Factors of x³ − 1 + y³ + 3xy
The factors of \[ x^3-1+y^3+3xy \] are
(a) \((x-1+y)(x^2+1+y^2+x+y-xy)\)
(b) \((x+y+1)(x^2+y^2+1-xy-x-y)\)
(c) \((x-1+y)(x^2-1-y^2+x+y+xy)\)
(d) \(3(x+y-1)(x^2+y^2-1)\)
Solution
\[ x^3+y^3+(-1)^3-3(x)(y)(-1) \]
Using identity:
\[ a^3+b^3+c^3-3abc =(a+b+c)(a^2+b^2+c^2-ab-bc-ca) \]
\[ = (x+y-1)(x^2+y^2+1-xy+x+y) \]
Therefore,
\[ \boxed{(a)\ (x+y-1)(x^2+y^2+1-xy+x+y)} \]