express \(( 1 + x ) ^ { 3 }\) in ascending powers of \(x\);
express \(( 1 + x ) ^ { 4 }\) in ascending powers of \(x\).
Hence, or otherwise:
express \(( 1 + 4 x ) ^ { 3 }\) in ascending powers of \(x\);
express \(( 1 + 3 x ) ^ { 4 }\) in ascending powers of \(x\).
Show that the expansion of
$$( 1 + 3 x ) ^ { 4 } - ( 1 + 4 x ) ^ { 3 }$$
can be written in the form
$$p x ^ { 2 } + q x ^ { 3 } + r x ^ { 4 }$$
where \(p , q\) and \(r\) are integers.