MATH SOLVE

4 months ago

Q:
# Complete the following statements in relation to the third term in the expansion of the binomial (2x+y^2)^5

Accepted Solution

A:

We want to expand (2x + y²)⁵.

From Pascal's Triangle (shown below), the coefficients of the expansion are

1,5,10,10,5,1.

n=0: 1

n=1: 1 1

n=2: 1 2 1

n=3: 1 3 3 1

n=4: 1 4 6 4 1

n=5: 1 5 10 10 5 1

Therefore the third term in the expansion is

10*(2x)³*(y²)² = 10*(8x³)*(y⁴) = 80x³y⁴

Alternatively, the third term is

₅C₃ (2x)³(y²)² = 10*(8x³)*(y⁴) = 80x³y⁴

Answer: 80x³y⁴

From Pascal's Triangle (shown below), the coefficients of the expansion are

1,5,10,10,5,1.

n=0: 1

n=1: 1 1

n=2: 1 2 1

n=3: 1 3 3 1

n=4: 1 4 6 4 1

n=5: 1 5 10 10 5 1

Therefore the third term in the expansion is

10*(2x)³*(y²)² = 10*(8x³)*(y⁴) = 80x³y⁴

Alternatively, the third term is

₅C₃ (2x)³(y²)² = 10*(8x³)*(y⁴) = 80x³y⁴

Answer: 80x³y⁴