Use the binomial theorem to expand (x2-2y)4

Sure, let's use the binomial theorem to expand (x² - 2y)⁴. The binomial theorem states that (a + b)ⁿ = Σ (n choose k) * a^(n-k) * b^k, where the sum is from k = 0 to n.

Here, a = x², b = -2y, and n = 4.

Let's calculate each term:

• For k = 0: (4 choose 0) * (x²)^(4-0) * (-2y)^0 = 1 * x⁸ * 1 = x⁸
• For k = 1: (4 choose 1) * (x²)^(4-1) * (-2y)^1 = 4 * x⁶ * -2y = -8x⁶y
• For k = 2: (4 choose 2) * (x²)^(4-2) * (-2y)^2 = 6 * x⁴ * 4y² = 24x⁴y²
• For k = 3: (4 choose 3) * (x²)^(4-3) * (-2y)^3 = 4 * x² * -8y³ = -32x²y³
• For k = 4: (4 choose 4) * (x²)^(4-4) * (-2y)^4 = 1 * 1 * 16y⁴ = 16y⁴

So, (x² - 2y)⁴ = x⁸ - 8x⁶y + 24x⁴y² - 32x²y³ + 16y⁴.