Final answer:
To expand (3x-4)⁴ using Pascal's Triangle, we combine the coefficients 1, 4, 6, 4, 1 with each term of the binomial raised to a power. The expanded form is 81x⁴ - 1296x³ + 864x² + 768x + 256.
Step-by-step explanation:
The binomial theorem states that (a + b)n = C(n, 0)anb0 + C(n, 1)an-1b1 + C(n, 2)an-2b2 + ... + C(n, n)a0bn, where C(n, k) represents the binomial coefficient.
To expand the expression (3x-4)⁴ using Pascal's Triangle, we first need to identify the coefficients for the expansion of a binomial raised to the fourth power, which are 1, 4, 6, 4, and 1. Now, we apply the binomial theorem: (a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴. Let's substitute a = 3x and b = -4 into the formula:
- First term: (3x)⁴ = 81x⁴
- Second term: 4·(3x)³·(-4) = -4· 81x³ × 4 = -1296x³
- Third term: 6·(3x)²·(-4)² = 6· 9x² × 16 = 864x²
- Fourth term: 4·(3x)·(-4)³ = -4· 3x × -64 = 768x
- Fifth term: (-4)⁴ = 256
Combining these terms, the expanded form of (3x-4)⁴ is 81x⁴ - 1296x³ + 864x² + 768x + 256.