Question

Expand the binomial (3x2 + 2y3)4. The coefficient of the third term in the expansion of the binomial (3x2 + 2y3)4 is

Answer 1

`(a+b)^n=\displaystyle\sum_(k=0)^n\binom nka^(n-k)b^k`

where
`\dbinom nk=(n!)/(k!(n-k)!)`. The third term of the expansion occurs when
`k=2`.

So the third term of the expansion of
`(3x^2+2y^3)^4` is


`\dbinom42(3x^2)^(4-2)(2y^3)^2=6(3^2x^4)(4y^6)=216x^4y^6`

The coefficient is 216.

Answer 2

Answer;

-The coefficient of the third term in the expansion of the binomial

(3x^2 +2y^3)^4 is 216.

Solution;

Expand the binomial (3x² + 2y³)^4

Coefficients for expansion to power 4 are; 1, 4, 6, 4, 1

Thus; (3x² + 2y³)^4

=1(3x²)^0(2y³)^4 + 4 (3x²)^1(2y³)³ + 6 (3x²)² (2y³)² + 4 (3x²)³(2y³) + 1(3x²)^4(2y³)^0

=81 x^8 +216 x^6 y^3 +216 x^4 y^6 +96 x^2 y^9 +16 y^12

The third term is 216 x^4 y^6.

The coefficient is 216.