Страница 37 номер 2.2.61, ГДЗ по алгебре за 7, 8 и 9 класс к задачнику Ткачевой
Разложить на множители (2.2.60—2.2.62): 1) \(9x^2 - y^2 + 2y - 1\); 2) \(x^2 + y^2 + 2x - 2y - 2xy\); 3) \((a + 1)(a^2 - b^2) - (a + b)(a^2 - 1)\); 4) \((a + b)(a^3 - b^3) - (a - b)(a^3 + b^3)\); 5) \(b^4 - b^2(a^2 + 1) + a^2\); 6) \(b^3 + (a - 1)b + a\).
1) \[\begin{aligned} &9x^2 - y^2 + 2y - 1 = 9x^2 - (y^2 - 2y + 1) = {} \\ &= (3x)^2 - (y - 1)^2 = (3x - y + 1)(3x + y - 1); \end{aligned}\]
2) \[\begin{aligned} &x^2 + y^2 + 2x - 2y - 2xy = (x^2 - 2xy + y^2) + (2x - 2y) = {} \\ &= (x - y)^2 + 2(x - y) = (x - y)(x - y + 2); \end{aligned}\]
3) \[\begin{aligned} &(a + 1)(a^2 - b^2) - (a + b)(a^2 - 1) = {} \\ &= (a + 1)(a - b)(a + b) - (a + b)(a - 1)(a + 1) = {} \\ &= (a + 1)(a + b)((a - b) - (a - 1)) = {} \\ &= (a + 1)(a + b)(1 - b); \end{aligned}\]
4) \[\begin{aligned} &(a + b)(a^3 - b^3) - (a - b)(a^3 + b^3) = {} \\ &= (a + b)(a - b)(a^2 + ab + b^2) - (a - b)(a + b)(a^2 - ab + b^2) = {} \\ &= (a + b)(a - b)((a^2 + ab + b^2) - (a^2 - ab + b^2)) = {} \\ &= (a + b)(a - b) \cdot 2ab = 2ab(a + b)(a - b); \end{aligned}\]
5) \[\begin{aligned} &b^4 - b^2(a^2 + 1) + a^2 = b^4 - a^2b^2 - b^2 + a^2 = {} \\ &= b^2(b^2 - 1) - a^2(b^2 - 1) = (b^2 - 1)(b^2 - a^2) = {} \\ &= (b - 1)(b + 1)(b - a)(b + a); \end{aligned}\]
6) \[\begin{aligned} &b^3 + (a - 1)b + a = b^3 + ab - b + a = (b^3 - b) + (ab + a) = {} \\ &= b(b^2 - 1) + a(b + 1) = b(b - 1)(b + 1) + a(b + 1) = {} \\ &= (b + 1)(b(b - 1) + a) = (b + 1)(b^2 - b + a). \end{aligned}\]
Ответ: 1) \((3x - y + 1)(3x + y - 1)\); 2) \((x - y)(x - y + 2)\); 3) \((a + 1)(a + b)(1 - b)\); 4) \(2ab(a + b)(a - b)\); 5) \((b - 1)(b + 1)(b - a)(b + a)\); 6) \((b + 1)(b^2 - b + a)\).
Разложить на множители (2.2.60—2.2.62): 1) \(9x^2 - y^2 + 2y - 1\); 2) \(x^2 + y^2 + 2x - 2y - 2xy\); 3) \((a + 1)(a^2 - b^2) - (a + b)(a^2 - 1)\); 4) \((a + b)(a^3 - b^3) - (a - b)(a^3 + b^3)\); 5) \(b^4 - b^2(a^2 + 1) + a^2\); 6) \(b^3 + (a - 1)b + a\).
Пользуемся вынесением общего множителя, способом группировки и формулами сокращённого умножения.
1) \[\begin{aligned} &9x^2 - y^2 + 2y - 1 = 9x^2 - (y^2 - 2y + 1) = {} \\ &= (3x)^2 - (y - 1)^2 = (3x - y + 1)(3x + y - 1); \end{aligned}\]
2) \[\begin{aligned} &x^2 + y^2 + 2x - 2y - 2xy = (x^2 - 2xy + y^2) + (2x - 2y) = {} \\ &= (x - y)^2 + 2(x - y) = (x - y)(x - y + 2); \end{aligned}\]
3) \[\begin{aligned} &(a + 1)(a^2 - b^2) - (a + b)(a^2 - 1) = {} \\ &= (a + 1)(a - b)(a + b) - (a + b)(a - 1)(a + 1) = {} \\ &= (a + 1)(a + b)((a - b) - (a - 1)) = {} \\ &= (a + 1)(a + b)(1 - b); \end{aligned}\]
4) \[\begin{aligned} &(a + b)(a^3 - b^3) - (a - b)(a^3 + b^3) = {} \\ &= (a + b)(a - b)(a^2 + ab + b^2) - (a - b)(a + b)(a^2 - ab + b^2) = {} \\ &= (a + b)(a - b)((a^2 + ab + b^2) - (a^2 - ab + b^2)) = {} \\ &= (a + b)(a - b) \cdot 2ab = 2ab(a + b)(a - b); \end{aligned}\]
5) \[\begin{aligned} &b^4 - b^2(a^2 + 1) + a^2 = b^4 - a^2b^2 - b^2 + a^2 = {} \\ &= b^2(b^2 - 1) - a^2(b^2 - 1) = (b^2 - 1)(b^2 - a^2) = {} \\ &= (b - 1)(b + 1)(b - a)(b + a); \end{aligned}\]
6) \[\begin{aligned} &b^3 + (a - 1)b + a = b^3 + ab - b + a = (b^3 - b) + (ab + a) = {} \\ &= b(b^2 - 1) + a(b + 1) = b(b - 1)(b + 1) + a(b + 1) = {} \\ &= (b + 1)(b(b - 1) + a) = (b + 1)(b^2 - b + a). \end{aligned}\]
Ответ: 1) \((3x - y + 1)(3x + y - 1)\); 2) \((x - y)(x - y + 2)\); 3) \((a + 1)(a + b)(1 - b)\); 4) \(2ab(a + b)(a - b)\); 5) \((b - 1)(b + 1)(b - a)(b + a)\); 6) \((b + 1)(b^2 - b + a)\).
