Question

Find \[\left( x + 1 \right)^6 + \left( x - 1 \right)^6\] . Hence, or otherwise evaluate \[\left( \sqrt{2} + 1 \right)^6 + \sqrt{2} - 1^6\] .

 

 

Answer 1

The expression  \[\left( x + 1 \right)^6 + \left( x - 1 \right)^6\]  can be written as \[(x + 1 )^6 + (x - 1 )^6 \]
\[ = 2[ ^{6}{}{C}_0 x^6 +^{6}{}{C}_2 x^4 + ^{6}{}{C}_4 x^2 + ^{6}{}{C}_6 x^0 ]\]
\[ = 2[ x^6 + 15 x^4 + 15 x^2 + 1]\]

By taking \[x = \sqrt{2}\] , we get:

\[(\sqrt{2} + 1 )^6 + (\sqrt{2} - 1 )^6 = 2[(\sqrt{2} )^6 + 15(\sqrt{2} )^4 + 15(\sqrt{2} )^2 + 1]\]

\[ = 2[8 + 15 \times 4 + 15 \times 2 + 1]\]
\[ = 2 \times (8 + 60 + 30 + 1)\]
\[ = 198\]