QUESTION 3 [5] \& 3.1. Write the first four terms of the expansion, \( (2 x+3)^{8} \)

To find the first four terms of the expansion \( (2x + 3)^8 \), we can use the Binomial Theorem, which states that \( (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \). Here, \( a = 2x \), \( b = 3 \), and \( n = 8 \). The first four terms are calculated as follows: 1. For \( k = 0 \): \[ \binom{8}{0} (2x)^8 (3)^0 = 1 \cdot 256x^8 \cdot 1 = 256x^8 \] 2. For \( k = 1 \): \[ \binom{8}{1} (2x)^7 (3)^1 = 8 \cdot 128x^7 \cdot 3 = 3072x^7 \] 3. For \( k = 2 \): \[ \binom{8}{2} (2x)^6 (3)^2 = 28 \cdot 64x^6 \cdot 9 = 16128x^6 \] 4. For \( k = 3 \): \[ \binom{8}{3} (2x)^5 (3)^3 = 56 \cdot 32x^5 \cdot 27 = 48384x^5 \] Thus, the first four terms of the expansion are: \[ 256x^8 + 3072x^7 + 16128x^6 + 48384x^5 \]