Standard product of two binomials

3

1. Find the first three terms in the expansion of \(( 3 - 2 x ) ^ { 5 }\) in ascending powers of \(x\).
2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 4 + x ) ^ { 2 } ( 3 - 2 x ) ^ { 5 }\).

2

1. Find the first three terms in the expansion, in ascending powers of \(x\), of \(( 2 + 3 x ) ^ { 4 }\).
2. Find the first three terms in the expansion, in ascending powers of \(x\), of \(( 1 - 2 x ) ^ { 5 }\).
3. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 2 + 3 x ) ^ { 4 } ( 1 - 2 x ) ^ { 5 }\).

1

1. Find the first three terms in the expansion, in ascending powers of \(x\), of \(( 1 + x ) ^ { 5 }\).
2. Find the first three terms in the expansion, in ascending powers of \(x\), of \(( 1 - 2 x ) ^ { 6 }\).
3. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 + x ) ^ { 5 } ( 1 - 2 x ) ^ { 6 }\).

3

1. Find the first three terms in ascending powers of \(x\) of the expansion of \(( 1 + 2 x ) ^ { 5 }\).
2. Find the first three terms in ascending powers of \(x\) of the expansion of \(( 1 - 3 x ) ^ { 4 }\).
3. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 + 2 x ) ^ { 5 } ( 1 - 3 x ) ^ { 4 }\).

3

1. Find the first 3 terms in the expansion, in ascending powers of \(x\), of \(\left( 2 + x ^ { 2 } \right) ^ { 5 }\).
2. Hence find the coefficient of \(x ^ { 4 }\) in the expansion of \(\left( 1 + x ^ { 2 } \right) ^ { 2 } \left( 2 + x ^ { 2 } \right) ^ { 5 }\).

8

1. Find the coefficient of \(x ^ { 8 }\) in the expansion of \(\left( x + 3 x ^ { 2 } \right) ^ { 4 }\).
2. Find the coefficient of \(x ^ { 8 }\) in the expansion of \(\left( x + 3 x ^ { 2 } \right) ^ { 5 }\).
3. Hence find the coefficient of \(x ^ { 8 }\) in the expansion of \(\left[ 1 + \left( x + 3 x ^ { 2 } \right) \right] ^ { 5 }\).

3

1. Find the first three terms of the expansion, in ascending powers of \(x\), of \(( 1 - 2 x ) ^ { 12 }\).
2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of $$( 1 + 3 x ) ( 1 - 2 x ) ^ { 12 } .$$

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

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

1. (a) Expand \(\left( 1 + \frac { 3 } { x } \right) ^ { 2 }\) simplifying each term.
   (b) Use the binomial expansion to find, in ascending powers of \(x\), the first four terms in the expansion of

$$\left( 1 + \frac { 3 } { 4 } x \right) ^ { 6 }$$ simplifying each term.
(c) Hence find the coefficient of \(x\) in the expansion of $$\left( 1 + \frac { 3 } { x } \right) ^ { 2 } \left( 1 + \frac { 3 } { 4 } x \right) ^ { 6 }$$

7

1. The first four terms of the binomial expansion of \(( 1 + 2 x ) ^ { 7 }\) in ascending powers of \(x\) are \(1 + a x + b x ^ { 2 } + c x ^ { 3 }\). Find the values of the integers \(a , b\) and \(c\).
2. Hence find the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( 1 - \frac { 1 } { 2 } x \right) ^ { 2 } ( 1 + 2 x ) ^ { 7 }\).

4

1. The expression \(( 1 - 2 x ) ^ { 4 }\) can be written in the form $$1 + p x + q x ^ { 2 } - 32 x ^ { 3 } + 16 x ^ { 4 }$$ By using the binomial expansion, or otherwise, find the values of the integers \(p\) and \(q\).
2. Find the coefficient of \(x\) in the expansion of \(( 2 + x ) ^ { 9 }\).
3. Find the coefficient of \(x\) in the expansion of \(( 1 - 2 x ) ^ { 4 } ( 2 + x ) ^ { 9 }\).

7

1. The expression \(( 1 - 2 x ) ^ { 5 }\) can be written in the form $$1 + p x + q x ^ { 2 } + r x ^ { 3 } + 80 x ^ { 4 } - 32 x ^ { 5 }$$ By using the binomial expansion, or otherwise, find the values of the coefficients \(p , q\) and \(r\).
2. Find the value of the coefficient of \(x ^ { 10 }\) in the expansion of \(( 1 - 2 x ) ^ { 5 } ( 2 + x ) ^ { 7 }\).
   [0pt] [5 marks]

1. i. Find the binomial expansion of \(( 2 + x ) ^ { 5 }\), simplifying the terms.
   ii. Hence find the coefficient of \(y ^ { 3 }\) in the expansion of \(\left( 2 + 3 y + y ^ { 2 } \right) ^ { 5 }\).
2. Let \(a = \log _ { 2 } x , b = \log _ { 2 } y\) and \(c = \log _ { 2 } z\).

Express \(\log _ { 2 } ( x y ) - \log _ { 2 } \left( \frac { z } { x ^ { 2 } } \right)\) in terms of \(a , b\) and \(c\).

7

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