Product of expansions

3 Find the term independent of \(x\) in each of the following expansions.

1. \(\left( 3 x + \frac { 2 } { x ^ { 2 } } \right) ^ { 6 }\)
2. \(\left( 3 x + \frac { 2 } { x ^ { 2 } } \right) ^ { 6 } \left( 1 - x ^ { 3 } \right)\)

4 Find the term that is independent of \(x\) in the expansion of

1. \(\left( x - \frac { 2 } { x } \right) ^ { 6 }\),
2. \(\left( 2 + \frac { 3 } { x ^ { 2 } } \right) \left( x - \frac { 2 } { x } \right) ^ { 6 }\).

1. (a) Find, in terms of \(a\), the first 3 terms, in ascending powers of \(x\), of the binomial expansion of

$$( 2 + a x ) ^ { 6 }$$ where \(a\) is a non-zero constant. Give each term in simplest form. $$f ( x ) = \left( 3 + \frac { 1 } { x } \right) ^ { 2 } ( 2 + a x ) ^ { 6 }$$ Given that the constant term in the expansion of \(\mathrm { f } ( x )\) is 576
(b) find the value of \(a\).

Find the term independent of \(x\) in the expansion of each of the following:

1. \(\left(x + \frac{3}{x^2}\right)^6\) [2]
2. \((4x^3 - 5)\left(x + \frac{3}{x^2}\right)^6\) [4]