Разложите на множители:
а) $a^3 + a^2 - a^2b - ab^2$;
б) $xy^2 + x^2y - x^3 - y^3$;
в) $n^4 + an^3 - n - a$;
г) $a^3 - 3a^2b + 3ab^2 - b^3$.
$a^3 + a^2 - a^2b - ab^2 = (a^3 + b^3) - (a^2b + ab^2) = (a + b)(a^2 - ab + b^2) - ab(a + b) = (a + b)(a^2 - ab + b^2 - ab) = (a + b)(a^2 - 2ab + b^2) = (a + b)(a - b)^2$
$xy^2 + x^2y - x^3 - y^3 = (xy^2 + x^2y) - (x^3 + y^3) = xy(x + y) - (x + y)(x^2 - xy + y^2) = (x + y)(xy - (x^2 - xy + y^2)) = (x + y)(xy - x^2 + xy - y^2) = (x + y)(-x^2 + 2xy - y^2) = -(x + y)(x^2 - 2xy + y^2) = -(x + y)(x - y)^2$
$n^4 + an^3 - n - a = (n^4 + an^3) - (n + a) = n^3(n + a) - (n + a) = (n + a)(n^3 - 1) = (n + a)(n - 1)(n^2 + n + 1)$
$a^3 - 3a^2b + 3ab^2 - b^3 = (a^3 - b^3) - (3a^2b - 3ab^2) = (a - b)(a^2 + ab + b^2) - 3ab(a - b) = (a - b)(a^2 + ab + b^2 - 3ab) = (a - b)(a^2 - 2ab + b^2) = (a - b)(a - b)^2 = (a - b)^3$
Пожауйста, оцените решение