Prove that for no integer n,n^6+3n^5-5n^4-15n^3+4n^2+12n+3 is a perfect square.
n^x means "n tothe power of x"

Asked by satishsingh881 | 22nd Sep, 2014, 11:03: PM

Expert Answer:

n to the power of 6 plus 3 n to the power of 5 minus 5 n to the power of 4 minus 15 n cubed plus 4 n squared plus 12 n plus 3 equals n to the power of 5 left parenthesis n plus 3 right parenthesis minus 5 n cubed left parenthesis n plus 3 right parenthesis plus 4 n left parenthesis n plus 3 right parenthesis plus 3 equals left parenthesis n plus 3 right parenthesis left parenthesis n to the power of 5 minus 5 n cubed plus 4 n right parenthesis plus 3 equals left parenthesis n plus 3 right parenthesis n left parenthesis n to the power of 4 minus 5 n squared plus 4 right parenthesis plus 3 equals left parenthesis n plus 3 right parenthesis n left parenthesis n to the power of 4 minus 4 n squared minus n squared plus 4 right parenthesis plus 3 space space space left square bracket F a c t o r i z i n g right square bracket equals left parenthesis n plus 3 right parenthesis n left parenthesis n squared left parenthesis n squared minus 4 right parenthesis minus 1 left parenthesis n squared minus 4 right parenthesis right parenthesis plus 3 equals left parenthesis n plus 3 right parenthesis n left parenthesis n squared minus 1 right parenthesis left parenthesis n squared minus 4 right parenthesis plus 3 equals left parenthesis n plus 3 right parenthesis n left parenthesis n plus 1 right parenthesis left parenthesis n minus 1 right parenthesis left parenthesis n plus 2 right parenthesis left parenthesis n minus 2 right parenthesis plus 3 equals left parenthesis n minus 2 right parenthesis left parenthesis n minus 1 right parenthesis n left parenthesis n plus 1 right parenthesis left parenthesis n plus 2 right parenthesis left parenthesis n plus 3 right parenthesis plus 3 S i n c e space t h e space a b o v e space f o r m space c a n n o t space b e space w r i t t e n space f o r space a n y space f o r m space o f space a space p e r f e c t space s q u a r e comma space t h e r e f o r e space i t space i s space n o t space a space p e r f e c t space s q u a r e.

Answered by Prasenjit Paul | 23rd Sep, 2014, 11:29: AM