Given $$ f(x) = (2 + 3x)^3, \, |x| < \frac{2}{3} $$ find the binomial expansion of f(x), in ascending powers of x, up to and including the term in x^3 - Edexcel - A-Level Maths Pure - Question 13 - 2013 - Paper 1

To find the binomial expansion of the function $f(x) = (2 + 3x)^3$, we apply the Binomial Theorem, which states:

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

In this case:

• Let ( a = 2 )
• Let ( b = 3x )
• Let ( n = 3 )

Now, we can expand:

$$(2 + 3x)^3 = \sum_{k=0}^{3} \binom{3}{k} (2)^{3-k} (3x)^{k}$$

Calculating each term for ( k = 0 ) to ( 3 ):

1. For ( k = 0 ): $\binom{3}{0} (2)^3 (3x)^0 = 1 \cdot 8 \cdot 1 = 8$

2. For ( k = 1 ): $\binom{3}{1} (2)^2 (3x)^1 = 3 \cdot 4 \cdot 3x = 36x$

3. For ( k = 2 ): $\binom{3}{2} (2)^1 (3x)^2 = 3 \cdot 2 \cdot 9x^2 = 54x^2$

4. For ( k = 3 ): $\binom{3}{3} (2)^0 (3x)^3 = 1 \cdot 1 \cdot 27x^3 = 27x^3$

Combining all these terms, we get:

$f(x) = 8 + 36x + 54x^2 + 27x^3$

Thus, the binomial expansion of ( f(x) ) is:

$f(x) = 8 + 36x + 54x^2 + 27x^3$