Expand \(( 1 + 2 x ) ^ { 6 }\), simplifying all the terms.
Hence find an expression for \(\mathrm { f } ( x ) = ( 1 + 2 x ) ^ { 6 } + ( 1 - 2 x ) ^ { 6 }\) in its simplest form.
Substituting \(x = 0.01\) into the first two terms of \(\mathrm { f } ( x )\) gives an approximate value, z for \(1.02 ^ { 6 } + 0.98 ^ { 6 }\). Find \(z\).
By considering the value of the third term, comment on the accuracy of \(z\) as an approximation for \(1.02 ^ { 6 } + 0.98 ^ { 6 }\).