Question

Expand (3x - 4y)⁴ using the binomial theorem.

a) 81x⁴ −256y⁴
b) 81x⁴ −256y³
c) 81x⁴ −256y²
d) 81x⁴ −256y

Answer 1

The correct expansion of (3x - 4y)⁴ using the binomial theorem is 81x⁴ - 432x³y + 864x²y² - 768xy³ + 256y⁴, which does not match any of the options given in the original question.

Step-by-step explanation:

To expand (3x - 4y)⁴ using the binomial theorem, we will apply the pattern which is:

(a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴.

Here, a = 3x and b = -4y. Now we will calculate each term:

1. a⁴ = (3x)⁴ = 81x⁴
2. 4a³b = 4(3x)³(-4y) = 4 × 27x³ × (-4y) = -432x³y
3. 6a²b² = 6(3x)²(-4y)² = 6 × 9x² × 16y² = 864x²y²
4. 4ab³ = 4(3x)(-4y)³ = 4 × 3x × (-64y³) = -768xy³
5. b⁴ = (-4y)⁴ = 256y⁴

Combining all of these, we get the expanded form:

81x⁴ - 432x³y + 864x²y² - 768xy³ + 256y⁴.

It's clear that the correct answer is not among the options a), b), c), or d) provided in the question.