Question

Fill in the missing coefficients in the expansion of the binomial (2x^2+y^2)^4

Answer 1

Given Expression
`(2x^2+y^2)^4`

To find: Expansion of expression

Consider,

`(2x^2+y^2)^4\\\\\implies((2x^2+y^2)^2)^2`

using identity,
`(a+b)^2=a^2+b^2+2ab` we get,

`\implies((2x^2)^2+(y^2)^2+2*(2x^2)*(y^2))^2`

`\implies(4x^4+y^4+4x^2y^2)^2` (using law of exponent,
`(x^a)^b=x^(ab)` )

Now using identinty,
`(x+y+z)^2=x^2+y^2+z^2+2xy+2yz+2xz` we get,

`\implies(4x^4)^2+(y^2)^2+(4x^2y^2)^2+2(4x^4)(y^2)+2(y^2)(4x^2y^2)+2(4x^2y^2)(4x^4)\\\\\implies16x^8+y^4+16x^4y^4+8x^4y^2+8x^2y^4+32x^8y^2`

Answer 2

The expansion of the binomial:
( 2 x² + y² ) ^4 =

`(2 x^(2) ) ^(4) + 4 * (2 x^(2) ) ^(3)* y^(2) + 6 * ( 2 x^(2) ) ^(2)*( y^(2)) ^(2) +4*2 x^(2) *( y^(2)) ^(3)+( y^(2)) ^(4)`=

`=16 x^(8)+32 x^(6) y^(2)+24 x^(4) y^(4) + 8 x^(2) y^(6)+ y^(8)`