Question

How do you expand `(2y^2+x)^7`?

Answer 1

`(2y^2+x)^7=128y^14+448y^12x+672y^10x^2+560y^8x^3+280y^6x^4+105y^4x^5+14y^2x^6+x^7`

Explanation:

We can expand using this general formula:

`(a+b)^n=(C_(n,0))a^nb^0+(C_(n,1))a^(n-1)b^1+...+(C_(n,n))a^0b^n`

And so we have:

`((color(white)(0),C,a,b,"Term"),(color(red)0,1,128y^14,1,128y^14),(color(red)1,7,64y^12,x,448y^12x),(color(red)2,21,32y^10,x^2,672y^10x^2),(color(red)3,35,16y^8,x^3,560y^8x^3),(color(red)4,35,8y^6,x^4,280y^6x^4),(color(red)5,21,4y^4,x^5,105y^4x^5),(color(red)6,7,2y^2,x^6,14y^2x^6),(color(red)7,1,1,x^7,x^7))`

And now we add the terms:

`(2y^2+x)^7=128y^14+448y^12x+672y^10x^2+560y^8x^3+280y^6x^4+105y^4x^5+14y^2x^6+x^7`