod 23$$
to:
$$x = 18, 19$$
One of the recommended ...
- 271
2
votes
1
answer
68
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Show that every greatest common divisor is a nonzero constant multiples of d(x)
Let $a(x), b(x), d(x)$ be polynomials
I need to show that every greatest common divisor $d(x)$ of $a(x)$ and $b(x)$ is a nonzero constant multiples of $d(x)$
I know it should be easy but i’m stuck, ...
- 135
2
votes
3
answers
136
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How to calculate the $\gcd(10^6 +1 , 10^2 +1)$?
I can't really figure out how to approach this question..
I have tried to factor 10^2 +1 out of 10^6 +1 , however, the '+1' part makes it difficult.
$10^6 + 1 = (10^2 +1) \cdot 10^3 + 899000$,
$10^...
- 339
10
votes
0
answers
267
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$\gcd(p_{n-1}, \ n^5 - n^3 + n^2 - n + 1) = 1$ where $p_n = n$th prime.
How can I prove in general that, for all $n\geq 2$:
$$
\gcd(p_{n-1}, \ n^5 - n^3 + n^2 - n + 1) = 1
$$
Seems to always be true:
...
1
vote
2
answers
64
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How do you compute the $\gcd(1+n+n^2,1+n+n^2+s+2ns+s^2)$
I would like to prove the following claim which I think is true:
Claim: Let $n,$ $m$ and $s$ be positive numbers. Fix $s$, then for every positive number $n$ the $\gcd(1+n+n^2,1+n+s+n^2+2ns+s^2)$ ...
- 3,758
0
votes
1
answer
43
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A query on GCD of polynomials.
I was figuring out the GCD of $p(x)=x^5 + x^4 + 2x^3 + 2x^2 + 2x + 1$ and $q(x)=x^5 + x^4 + x^3 -x^2 -x -1$ and it turns out to be $g(x)=x^2+x+1$. But when I substitute $x=1$ in the above polynomials ...
- 2,411
3
votes
1
answer
258
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I get a wrong answer for the gcd of two polynomials
Hello first post here,
I am trying to get the gcd of the two polynomials using the euclidean algorithm, but as result I get a fraction with huge numbers, instead of 1, which I get as result after ...
6
votes
1
answer
365
views
Denominator in rational gcd of integer polynomials
A recent question tells us that even if two polynomials $f,g\in \mathbb Z[X]$ have no common factor as polynomials, their values at integer points may have common factors. That question gives this ...
- 218k
0
votes
2
answers
249
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Finding the GCD of two polynomials.
Hello I'm trying to find the gcd of these two polynomials:
$$x^4-x^3-4x^2-x+5$$ $$x^2+x-2$$
And then I want to express the gcd of these two polynomials in terms of themselves multiplied by other ...
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