f(x) = (3 + 2x)^3, |x| < \frac{1}{2} - Edexcel - A-Level Maths Pure - Question 3 - 2007 - Paper 8

To find the binomial expansion of ( f(x) = (3 + 2x)^3 ), we will use the binomial theorem, which states that:

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

In this case, let ( a = 3 ), ( b = 2x ), and ( n = 3 ). Therefore, the expansion is:

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

Calculating each term:

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

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

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

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

Now, summing these terms, we get:

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