ГДЗ до підручника «Алгебра» А.Г. Мерзляка. 7 клас

731. 1) (a + b + c)3 - a3 - b3 - c3 = 3(a + b) • (b + c)(a + c) — тотожність;

(a + b + c)3 - a3 - b3 - c3 = ((a + b + c)3 - a3) - (b3 + c3) - (a + b + c - a)(a + b + c)2 + a(a + b + c) + a2) - (b + c)(b2 - bc + c2) = (b + c)(a2 + b2 + c2 + 2ab + 2bc + 2ac + a2 + ab + ac + a2) - (b + c)(b2 - bc + c2) = (b + c)(3a2 + b2 + c2 + 3ab + 3ac + 2bc - b2 + bc - c2) = (b + c)(3a2 + 3bc + 3ab + 3ac) = (b + c) • (3a2 + 3ab) + (3bc + 3ac) = (b + c)(3a(a + b) +

3c(b + a)) = (b + c)(a + b)(3a + 3c) = 3(a + b) • (b + c)(a + c);

3(a + b)(b + c)(a + c) = 3(a + b)(b + c)(a + c) — правильна рівність.

2) (a - b)3 + (b - c)3 - (a - c)3 = -3(a - b)(b - c) • (a - c) — тотожність;

(a - b)3 + (b - c)3 - (a - c)3 = (a - b + b - c) • ((a - b)2 - (a - b)(b - c) + (b - c)2) - (a - c)3 = (a - c)(a2 - 2ab + b2 - ab + b2 + ac - bc + b2 - 2bc + c2) - (a - c)3 = (a - c)(a2 + 3b2 + c2 - 3ab - 3bc + ac) - (a - c)3) = (a - c) • (a2 + 3b2 + c2 - 3ab - 3bc + ac - a2 + 2ac - c2) = (a - c)(3b2 - 3ab - 3bc + 3ac) = 3(a - c) • ((b2 - ab) - (bc - ac)) = 3(a - c)(b(b - a) - c(b - a)) = 3(a + c)(b - a)(b - c) = -3(a - c) • (a - b)(b - c);

-3(a - c)(a - b)(b - c) = -3(a - c)(a - b)(b - c) — правильна рівність.

732. 1) (х - у)(х + у) + 2(х + 3у) - 8 = х2 - у2 + 2x + 6y - 8 = (x2 + 2x + 1) - (y2 - 6y + 9) = (x + 1)2 - (у - 3)2 = (x + 1 - у + 3)(x + 1 + y - 3) = (x - y + 4)(x + у - 2).

2) (2a - 3b)(2a + 3b) - 4(a + 3b) - 3 = 4a2 - 9b2 - 4a - 12b - 3 = 4a2 - 4a + 1 - 1 - 9b2 - 12b - 3 = (4a2 - 4a + 1) - (9b2 + 12b + 4) = (2a - 1)2 - (3b + 2)2 = (2a - 1 - 3b - 2) • (2a + 3b + 1) = (2a - 3b - 3)(2a + 3b + 1).

733. 1) (5x - y2)(5x + y2) - 2(15x - 7y2) - 40 = 25x2 - y4 - 30x + 14y2 - 40 = (25x2 - 30x + 9) + (y4 + 14y2 - 49) = (5x - 3)2 - (y2 - 7)2 = (5x - 3 - y2 + 7)(5x - 3 + y2 - 7) = (5x - y2 + 4)(5x + y2 - 10).

2) (3m - 2n)(12m + 5n) + 3m(3n + 4) - 2(3n2 - 20n + 12) = 36m2 + 15mn - 24mn - 10n2 + 9mn + 12m - 6n2 + 40n - 24 = 36m2 - 16n2 + 12m + 40n - 24 = 36m2 + 12m + 1 - 16n2 + 40n - 24 - 1 = (6m + 1)2 - (16n2 - 40n + 25) = (6m + 1)2 - (4n - 5)2 = (6m + 1 - 4n + 5) • (6m + 1 + 4n - 5) = (6m - 4n + 6)(6m + 4n - 4) = 2(3m - 2n + 3) • 2(3m + 2n - 2) = 4(3m - 2n + 3)(3m + 2n - 2).

734. 1) x2 - 10x + 24 = (x2 - 10x + 25) - 1 = (x - 5)2 - 12 = (x - 5 - 1)(x - 5 + 1) = (x - 6)(x - 4).

2) a2 + 4a - 32 = (a2 + 4a + 4) - 32 - 4 = (a + 2)2 - 36 = (a + 2)2 - 62 = (a + 2 - 6)(a + 2 + 6) = (a - 4)(a + 8).

3) b2 - 3b - 4 = (b2 - 3b + 2,25) - 4 - 2,25 = (b - 1,5)2 - 2,52 = (b - 1,5 - 2,5)(b - 1,5 + 2,5) = (b - 4)(b + 1).

4) 4a2 - 12a + 5 = 4a2 - 12a + 9 - 9 + 5 = (2a - 3)2 - 4 = (2a - 3 - 2)(2a - 3 + 2) = (2a - 5)(2a - 1).

5) 9x2 - 24xy + 7y2 = 9x2 - 24xy + 16y2 - 16y2 + 7y2 = (3x - 4y)2 - 9y2 = (3x - 4y - 3y)(3x - 4y + 3y) = (3x - 7y)(3x - y).

6) 36m2 - 60mn + 21n2 = 36m2 - 60mn + 25n2 - 25n2 + 21n2 = (6m - 5n)2 - 4n2 = (6m - 5n - 2n)(6m - 5n + 2n) = (6m - 7n) • (6m - 3n) = 3(6m - 7n)(2m - n).

735. 1) x2 - 4x + 3 = x2 - 4x + 4 - 1 = (x - 2)2 - 12 = (x - 2 - 1)(x - 2 + 1) = (x - 3)(x - 1).

2) a2 + 2a - 24 = a2 + 2a + 1 - 25 = (a + 1)2 - 52 = (a + 1 - 5)(a + 1 + 5) = (a - 4)(a + 6).

3) y2 + 12y + 35 = y2 + 12y + 36 - 1 = (y + 6)2 - 12 = (y + 6 - 1)(y + 6 + 1) = (y + 5)(y + 7).

4) x2 + x - 6 = x2 + x + 0,25 - 6,25 = (x + 0,5)2 - 2,52 = (x + 0,5 - 2,5)(x + 0,5 + 2,5) = (x - 2)(x + 3).

5) c2 + 8cd + 15d2 = c2 + 8cd + 16d2 - d2 = (c + 4d2) - d2 = (c + 4d - d)(c + 4d + d) = (c + 3d)(c + 5d).

6) 9x2 - 30xy + 16y2 = 9x2 - 30xy + 25y2 - 9y2 = (3x - 5y)2 - (3y)2 = (3x - 5y - 3y) • (3x - 5y + 3y) = (3x - 8y)(3x - 2y).

736. 1) x1x22 - x12x2 = x1x2(x2 - x1) = 5 • (-8) = -40;

2) x12 + x22 = (x1 - x2)2 + 2x1x2 = 82 + 2 • 5 = 64 + 10 = 74;

3) (x1 + x2)2 = x12 + x22 + 2x1x2 = (x1 - x2)2 + 2x1x2 + 2x1x2 = (x1 - x2)2 + 4x1x2 = 82 + 4 • 5 = 64 + 20 = 84;

4) x13 - x23 = (x1 - x2)(x1 + x1x2 + x22) = 8(5 + (x12 + x22)) = 8(5 + 74) = 8 • 79 = 632.

737. 1) x3y2 + x2y3 = x2y2(x + y) = (xy)2 • (x + y) = (-3)2 • 6 = 9 • 6 = 54.

2) (x - y)2 = x2 - 2xy + y2 = x2 + y2 - 2xy = (х + y)2 - 4хy = 62 - 4 • (-3) = 36 + 12 = 48.

3) x4 + y4 = (x2 + y2)2 - 2x2y2 = ((x + y)2 - 2xy)2 - 2(xy)2 = (x + y)4 - 4xy(x + y)2 + 4x2y2 - 2(xy)2 = (x + y)4 - 4xy(x + y)2 + 2x2y2 = 64 - 4 • (-3) • 62 + 2 • (-3)2 = 1296 + 432 + 18 = 1746.

738. (2n - 1)3 - 4n2 + 2n + 1 = (2n - 1)2 • (2n - 1) - 4n2 + 2n + 1 = (4n2 - 4n + 1) • (2n - 1) - 4n2 + 2n + 1 = 8n3 - 8n2 + 2n - 4n2 + 4n - 1 - 4n2 + 2n + 1 = 8n3 - 16n2 + 8n = 8n3 + 8n - 16n2 = 8n(n2 + 1) - 16n2; ділиться на 16, бо 8n(n2 + 1) ділиться на 16 і 16n2 ділиться на 16, бо 16 ділиться на 16.

Якщо n — непарне, то n2 + 1 — парне. Якщо n — парне, то n2 + 1 — непарне. Тобто один з множників n і n2 + 1 ділиться на 2.

Тоді 8n(n2 + 1) ділиться на 16.

739. 1) х4 - 5х2 + 4 = х4 - 4х2 - х2 + 4 = х2(х2 - 4)-(х2 - 4) = (х2 - 4)(х2 - 1) = (х - 2) • (х + 2)(х - 1)(х + 1).

2) x4 + х2 + 1 = х4 + 2х2 + 1 - х2 = (х2 + 1)2 - x2 = (х2 + 1 - х)(х2 + 1 + х).

3) 4х4 - 12х2 + 1 = (4х4 + 4х2 + 1) - 16х2 = (2х2 + 1)2 - (4х)2 = (2х2 + 1 - 4х)(2х2 + 1 + 4х).

4) х5 + х + 1 = x5 - x2 + х2 + х + 1 = х2(х8 - 1) + (х2 + х + 1) = х2(х - 1)(х2 + х + 1) + (х2 + х + 1) = (x2 + x + 1)(х2(х - 1) + 1) = (x2 + x + 1) (х3 - х2 + 1).

5) х4 + 4 = х4 + 4х2 + 4 - 4х2 = (х2 + 2)2 - (2х)2 = (х2 - 2х + 2)(х2 + 2х + 2).

6) x8 + x4 - 2 = x8 - х4 + 2х4 - 2 = х4(х4 - 1) + 2(х4 - 1) = (x4 - 1)(х4 + 2) = (х2 - 1)(х2 + 1) • (х4 + 2) = (х - 1)(х + 1)(х2 + 1)(х4 + 2).

740. 1) х4 + 5х2 + 9 = (х4 + 6х2 + 9) - х2 = (х2 + 3)2 - х2 = (х2 + 3 - х)(х2 + 3 + х).

2) х4 - 8х2 + 4 = х4 - 4х2 + 4 - 4х2 = (х2 - 2)2 - (2х)2 = (х2 - 2х - 2)(х2 + 2х - 2).

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