Prove that for no integer n, n6 +3n5_5n4_15n3+4n2+12n+3 is a perfect square.

Hello StudentPlease fidn the solnn6 +3n5_5n4_15n3+4n2+12n+3 == n5(n+3) –5n3(n+3)+4n(n+3) + 3=(n+3)[n5 –5n3+4n] + 3=n(n+3)[n4-5n2+4] + 3=n(n+3)(n2-4)(n2-1) + 3=(n-2)(n-1)n(n+1)(n+2)(n+3) + 3So this can not be perfect squareThansk

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Grade 1010 grade maths

Prove that for no integer n, n⁶+3n^5_5n^4_15n³+4n²+12n+3 is a perfect square.

Ahalya

11 Years agoGrade 10

1 Answer

Nishant Vora

Approved Tutor Answer11 Years ago

Hello Student
Please fidn the soln

n6 +3n5_5n4_15n3+4n2+12n+3 =
= n⁵(n+3) –5n³(n+3)+4n(n+3) + 3
=(n+3)[n⁵ –5n³+4n] + 3
=n(n+3)[n⁴-5n²+4] + 3
=n(n+3)(n²-4)(n²-1) + 3
=(n-2)(n-1)n(n+1)(n+2)(n+3) + 3

So this can not be perfect square

Thansk