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

The expansion of the binomial:
( 2 x² + y² ) ^4 =
[tex](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} [/tex]=
[tex]=16 x^{8}+32 x^{6} y^{2}+24 x^{4} y^{4} + 8 x^{2} y^{6}+ y^{8} [/tex]

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aquialaska aquialaska · Answer 2 · 2019-03-19T12:07:21+01:00

Answer:

Given Expression [tex](2x^2+y^2)^4[/tex]

To find: Expansion of expression

Consider,

[tex](2x^2+y^2)^4\\\\\implies((2x^2+y^2)^2)^2[/tex]

using identity, [tex](a+b)^2=a^2+b^2+2ab[/tex] we get,

[tex]\implies((2x^2)^2+(y^2)^2+2\times(2x^2)\times(y^2))^2[/tex]

[tex]\implies(4x^4+y^4+4x^2y^2)^2[/tex] (using law of exponent, [tex](x^a)^b=x^{ab}[/tex] )

Now using identinty, [tex](x+y+z)^2=x^2+y^2+z^2+2xy+2yz+2xz[/tex] we get,

[tex]\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[/tex]