Factoring/Reducing a polynomial $x^4 -2x^3 + 2x^2 + x + 4$
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The problem asks to determine whether or not $x^4 -2x^3 + 2x^2 + x + 4$ is reducible in $mathbb{Q}[x]$. I tried using the fact that if it is reducible (solution manual said it is) then it is reducible in $mathbb{Z}[x]$. That didn't work for me. Eisenstein's criteria isn't applicable.
abstract-algebra polynomials irreducible-polynomials
edited Dec 2 '18 at 4:05
Martin Sleziak
44.7k9117272
asked Nov 18 '15 at 13:42
Yunus SyedYunus Syed
59028
$endgroup$
add a comment |
$begingroup$
The problem asks to determine whether or not $x^4 -2x^3 + 2x^2 + x + 4$ is reducible in $mathbb{Q}[x]$. I tried using the fact that if it is reducible (solution manual said it is) then it is reducible in $mathbb{Z}[x]$. That didn't work for me. Eisenstein's criteria isn't applicable.
abstract-algebra polynomials irreducible-polynomials
edited Dec 2 '18 at 4:05
Martin Sleziak
44.7k9117272
asked Nov 18 '15 at 13:42
Yunus SyedYunus Syed
59028
$endgroup$
$begingroup$
How exactly did you try to use the fact that if it is reducible in $Bbb{Q}[x]$, then it is reducible in $Bbb{Z}[X]$? If you share your efforts, other users can give you an answer more suitable to your needs.
$endgroup$
– Servaes
Nov 18 '15 at 13:49
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See also: Can I use Eisenstein's criterion to show $x^{4}-2x^{3}+2x^{2}+x+4$ is reducible over $mathbb{Q}$?
$endgroup$
– Martin Sleziak
Dec 2 '18 at 4:06
add a comment |
$begingroup$
The problem asks to determine whether or not $x^4 -2x^3 + 2x^2 + x + 4$ is reducible in $mathbb{Q}[x]$. I tried using the fact that if it is reducible (solution manual said it is) then it is reducible in $mathbb{Z}[x]$. That didn't work for me. Eisenstein's criteria isn't applicable.
abstract-algebra polynomials irreducible-polynomials
edited Dec 2 '18 at 4:05
Martin Sleziak
44.7k9117272
asked Nov 18 '15 at 13:42
Yunus SyedYunus Syed
59028
$endgroup$
The problem asks to determine whether or not $x^4 -2x^3 + 2x^2 + x + 4$ is reducible in $mathbb{Q}[x]$. I tried using the fact that if it is reducible (solution manual said it is) then it is reducible in $mathbb{Z}[x]$. That didn't work for me. Eisenstein's criteria isn't applicable.
abstract-algebra polynomials irreducible-polynomials
abstract-algebra polynomials irreducible-polynomials
edited Dec 2 '18 at 4:05
Martin Sleziak
44.7k9117272
asked Nov 18 '15 at 13:42
Yunus SyedYunus Syed
59028
edited Dec 2 '18 at 4:05
Martin Sleziak
44.7k9117272
asked Nov 18 '15 at 13:42
Yunus SyedYunus Syed
59028
edited Dec 2 '18 at 4:05
Martin Sleziak
44.7k9117272
edited Dec 2 '18 at 4:05
Martin Sleziak
44.7k9117272
edited Dec 2 '18 at 4:05
Martin Sleziak
44.7k9117272
44.7k9117272
asked Nov 18 '15 at 13:42
Yunus SyedYunus Syed
59028
asked Nov 18 '15 at 13:42
Yunus SyedYunus Syed
59028
asked Nov 18 '15 at 13:42
Yunus SyedYunus Syed
59028
59028
$begingroup$
How exactly did you try to use the fact that if it is reducible in $Bbb{Q}[x]$, then it is reducible in $Bbb{Z}[X]$? If you share your efforts, other users can give you an answer more suitable to your needs.
$endgroup$
– Servaes
Nov 18 '15 at 13:49
$begingroup$
See also: Can I use Eisenstein's criterion to show $x^{4}-2x^{3}+2x^{2}+x+4$ is reducible over $mathbb{Q}$?
$endgroup$
– Martin Sleziak
Dec 2 '18 at 4:06
add a comment |
$begingroup$
How exactly did you try to use the fact that if it is reducible in $Bbb{Q}[x]$, then it is reducible in $Bbb{Z}[X]$? If you share your efforts, other users can give you an answer more suitable to your needs.
$endgroup$
– Servaes
Nov 18 '15 at 13:49
$begingroup$
See also: Can I use Eisenstein's criterion to show $x^{4}-2x^{3}+2x^{2}+x+4$ is reducible over $mathbb{Q}$?
$endgroup$
– Martin Sleziak
Dec 2 '18 at 4:06
$begingroup$
How exactly did you try to use the fact that if it is reducible in $Bbb{Q}[x]$, then it is reducible in $Bbb{Z}[X]$? If you share your efforts, other users can give you an answer more suitable to your needs.
$endgroup$
– Servaes
Nov 18 '15 at 13:49
$begingroup$
How exactly did you try to use the fact that if it is reducible in $Bbb{Q}[x]$, then it is reducible in $Bbb{Z}[X]$? If you share your efforts, other users can give you an answer more suitable to your needs.
$endgroup$
– Servaes
Nov 18 '15 at 13:49
$begingroup$
See also: Can I use Eisenstein's criterion to show $x^{4}-2x^{3}+2x^{2}+x+4$ is reducible over $mathbb{Q}$?
$endgroup$
– Martin Sleziak
Dec 2 '18 at 4:06
$begingroup$
See also: Can I use Eisenstein's criterion to show $x^{4}-2x^{3}+2x^{2}+x+4$ is reducible over $mathbb{Q}$?
$endgroup$
– Martin Sleziak
Dec 2 '18 at 4:06
add a comment |
1 Answer
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Since it is a monic polynomial, then the only possible rational roots are integer factors of its constant term. If none of those works, then it has no linear factor, so the only possible way to factor it is in the form $$(x^2+ax+b)(x^2+cx+d)$$ for some $a,b,c,d.$ Try expanding this product, equating the coefficients, and coming up with $a,b,c,d$ that fit the bill.
answered Nov 18 '15 at 13:58
Cameron BuieCameron Buie
85.1k771155
$endgroup$
$begingroup$
I just realized what I did wrong
$endgroup$
– Yunus Syed
Nov 18 '15 at 14:07
add a comment |
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1 Answer
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1 Answer
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active
oldest
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oldest
votes
active
oldest
votes
$begingroup$
Since it is a monic polynomial, then the only possible rational roots are integer factors of its constant term. If none of those works, then it has no linear factor, so the only possible way to factor it is in the form $$(x^2+ax+b)(x^2+cx+d)$$ for some $a,b,c,d.$ Try expanding this product, equating the coefficients, and coming up with $a,b,c,d$ that fit the bill.
answered Nov 18 '15 at 13:58
Cameron BuieCameron Buie
85.1k771155
$endgroup$
$begingroup$
I just realized what I did wrong
$endgroup$
– Yunus Syed
Nov 18 '15 at 14:07
add a comment |
$begingroup$
Since it is a monic polynomial, then the only possible rational roots are integer factors of its constant term. If none of those works, then it has no linear factor, so the only possible way to factor it is in the form $$(x^2+ax+b)(x^2+cx+d)$$ for some $a,b,c,d.$ Try expanding this product, equating the coefficients, and coming up with $a,b,c,d$ that fit the bill.
answered Nov 18 '15 at 13:58
Cameron BuieCameron Buie
85.1k771155
$endgroup$
$begingroup$
I just realized what I did wrong
$endgroup$
– Yunus Syed
Nov 18 '15 at 14:07
add a comment |
$begingroup$
Since it is a monic polynomial, then the only possible rational roots are integer factors of its constant term. If none of those works, then it has no linear factor, so the only possible way to factor it is in the form $$(x^2+ax+b)(x^2+cx+d)$$ for some $a,b,c,d.$ Try expanding this product, equating the coefficients, and coming up with $a,b,c,d$ that fit the bill.
answered Nov 18 '15 at 13:58
Cameron BuieCameron Buie
85.1k771155
$endgroup$
Since it is a monic polynomial, then the only possible rational roots are integer factors of its constant term. If none of those works, then it has no linear factor, so the only possible way to factor it is in the form $$(x^2+ax+b)(x^2+cx+d)$$ for some $a,b,c,d.$ Try expanding this product, equating the coefficients, and coming up with $a,b,c,d$ that fit the bill.
answered Nov 18 '15 at 13:58
Cameron BuieCameron Buie
85.1k771155
answered Nov 18 '15 at 13:58
Cameron BuieCameron Buie
85.1k771155
answered Nov 18 '15 at 13:58
Cameron BuieCameron Buie
85.1k771155
answered Nov 18 '15 at 13:58
Cameron BuieCameron Buie
85.1k771155
85.1k771155
$begingroup$
I just realized what I did wrong
$endgroup$
– Yunus Syed
Nov 18 '15 at 14:07
add a comment |
$begingroup$
I just realized what I did wrong
$endgroup$
– Yunus Syed
Nov 18 '15 at 14:07
$begingroup$
I just realized what I did wrong
$endgroup$
– Yunus Syed
Nov 18 '15 at 14:07
$begingroup$
I just realized what I did wrong
$endgroup$
– Yunus Syed
Nov 18 '15 at 14:07
add a comment |
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$begingroup$
How exactly did you try to use the fact that if it is reducible in $Bbb{Q}[x]$, then it is reducible in $Bbb{Z}[X]$? If you share your efforts, other users can give you an answer more suitable to your needs.
$endgroup$
– Servaes
Nov 18 '15 at 13:49
$begingroup$
See also: Can I use Eisenstein's criterion to show $x^{4}-2x^{3}+2x^{2}+x+4$ is reducible over $mathbb{Q}$?
$endgroup$
– Martin Sleziak
Dec 2 '18 at 4:06