4. (a) Find the first 4 terms, in ascending powers of x, of the binomial expansion of (1 + a x )^{3}, where a is a constant - Edexcel - A-Level Maths Pure - Question 6 - 2010 - Paper 3

To find the first four terms of the binomial expansion of the expression (1 + a x)^{3}, we can make use of the Binomial Theorem, which states:

$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}$

In our case, let a = 1, b = ax, and n = 3:

1. The first term, when k = 0: $\binom{3}{0} (1)^{3} (a x)^{0} = 1$

2. The second term, when k = 1: $\binom{3}{1} (1)^{2} (a x)^{1} = 3 (a x) = 3a x$

3. The third term, when k = 2: $\binom{3}{2} (1)^{1} (a x)^{2} = 3 (a^{2} x^{2}) = 3a^{2} x^{2}$

4. The fourth term, when k = 3: $\binom{3}{3} (1)^{0} (a x)^{3} = (a^{3} x^{3}) = a^{3} x^{3}$

Therefore, the first four terms of the expansion are:

$1, \, 3a x, \, 3a^{2} x^{2}, \, a^{3} x^{3}$