Binomial times quadratic coefficient

1

1. Expand \(( 1 + 3 x ) ^ { 6 }\) in ascending powers of \(x\) up to, and including, the term in \(x ^ { 2 }\).
2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 - 7 x + x ^ { 2 } \right) ( 1 + 3 x ) ^ { 6 }\).

1. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of

$$\left( 1 - \frac { 1 } { 2 } x \right) ^ { 10 }$$ giving each term in its simplest form.
(b) Hence find the coefficient of \(x ^ { 3 }\) in the expansion of $$\left( 3 + 5 x - 2 x ^ { 2 } \right) \left( 1 - \frac { 1 } { 2 } x \right) ^ { 10 }$$

3

1. Find and simplify the first four terms in the binomial expansion of \(\left( 1 + \frac { 1 } { 2 } x \right) ^ { 10 }\) in ascending powers of \(x\).
2. Hence find the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( 3 + 4 x + 2 x ^ { 2 } \right) \left( 1 + \frac { 1 } { 2 } x \right) ^ { 10 }\).