Страница 35 номер 2.2.48, ГДЗ по алгебре за 7, 8 и 9 класс к задачнику Ткачевой
Разложить на множители: 1) \(x^6 - 0{,}04y^4\); 2) \(\frac{x^8}{9} - y^{10}\); 3) \(121a^2b^4c^6 - 144\); 4) \(169x^4y^6z^8 - 196\); 5) \((4a^2 + b^2)^2 - (2a^2 - b^2)^2\); 6) \((9m^2 - 5n^2)^2 - (6m^2 + n^2)^2\); 7) \((2a - b + c)^2 - (3a - 2b - c)^2\); 8) \((6x + y - 2z)^2 - (x - y + z)^2\); 9) \(a^{2n} - b^{2n}\); 10) \(x^{4n} - y^{4n}\); 11) \(64m^{6n} - 1\); 12) \(81c^{8n} - 4\).
1) \(x^6 - 0{,}04y^4 = (x^3)^2 - (0{,}2y^2)^2 = (x^3 - 0{,}2y^2)(x^3 + 0{,}2y^2)\);
2) \(\frac{x^8}{9} - y^{10} = \left(\frac{x^4}{3}\right)^2 - (y^5)^2 = \left(\frac{x^4}{3} - y^5\right)\left(\frac{x^4}{3} + y^5\right)\);
3) \(121a^2b^4c^6 - 144 = (11ab^2c^3)^2 - 12^2 = (11ab^2c^3 - 12)(11ab^2c^3 + 12)\);
4) \(169x^4y^6z^8 - 196 = (13x^2y^3z^4)^2 - 14^2 = (13x^2y^3z^4 - 14)(13x^2y^3z^4 + 14)\);
5) \[\begin{aligned} &(4a^2 + b^2)^2 - (2a^2 - b^2)^2 = {} \\ &= (4a^2 + b^2 - 2a^2 + b^2)(4a^2 + b^2 + 2a^2 - b^2) = {} \\ &= (2a^2 + 2b^2) \cdot 6a^2 = 12a^2(a^2 + b^2). \end{aligned}\]
6) \[\begin{aligned} &(9m^2 - 5n^2)^2 - (6m^2 + n^2)^2 = {} \\ &= (9m^2 - 5n^2 - 6m^2 - n^2)(9m^2 - 5n^2 + 6m^2 + n^2) = {} \\ &= (3m^2 - 6n^2)(15m^2 - 4n^2) = 3(m^2 - 2n^2)(15m^2 - 4n^2). \end{aligned}\]
7) \[\begin{aligned} &(2a - b + c)^2 - (3a - 2b - c)^2 = {} \\ &= (2a - b + c - 3a + 2b + c)(2a - b + c + 3a - 2b - c) = {} \\ &= (b - a + 2c)(5a - 3b). \end{aligned}\]
8) \[\begin{aligned} &(6x + y - 2z)^2 - (x - y + z)^2 = {} \\ &= (6x + y - 2z - x + y - z)(6x + y - 2z + x - y + z) = {} \\ &= (5x + 2y - 3z)(7x - z). \end{aligned}\]
9) \(a^{2n} - b^{2n} = (a^n)^2 - (b^n)^2 = (a^n - b^n)(a^n + b^n)\);
10) \[\begin{aligned} &x^{4n} - y^{4n} = (x^{2n})^2 - (y^{2n})^2 = (x^{2n} - y^{2n})(x^{2n} + y^{2n}) = {} \\ &= (x^n - y^n)(x^n + y^n)(x^{2n} + y^{2n}). \end{aligned}\]
11) Оба множителя раскладываются дальше — по формулам разности и суммы кубов:
\[\begin{aligned} &64m^{6n} - 1 = (8m^{3n})^2 - 1^2 = (8m^{3n} - 1)(8m^{3n} + 1) = {} \\ &= \left((2m^n)^3 - 1^3\right)\left((2m^n)^3 + 1^3\right) = {} \\ &= (2m^n - 1)(4m^{2n} + 2m^n + 1)(2m^n + 1)(4m^{2n} - 2m^n + 1). \end{aligned}\]
12) \(81c^{8n} - 4 = (9c^{4n})^2 - 2^2 = (9c^{4n} - 2)(9c^{4n} + 2)\).
Ответ: 1) \((x^3 - 0{,}2y^2)(x^3 + 0{,}2y^2)\); 2) \(\left(\frac{x^4}{3} - y^5\right)\left(\frac{x^4}{3} + y^5\right)\); 3) \((11ab^2c^3 - 12)(11ab^2c^3 + 12)\); 4) \((13x^2y^3z^4 - 14)(13x^2y^3z^4 + 14)\); 5) \(12a^2(a^2 + b^2)\); 6) \(3(m^2 - 2n^2)(15m^2 - 4n^2)\); 7) \((b - a + 2c)(5a - 3b)\); 8) \((5x + 2y - 3z)(7x - z)\); 9) \((a^n - b^n)(a^n + b^n)\); 10) \((x^n - y^n)(x^n + y^n)(x^{2n} + y^{2n})\); 11) \((2m^n - 1)(4m^{2n} + 2m^n + 1)(2m^n + 1)(4m^{2n} - 2m^n + 1)\); 12) \((9c^{4n} - 2)(9c^{4n} + 2)\).
Разложить на множители: 1) \(x^6 - 0{,}04y^4\); 2) \(\frac{x^8}{9} - y^{10}\); 3) \(121a^2b^4c^6 - 144\); 4) \(169x^4y^6z^8 - 196\); 5) \((4a^2 + b^2)^2 - (2a^2 - b^2)^2\); 6) \((9m^2 - 5n^2)^2 - (6m^2 + n^2)^2\); 7) \((2a - b + c)^2 - (3a - 2b - c)^2\); 8) \((6x + y - 2z)^2 - (x - y + z)^2\); 9) \(a^{2n} - b^{2n}\); 10) \(x^{4n} - y^{4n}\); 11) \(64m^{6n} - 1\); 12) \(81c^{8n} - 4\).
Разность квадратов двух выражений равна произведению разности этих выражений и их суммы: \(a^2 - b^2 = (a - b)(a + b)\).
1) \(x^6 - 0{,}04y^4 = (x^3)^2 - (0{,}2y^2)^2 = (x^3 - 0{,}2y^2)(x^3 + 0{,}2y^2)\);
2) \(\frac{x^8}{9} - y^{10} = \left(\frac{x^4}{3}\right)^2 - (y^5)^2 = \left(\frac{x^4}{3} - y^5\right)\left(\frac{x^4}{3} + y^5\right)\);
3) \(121a^2b^4c^6 - 144 = (11ab^2c^3)^2 - 12^2 = (11ab^2c^3 - 12)(11ab^2c^3 + 12)\);
4) \(169x^4y^6z^8 - 196 = (13x^2y^3z^4)^2 - 14^2 = (13x^2y^3z^4 - 14)(13x^2y^3z^4 + 14)\);
5) \[\begin{aligned} &(4a^2 + b^2)^2 - (2a^2 - b^2)^2 = {} \\ &= (4a^2 + b^2 - 2a^2 + b^2)(4a^2 + b^2 + 2a^2 - b^2) = {} \\ &= (2a^2 + 2b^2) \cdot 6a^2 = 12a^2(a^2 + b^2). \end{aligned}\]
6) \[\begin{aligned} &(9m^2 - 5n^2)^2 - (6m^2 + n^2)^2 = {} \\ &= (9m^2 - 5n^2 - 6m^2 - n^2)(9m^2 - 5n^2 + 6m^2 + n^2) = {} \\ &= (3m^2 - 6n^2)(15m^2 - 4n^2) = 3(m^2 - 2n^2)(15m^2 - 4n^2). \end{aligned}\]
7) \[\begin{aligned} &(2a - b + c)^2 - (3a - 2b - c)^2 = {} \\ &= (2a - b + c - 3a + 2b + c)(2a - b + c + 3a - 2b - c) = {} \\ &= (b - a + 2c)(5a - 3b). \end{aligned}\]
8) \[\begin{aligned} &(6x + y - 2z)^2 - (x - y + z)^2 = {} \\ &= (6x + y - 2z - x + y - z)(6x + y - 2z + x - y + z) = {} \\ &= (5x + 2y - 3z)(7x - z). \end{aligned}\]
9) \(a^{2n} - b^{2n} = (a^n)^2 - (b^n)^2 = (a^n - b^n)(a^n + b^n)\);
10) \[\begin{aligned} &x^{4n} - y^{4n} = (x^{2n})^2 - (y^{2n})^2 = (x^{2n} - y^{2n})(x^{2n} + y^{2n}) = {} \\ &= (x^n - y^n)(x^n + y^n)(x^{2n} + y^{2n}). \end{aligned}\]
11) Оба множителя, полученные по формуле разности квадратов, раскладываются дальше — по формулам разности и суммы кубов:
\[\begin{aligned} &64m^{6n} - 1 = (8m^{3n})^2 - 1^2 = (8m^{3n} - 1)(8m^{3n} + 1) = {} \\ &= \left((2m^n)^3 - 1^3\right)\left((2m^n)^3 + 1^3\right) = {} \\ &= (2m^n - 1)(4m^{2n} + 2m^n + 1)(2m^n + 1)(4m^{2n} - 2m^n + 1). \end{aligned}\]
12) \(81c^{8n} - 4 = (9c^{4n})^2 - 2^2 = (9c^{4n} - 2)(9c^{4n} + 2)\).
Ответ: 1) \((x^3 - 0{,}2y^2)(x^3 + 0{,}2y^2)\); 2) \(\left(\frac{x^4}{3} - y^5\right)\left(\frac{x^4}{3} + y^5\right)\); 3) \((11ab^2c^3 - 12)(11ab^2c^3 + 12)\); 4) \((13x^2y^3z^4 - 14)(13x^2y^3z^4 + 14)\); 5) \(12a^2(a^2 + b^2)\); 6) \(3(m^2 - 2n^2)(15m^2 - 4n^2)\); 7) \((b - a + 2c)(5a - 3b)\); 8) \((5x + 2y - 3z)(7x - z)\); 9) \((a^n - b^n)(a^n + b^n)\); 10) \((x^n - y^n)(x^n + y^n)(x^{2n} + y^{2n})\); 11) \((2m^n - 1)(4m^{2n} + 2m^n + 1)(2m^n + 1)(4m^{2n} - 2m^n + 1)\); 12) \((9c^{4n} - 2)(9c^{4n} + 2)\).
