Binomial times linear coefficient

1 Find the coefficient of \(x ^ { 2 }\) in the expansion of $$( 2 - 5 x ) ( 1 + 3 x ) ^ { 10 }$$

4 Find the coefficient of \(x ^ { 3 }\) in the expansion of

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

3

1. Find the coefficients of \(x ^ { 2 }\) and \(x ^ { 3 }\) in the expansion of \(( 2 - x ) ^ { 6 }\).
2. Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 3 x + 1 ) ( 2 - x ) ^ { 6 }\).

1

1. Find the first 3 terms in the expansion, in ascending powers of \(x\), of \(\left( 1 - 2 x ^ { 2 } \right) ^ { 8 }\).
2. Find the coefficient of \(x ^ { 4 }\) in the expansion of \(\left( 2 - x ^ { 2 } \right) \left( 1 - 2 x ^ { 2 } \right) ^ { 8 }\).

4

1. Find the first 3 terms in the expansion of \(\left( 2 x - x ^ { 2 } \right) ^ { 6 }\) in ascending powers of \(x\).
2. Hence find the coefficient of \(x ^ { 8 }\) in the expansion of \(( 2 + x ) \left( 2 x - x ^ { 2 } \right) ^ { 6 }\).

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

$$\left( 1 - \frac { 1 } { 6 } x \right) ^ { 9 }$$ giving each term in simplest form.
(b) Hence find the coefficient of \(x ^ { 3 }\) in the expansion of $$( 10 x + 3 ) \left( 1 - \frac { 1 } { 6 } x \right) ^ { 9 }$$ giving the answer in simplest form.

3. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 2 - 3 x ) ^ { 6 }$$ giving each term in its simplest form.
(b) Hence, or otherwise, find the first 3 terms, in ascending powers of \(x\), of the expansion of $$\left( 1 + \frac { x } { 2 } \right) ( 2 - 3 x ) ^ { 6 }$$

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

$$( 2 - 9 x ) ^ { 4 }$$ giving each term in its simplest form. $$f ( x ) = ( 1 + k x ) ( 2 - 9 x ) ^ { 4 } , \text { where } k \text { is a constant }$$ The expansion, in ascending powers of \(x\), of \(\mathrm { f } ( x )\) up to and including the term in \(x ^ { 2 }\) is $$A - 232 x + B x ^ { 2 }$$ where \(A\) and \(B\) are constants.
(b) Write down the value of \(A\).
(c) Find the value of \(k\).
(d) Hence find the value of \(B\).

1

1. Find and simplify the first three terms, in ascending powers of \(x\), in the binomial expansion of \(( 1 + 2 x ) ^ { 7 }\).
2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 2 - 5 x ) ( 1 + 2 x ) ^ { 7 }\).

1. Find the coefficient of \(x ^ { 2 }\) in the expansion of

$$( 1 + x ) ( 1 - x ) ^ { 6 }$$