2. (a) Use the binomial theorem to find all the terms of the expansion of $(2 + 3x)^4$ Give each term in its simplest form - Edexcel - A-Level Maths Pure - Question 4 - 2013 - Paper 4

To expand $(2 - 3x)^4$, we again apply the binomial theorem:

(a - b)^n = inom{n}{0}a^n(-b)^0 + inom{n}{1}a^{n-1}(-b)^1 + inom{n}{2}a^{n-2}(-b)^2 + ... + inom{n}{n}a^0(-b)^n

In this case:

• $a = 2$
• $b = 3x$
• $n = 4$

Applying the theorem, we have:

1. For $k = 0$: inom{4}{0} (2)^4 (-3x)^0 = 1 imes 16 = 16

2. For $k = 1$: inom{4}{1} (2)^{3} (-3x)^{1} = 4 imes 8 imes (-3x) = -96x

3. For $k = 2$: inom{4}{2} (2)^{2} (-3x)^{2} = 6 imes 4 imes 9x^2 = 216x^2

4. For $k = 3$: inom{4}{3} (2)^{1} (-3x)^{3} = 4 imes 2 imes (-27x^3) = -216x^3

5. For $k = 4$: inom{4}{4} (2)^{0} (-3x)^{4} = 1 imes 81x^4 = 81x^4

Thus, the expansion is: $16 - 96x + 216x^2 - 216x^3 + 81x^4$

Therefore, the terms in ascending order of $x$ are:

• $16$
• $-96x$
• $216x^2$
• $-216x^3$
• $81x^4$