The polynomial expression $m^5+3m^4n-5m^3n^2-15m^2n^3+4mn^4+12n^5$ is proven to be never equal to 33. The discussion emphasizes the importance of rigorous mathematical proof in establishing the inequality of polynomial expressions. The contributor, kaliprasad, received commendation for their clear and effective demonstration of this fact.
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anemone said:Prove that $m^5+3m^4n-5m^3n^2-15m^2n^3+4mn^4+12n^5$ is never equal to 33.
kaliprasad said: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 different and 33 is product of atmost 4 mumbers $( (- 1) * 1 * (-3) * 11$ and hence product
cannot be 33