Question

Use Pascal's Triangle to expand (2z-4y²)⁴. Express your answer in simplest form.

Answer 1

To expand (2z-4y²)⁴ using Pascal's Triangle, we utilize the coefficients from the 4th row (1, 4, 6, 4, 1) and apply them in the binomial theorem to get the expanded form 16z⁴ - 128z³y² + 384z²y⁴ - 256zy⁶ + 256y⁸.

Step-by-step explanation:

To expand the binomial (2z-4y²)⁴ using Pascal's Triangle, we first look at the 4th row of Pascal's Triangle, which corresponds to the coefficients needed for the expansion of a binomial raised to the fourth power. The coefficients from Pascal's Triangle are 1, 4, 6, 4, 1.

The binomial theorem states that (a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴. Applying this to our binomial, where a = 2z and b = -4y², gives us:

1. (2z)⁴ = 16z⁴
2. 4 × (2z)³ × (-4y²) = -128z³y²
3. 6 × (2z)² × (-4y²)² = 384z²y⁴
4. 4 × (2z) × (-4y²)³ = -256zy⁶
5. (-4y²)⁴ = 256y⁸

Combining these terms gives us the final expanded form of the binomial:

16z⁴ - 128z³y² + 384z²y⁴ - 256zy⁶ + 256y⁸