MHB Prove $m^5+3m^4n-5m^3n^2-15m^2n^3+4mn^4+12n^5$ ≠ 33

[anemone]

Prove that $m^5+3m^4n-5m^3n^2-15m^2n^3+4mn^4+12n^5$ is never equal to 33.

 

[kaliprasad]

Prove that $m^5+3m^4n-5m^3n^2-15m^2n^3+4mn^4+12n^5$ is never equal to 33.

Spoiler

First let us factor $m^5+3m^4n-5m^3n^2-15m^2n^3+4mn^4+12n^5$
$m^5+3m^4n-5m^3n^2-15m^2n^3+4mn^4+12n^5$
= $m^4(m+3n) - 5m^2n^2(m+3n) + 4n^4(m+3n)$
= $(m+3n)(m^4-5m^2n^2+4n^4)$
= $(m+3n)(m^2-4n^2)(m^2-n^2)$
= $(m+3n)(m-2n)(m+2n)(m-n)(m+n)$
= $(m-2n)(m-n)(m+n)(m+2n)(m+3n)$
for n = 0 the value is $m^5$ and 33 is not a $5^{th}$ power
for n not zero above 5 values are differnt and 33 is product of atmost 4 mumbers $( (- 1) * 1 * (-3) * 11$ and hence product
cannot be 33

 

[anemone]

Spoiler

First let us factor $m^5+3m^4n-5m^3n^2-15m^2n^3+4mn^4+12n^5$
$m^5+3m^4n-5m^3n^2-15m^2n^3+4mn^4+12n^5$
= $m^4(m+3n) - 5m^2n^2(m+3n) + 4n^4(m+3n)$
= $(m+3n)(m^4-5m^2n^2+4n^4)$
= $(m+3n)(m^2-4n^2)(m^2-n^2)$
= $(m+3n)(m-2n)(m+2n)(m-n)(m+n)$
= $(m-2n)(m-n)(m+n)(m+2n)(m+3n)$
for n = 0 the value is $m^5$ and 33 is not a $5^{th}$ power
for n not zero above 5 values are differnt and 33 is product of atmost 4 mumbers $( (- 1) * 1 * (-3) * 11$ and hence product
cannot be 33

Very well done, kaliprasad!(Cool)