Question

simplify x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+y^5

Answer 1

The given expression is

$x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+y^5=x^5+y^5+5x^4y+5xy^4+10x^3y^2+10x^2y^3\\ x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+y^5=x^5+y^5+5xy(x^3+y^3)+10x^2y^2(x+y)\\ x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+y^5=(x^3+y^3)(x^2+y^2)-x^2y^2(x+y)+5xy(x^3+y^3)+10x^2y^2(x+y)$

$x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+y^5=(x^3+y^3)(x^2+y^2+5xy)+9x^2y^2(x+y)\\ x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+y^5=(x+y)(x^2+y^2-xy)(x^2+y^2+5xy)+9x^2y^2(x+y)\\ x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+y^5=(x+y)[(x^2+y^2-xy)(x^2+y^2+5xy)+9x^2y^2]$

$x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+y^5=(x+y)[(x^2+y^2)^2+4xy(x^2+y^2)+4x^2y^2]\\ x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+y^5=(x+y)[(x^2+y^2+2xy)^2]\\ x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+y^5=(x+y)[(x+y)^4]\\ x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+y^5=(x+y)^5$

That is the simplified expression.