Question

Expand the binomial (2x^2+y^2)^4

Answer 1

(2x² + y²)⁴

By Binomial Expansion:

⁴C₀(2x²)⁴(y²)⁰ + ⁴C₁(2x²)³(y²)¹ + ⁴C₂(2x²)²(y²)² + ⁴C₃(2x²)¹(y²)³ + ⁴C₄(2x²)⁰(y²)⁴

From Pascal's Triangle:

Note the C is the values from Combinations.

For power 4, the factors are:

1 , 4, 6, 4, 1

1*(2x²)⁴(y²)⁰ + 4*(2x²)³(y²)¹ + 6*(2x²)²(y²)² + 4*(2x²)¹(y²)³ + 1*(2x²)⁰(y²)⁴

Note if it raised to power zero = 1.

(2⁴x² ˣ ⁴) + 4*2³x² ˣ ³y² ˣ ¹ + 6*2²x² ˣ ² y² ˣ ² + 4*2x² ˣ ¹y² ˣ ³ + y² ˣ ⁴

16x⁸ + 32x⁶y² + 24x⁴y⁴ + 8x²y⁶ + y⁸

I hope this helped.

Answer 2

Hello,

(u+v)^4= u^4+4^3v+6u²v²+4uv^3+v^4

with u=2x² and v=y²

(2x²+y²)^4=
16x^8 + 32x^6*y² +24x^4y^4 +8x²y^6 + y^8