Find (x + 1)^6 + (x - 1)^6. Hence or otherwise evaluate ( 2 + 1)^6 + ( 2 - 1)^6

Find (x + 1)^6 + (x - 1)^6. Hence or otherwise evaluate ( 2 + 1)^…

Question

Find $(x + 1)^6 + (x - 1)^6$. Hence or otherwise evaluate ${(\sqrt 2 + 1)^6} + {(\sqrt 2 - 1)^6}$

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Answer

$(x + 1)^6 + (x - 1)^6 = {[^6}{C_0}{x^6}{ + ^6}{C_1}{x^5}{ + ^6}{C_2}{x^4}{ + ^6}{C_3}{x^3}{ + ^6}{C_4}{x^2}{ + ^6}{C_5}x{ + ^6}{C_6}]$
$ + {[^6}{C_0}{x^6}{ + ^6}{C_1}{x^5}( - 1){ + ^6}{C_2}{x^4}{( - 1)^2}{ + ^6}{C_3}{x^3}{( - 1)^3}$${ + ^6}{C_4}{x^2}{( - 1)^4}{ + ^6}{C_5}x{( - 1)^5}{ + ^6}{C_6}{( - 1)^6}]$
$= [x^6 + 6x^5 + 15x^4 + 20x^3 + 15x$^2$+ 6x + 1] + [x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1]$
$= 2x^6 + 30x^4 + 30x^2 + 2$
$= 2(x^6 + 15x^4 + 15x^2 + 1)$
Putting $x = \sqrt 2 $
${(\sqrt 2 + 1)^6} + {(\sqrt 2 - 1)^6} = 2[{(\sqrt 2 )^6} + 15{(\sqrt 2 )^4} + 15{(\sqrt 2 )^2} + 1]$
$ = 2\left[ {8 + 15 \times 4 + 15 \times 2 + 1} \right]$
$= 2 [8 + 60 + 30 + 1]$
$ = 2 \times 99 = 198$

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