Standard binomial expansion

Find expansion of

2. (a) Use the binomial theorem to find all the terms of the expansion of $$( 2 + 3 x ) ^ { 4 }$$ Give each term in its simplest form.
(b) Write down the expansion of $$( 2 - 3 x ) ^ { 4 }$$ in ascending powers of \(x\), giving each term in its simplest form.

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

$$\left( 3 - \frac { 1 } { 3 } x \right) ^ { 5 }$$ giving each term in its simplest form. \includegraphics[max width=\textwidth, alt={}, center]{752efc6c-8d0e-46a6-b75d-5125956969d8-03_104_107_2631_1774}

7 Find and simplify the binomial expansion of \(( 3 x - 2 ) ^ { 4 }\).

1 Find and simplify the binomial expansion of \(( 3 x - 2 ) ^ { 4 }\).

9 Expand \(\left( 1 + \frac { 1 } { 2 } x \right) ^ { 4 }\), simplifying the coefficients.

18 Find the binomial expansion of \(( 2 + x ) ^ { 4 }\), writing each term as simply as possible.

1 Find the binomial expansion of \(( 3 x - 2 ) ^ { 4 }\).

6 Find the binomial expansion of \(( 1 - 5 x ) ^ { 4 }\), expressing the terms as simply as possible.

4 Using factorials, show that \(\binom { n } { r - 1 } + \binom { n } { r } = \binom { n + 1 } { r }\). Hence prove by mathematical induction that $$( a + x ) ^ { n } = \binom { n } { 0 } a ^ { n } + \binom { n } { 1 } a ^ { n - 1 } x + \ldots + \binom { n } { r } a ^ { n - r } x ^ { r } + \ldots + \binom { n } { n } x ^ { n }$$ for every positive integer \(n\).

2 Find the binomial expansion of \(( 3 - 2 x ) ^ { 3 }\).

5

1. 1. Describe the geometrical transformation that maps the graph of \(y = \left( 1 + \frac { x } { 3 } \right) ^ { 6 }\) onto the graph of \(y = ( 1 + 2 x ) ^ { 6 }\).
   2. The curve \(y = \left( 1 + \frac { x } { 3 } \right) ^ { 6 }\) is translated by the vector \(\left[ \begin{array} { l } 3 \\ 0 \end{array} \right]\) to give the curve \(y = \mathrm { g } ( x )\). Find an expression for \(\mathrm { g } ( x )\), simplifying your answer.
2. The first four terms in the binomial expansion of \(\left( 1 + \frac { x } { 3 } \right) ^ { 6 }\) are \(1 + a x + b x ^ { 2 } + c x ^ { 3 }\). Find the values of the constants \(a , b\) and \(c\), giving your answers in their simplest form.

10

1. Write down and simplify the first four terms in the expansion of \(( x + y ) ^ { 7 }\).
   Give your answer in ascending powers of \(x\).
2. Given that the terms in \(x ^ { 2 } y ^ { 5 }\) and \(x ^ { 3 } y ^ { 4 }\) in this expansion are equal, find the value of \(\frac { x } { y }\).
3. A hospital consultant has seven appointments every day. The number of these appointments which start late on a randomly chosen day is denoted by \(L\).
   The variable \(L\) is modelled by the distribution \(\mathrm { B } \left( 7 , \frac { 3 } { 8 } \right)\). Show that, in this model, the hospital consultant is equally likely to have two appointments start late or three appointments start late.

Find the first three terms, in ascending powers of \(x\), of the binomial expansion of \((3 + 2x)^5\), giving each term in its simplest form. [4]

Find the binomial expansion of \((2 + x)^4\), writing each term as simply as possible. [4]

Expand \((1 + \frac{1}{2}x)^4\), simplifying the coefficients. [4]

Expand \((1-2x)^4\) in ascending powers of \(x\), simplifying the coefficients. [5]

Expand \((3 - 2x)^4\) in ascending powers of \(x\) and simplify each coefficient. [4]

Expand \((x - 2y)^5\). [3]