Product with reciprocal term binomial

1

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

2

1. Find the first 3 terms in the expansion of \(\left( 2 x - \frac { 3 } { x } \right) ^ { 5 }\) in descending powers of \(x\).
2. Hence find the coefficient of \(x\) in the expansion of \(\left( 1 + \frac { 2 } { x ^ { 2 } } \right) \left( 2 x - \frac { 3 } { x } \right) ^ { 5 }\).

2

1. Find the first three terms, in descending powers of \(x\), in the expansion of \(\left( x - \frac { 2 } { x } \right) ^ { 6 }\).
2. Find the coefficient of \(x ^ { 4 }\) in the expansion of \(\left( 1 + x ^ { 2 } \right) \left( x - \frac { 2 } { x } \right) ^ { 6 }\).

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

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

2 Find the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 + x ^ { 2 } \right) \left( \frac { x } { 2 } - \frac { 4 } { x } \right) ^ { 6 }\).

1

1. Find the coefficient of \(x\) in the expansion of \(\left( 2 x - \frac { 1 } { x } \right) ^ { 5 }\).
2. Hence find the coefficient of \(x\) in the expansion of \(\left( 1 + 3 x ^ { 2 } \right) \left( 2 x - \frac { 1 } { x } \right) ^ { 5 }\).

6. (a) Find, in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), the binomial expansion of $$\left( 1 + \frac { 1 } { 4 } x \right) ^ { 12 }$$ giving each term in its simplest form.
(b) Hence find the coefficient of \(x\) in the expansion of $$\left( 3 + \frac { 2 } { x } \right) ^ { 2 } \left( 1 + \frac { 1 } { 4 } x \right) ^ { 12 }$$

3. (a) Find the first 4 terms, in ascending powers of \(x\), in the binomial expansion of $$\left( 1 + \frac { x } { 4 } \right) ^ { 12 }$$ giving each coefficient in its simplest form.
(b) Find the term independent of \(x\) in the expansion of $$\left( \frac { x ^ { 2 } + 8 } { x ^ { 5 } } \right) \left( 1 + \frac { x } { 4 } \right) ^ { 12 }$$

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

$$\left( 2 - \frac { x } { 4 } \right) ^ { 10 }$$ giving each term in its simplest form.
(b) Hence find the constant term in the series expansion of $$\left( 3 - \frac { 1 } { x } \right) ^ { 2 } \left( 2 - \frac { x } { 4 } \right) ^ { 10 }$$

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

$$\left( 3 - \frac { 2 x } { 9 } \right) ^ { 8 }$$ giving each term in simplest form. $$f ( x ) = \left( \frac { x - 1 } { 2 x } \right) \left( 3 - \frac { 2 x } { 9 } \right) ^ { 8 }$$ (b) Find the coefficient of \(x ^ { 2 }\) in the series expansion of \(\mathrm { f } ( x )\), giving your answer as a simplified fraction.

1. Find, in simplest form, the coefficient of \(x ^ { 5 }\) in the expansion of

$$\left( 5 + 8 x ^ { 2 } \right) \left( 3 - \frac { 1 } { 2 } x \right) ^ { 6 }$$

8

1. Expand \(\left( 1 + \frac { 4 } { x } \right) ^ { 2 }\).
2. The first four terms of the binomial expansion of \(\left( 1 + \frac { x } { 4 } \right) ^ { 8 }\) in ascending powers of \(x\) are \(1 + a x + b x ^ { 2 } + c x ^ { 3 }\). Find the values of the constants \(a , b\) and \(c\).
3. Hence find the coefficient of \(x\) in the expansion of \(\left( 1 + \frac { 4 } { x } \right) ^ { 2 } \left( 1 + \frac { x } { 4 } \right) ^ { 8 }\).

In all questions you must show all stages of your working, justifying solutions and not relying solely on calculator technology.

1. Find the first four terms, in ascending powers of \(x\), of the binomial expansion of \(\left(1+\frac{x}{2}\right)^7\), giving each coefficient in exact simplified form. [3]
2. Hence determine the coefficient of \(x\) in the expansion of $$\left(1+\frac{2}{x}\right)^2\left(1+\frac{x}{2}\right)^7.$$ [3]