§ 12. Розкладання многочленів на множники способом групування

391. ca - 2c + 5a - 10 = c(a - 2) + 5(a - 2) = (а - 2)(с + 5).

392. xy + yt - 2x - 2t = (xy + 2x) + (yt - 2t) = x(y - 2) + t(y - 2) = (y - 2)(x + t).

393. ab - cd - ad + cb = (ab - ad) + (cb - cd) = a(b - d) + c(b - d) = (b - d)(a + c).

394. 1) a(b + c) + 3b + 3c = a(b + c) + 3(b + c) = (b + c)(a + 3);

2) p(x - y) + 7x - 7y = p(x - y) + 7(x - y) = (x - y)(p + 7);

3) m(t - 5) + t - 5 = (t - 5)(m + 1);

4) b(m - c) + c - m = (m - c)(b - 1).

395. 1) c(x - y) + 3x - 3y = c(x - y) + 3(x - y) = (x - y)(c + 3);

2) a(c + m) + 9c + 9m = a(c + m) + 9(c + m) = (c + m)(a + 9);

3) x(c + 5) + c + 5 = (c + 5)(x + 1);

4) у(p - 3) + 3 - p = у(p - 3) - (p - 3) = (p - 3)(p - 1).

396. 1) ax + ay + 6x + 6y = a(x + y) + 6(x + y) = (x + y)(a + 6);

2) 5m - 5n + pm - pn = 5(m - n) + p(m - n) = (m - n)(5 + p);

3) 9p + mn + 9n + mp = 9(p + n) + m(p + n) = (p + n)(9 + m);

4) ab + ac - b - c = a(b + c) - (b + c) = (b + c)(a - 1);

5) 1 - by - y + b = (1 - y) + (b - by) = (1 - y) + b(1 - y) = (1 - y)(1 + b);

6) ma + 2a - 2m - 4 = a(m + 2) - 2(m + 2) = (m + 2)(a - 2).

397. 1) ab + 5a + bm + 5m = a(b + 5) + m(b + 5) = (b + 5)(a + m);

2) mp - b + bp - m = m(p - 1) + b(p - 1) = (p - 1)(m + b);

3) am - b + m - ab = m(a + 1) - b(a + 1) = (a + 1)(m - b);

4) cm - 3dm + cp - 3dp = m(c - 3d) + p(c - 3d) = (c - 3d)(m + p).

398. 1) ab - ac + 2b - 2c = a(b - c) + 2(b - c) = (b - c)(a + 2).

Якщо a = -1, b = 5,7, c = 6,7, то

(5,7 - 6,7) • (-1 + 2) = -1 • 1 = -1.

399. 1) 5x - 5y + xt - yt = 5(x - y) + t(x - y) = (x - p)(5 + 1).

Якщо x = 7,2, у = 6,2, t = -4,5, то

(7,2 - 6,2) • (5 - 4,5) = 1 • 0,5 = 0,5.

400. 1) a³ + a² + a + 1 = a²(a + 1) + (a + 1) = (a + 1)(a²+ 1);

2) b⁵ - b³ - b² + 1 = b³(b² - 1) - (b² - 1) = (b²- 1)(b³- 1);

3) c⁴ + 3c³ - c - 3 = c³(c + 3) - (c + 3) = (c + 3)(c³- 1);

4) a⁶- 5a⁴- 3a²+ 15 = a⁴(a²- 5) - 3(a²- 5) = (a²- 5)(a⁴- 3);

5) m²- mn - 8m + 8n = (m²- mn) + (-8m + 8n) = m(m - n) - 3(m - n) = (m - n)(m - 3);

6) ab - 9b + b² - 9a = (ab - 9a) + (b² - 9b) = a(b - 9) + b(b - 9) = (b - 9)(a + b);

7) 7t - ta + 7a - t² = (7t - t²) + (7a - ta) = t(7 - t + a(7 - t) = (7 - t)(t + a);

8) xy - ty - y² + xt = xy + xt + (-ty - y²) = x(y + t) - y(t + у) = (y + t)(x - y).

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