§ 21. Способ групировки

№ 575

а) 2x – x

2

= x(2 – x); – 3ax + 2x

2

= x(2x – 3a);

2ax

2

– 3a2

x = ax(2x – 3a); 4xy – 2x

2

y = 2xy(2 – x).

Ответ: 2x – x

2

, 4xy – 2x

2

y; 2x

2

– 3ax, 2ax

2

– 3a2

x.

б) ab – 3b2

= b(a – 3b); a2

– 3ab = a(a – 3b);

5 + 10x = 5(1 + 2x); a + 2ax = a(1 + 2x).

Ответ: ab – 3b2

, a2

– 3ab; 5 + 10x, a + 2ax.

в) n2

– nm = n(n – m); 6a2

– 9ab = 3a(2a – 3b);

mn – n2

= n(m – n); 2ab – 3b2

= b(2a – 3b).

Ответ: 6a2

– 9ab, 2ab – 3b2

.

г) 4x – 8 = 4(x – 2); x

2

– 2x = x(x – 2);

–5 – 15m = – 5(m + 3); 21mn + 7n = 7n(3m + 1).

Ответ: 4x – 8, x

2

– 2x.

№ 576

а) 2by – bz = b(2y – z); 4ax – az = a(4x – z);

2ay – az = a(2y – z); 4bx – bz = b(4x – z).

Ответ: 2by – bz, 2ay – az; 4ax – az, 4bx – bz.

б) 6ax – 3x = 3x(2a – 1); – 2a – 1;

3by – 3y = 3y(b – 1); c – cb = c(1 – b).

Ответ: таких двучленов нет.

в) a3

– 2a2

= a2

(a – 2); 4ab – 2a2

b = 2ab(2 – a);

5ac

2

– 10ac = 5ac(c – 2); 3a – 6 = 3(a –2).

Ответ: a3

– 2a2

, 3a – 6.

г) 3mn2

– 6m2

n = 3mn(n – 2m); abn – 2abm = ab(n – 2m);

a2

x

3

– 9a2

x = a2

x(x

2

– 9); 9x

2

– x

4

= x

2

(9 – x

2

).

Ответ: 3mn2

– 6m2

n, abn – 2abm.

№ 577

а) 3a + 3 – na – n = 3(a + 1) – n(a +1) = (a + 1)(3 – n);

б) 6mx – 2m + 9x – 3 = 2m(3x – 1) + 3(3x – 1) = (2m + 3)(3x – 1);

в) ax – 3x – 4a + 12 = x(a – 3) – 4(a – 3) =(x – 4)(a – 3);

г) 2mx – 3m – 4x + 6 = m(2x – 3) – 2(2x – 3) = (m – 2)(2x – 3).

№ 578

а) 7kn – 6k +14n – 12 = k(7n – 6) + 2(7n – 6) = (k + 2)(7n – 6);

б) 5a2

– 5ax – 7a + 7x = 5a(a – x) – 7(a – x) = (5a – 7)(a – x); 94

в) 9m2

– 9mn – 5m + 5n = 9m(m – n) – 5(m – n) = (9m – n)(m – n);

г) 6a2

– 2ab – 3ac + bc = 2a(3a – b) – c(3a – b) = (2a – c)(3a – b).

№ 579

а) 7c

2

– c – c

3

+ 7 = c

2

(7 – c) + 7 – c = (c

2

+1)(7 – c);

б) x

3

+ 28 – 14x

2

– 2x = x(x

2

– 2) – 14(x

2

– 2) = (x – 14)(x

2

– 2);

в) 2b3

– 6 – 4b2

+ 3b = 2b2

(b – 2) + 3(b – 2) = (2b3

+ 3)(b – 2);

г) x

3

– 6 + 2x – 3x

2

= x

2

(x – 3) + 2(x – 3) = (x

2

+ 2)(x – 3).

№ 580

а) 16ab2

+5b2

c+10c

3

+32ac

2

=16a(b2

+2c

2

)+5c(b2

+2c

2

) = (16a + 5c)(b2

+ 2c

2

);

б) 18a2

+27ab + 14ac + 21bc = 9a(2a + 3b) + 7c(2a + 3b)=(9a+7c)(2a + 3b);

в) 20n2

– 35a – 14an + 50n = 10n(2n + 5) – 7a(2n + 5) = (10n – 7a)(2n + 5);

г) 2x

2

yz–15yz–3xz

2

+10xy

2

= xz(2xy – 3z) + 5y(2xy – 3z) = (xz + 5y)(2xy – 3z).

№ 581

ax – 2a – 3x + 6 = x(a – 3) – 2(a – 3) = (x – 2)(a – 3);

а) 4; б) 0; в) – 2,25; г) 0.

№ 582

2a + b + 2a2

+ ab = (2a + b) + a(2a + b) = (2a + b)(a + 1);

а) (– 2 + 998)(– 1 + 1) = 0;

б) (91 – 3)(45,5 + 1) = 46,5 · 88 = 4092;

в) (2 · 7,4 + (– 2))(7,4 + 1) = 12,8 · 8,4 = 107,52;

г)

2

18

3

⎛⎞

−+ ⎜⎟

⎝⎠

(– 0,5 + 1) = 7

2

3

· 0,5 =

23 1 5

3

32 6

⋅= .

№ 583

5ab – 7b + 5a2

– 7a = b(5a – 7) + a(5a – 7) = (a + b)(5a – 7);

а) (0,2+1)(1 – 7) = 1,2 · (– 6) = – 7,2; б) (2,5–0,5)(12,5 – 7) = 2 · 5,5 = 11;

в) (3,7 – 3,7)(5 · 3,7 – 7) = 0; г) ((1,4 + 11)(7 – 7) = 0.

№ 584

а) 40a3

b2

c + 21bc – 56ac

2

– 15a2

b3

= 8ac(5a2

b2

– 7c) + 3b(7c – 5a2

b2

) =

= (8ac – 3b)(5a2

b2

– 7c);

б) 16xy

2

–5y

2

z–10z

3

+32xz

2

= 16x(y

2

+ 2z

2

) – 5z(y

2

+ 2z

2

)=(16x – 5z)(y

2

+ 2z

2

);

в) 30x

2

+ 10c – 25cx – 12x = 5x(6x – 5c) + 2(5c – 6x) = (5x – 2)(6x – 5c);

г) 18x

2

z – 10kxy + 20k

2

y – 36kxz = 18xz(x – 2k) – 10ky(x – 2k) =

= (18xz – 10ky)(x – 2k) = 2(9xz – 5ky)(x – 2k).

№ 585

а) ax

2

–ay–bx

2

+ cy + by – cx

2

= a(x

2

–y)–b(x

2

–y)– c(x

2

– y)=(x

2

– y)(a – b – c);

б) xy

2

–by

2

– ax + ab + y

2

– a = x(y

2

– a) – b(y

2

– a)+y

2

–a=(y

2

– a)(x – b + 1);

в) ax + bx + cx + ay + by + cy = (x + y)(a + b + c);

г) ab – a2

b2

+ a3

b3

– c + abc – ca2

b2

=

= ab(1 – ab +a2

b2

) – c(1 – ab + a2

b2

) = (ab – c)(1 – ab + a2

b2

). 95

№ 586

21a2

b – 4b – 12a + 7ab2

= 7ab(3a + b) – 4(b + 3a) = (3a + b)(7ab – 4);

а) (– 1 + 2)

14

4

3

⎛⎞

−− ⎜⎟

⎝⎠

=

14 12 26 2

18

333

−− ⎛⎞

⋅=−=− ⎜⎟

⎝⎠

;

б) ()

1

12 4 4

7

⎛⎞

+− ⎜⎟

⎝⎠

=0;

в)

8181 81

3743

7272 72

⎛⎞⎛ ⎞⎛⎞

⋅+ ⋅⋅ − = ⋅+ ⋅

⎜⎟⎜ ⎟⎜⎟

⎝⎠⎝ ⎠⎝⎠

(4 – 4) = 0;

г) (– 2 + 3)

2

37 4

3

⎛⎞

−⋅⋅− ⎜⎟

⎝⎠

= 1 · (– 14 – 4) = – 18.

№ 587

а) 2,7 · 6,2 – 9,3 · 1,2 + 6,2 · 9,3 – 1,2 · 2,7 =

= 2,7(6,1 – 1,2) – 9,3(1,2 – 6,2) = (6,2 – 1,2)(2,7 + 9,3) = 12 · 5 = 60;

б) 125 · 48 – 31 · 82 – 31 · 43 – 125 · 83 = 125(48 – 83) – 31(82 + 53) =

= 125(48 – 83 – 31) = 125 · (– 66) = – 8250;

в) 14,9 · 1,25 + 0,75 · 1,1 + 14,9 · 0,75 + 1,1 · 25 =

= 14,9(1,25 + 0,75) + 1,1(25 + 0,75) = 14,9 · 2 + 1,1 · 25,72 =

= 29,8 + 28,325 = 58,125;

г) ()

11 2 14 22 11 4

3 4 4,2 3 2 2,8 2,8 4,2 3 4 2

35 3 35 33 35 5

⎛⎞

⋅+⋅+⋅+⋅= + + + = ⎜⎟

⎝⎠

11

37377428

33

=⋅+⋅=⋅= .

№ 588

а) 109 · 9,17 – 5,37 · 72 – 37 · 9,17 + 1,2 · 72 =

= 9,17(109 – 37) – 72(5,37 – 1,2) = 9,17 · 72 – 72 · 4,17 = 72 · 5 = 360; б)

19,9 · 18 – 19,9 · 16 + 30,1 · 18 – 30,1 · 16 =

= (19,9 + 30,1)(18 – 16) = 50 · 2 = 100;

в) 15,5 · 20,8 + 51,5 · 9,2 – 3,5 · 20,8 – 3,5 · 9,2 =

= 15,5(20,8 + 9,2) – 3,5(20,8 + 9,2) = 30 · 12 = 360;

г) 77,3 · 13 + 8 · 37,3 – 77,3 · 8 – 13 · 37,3 =

= 13(77,3 – 37,3) + 8(37,3 – 77,3) = 40 · 5 = 200.

№ 589

а) x

2

+ 6x + 8 = x

2

+ 2x + 4x + 8 = x(x + 2) + 4(x + 2) = (x + 4)(x + 2);

б) x

2

– 8x + 15 = x

2

– 3x – 5x + 15 = x(x – 3) – 5(x – 3) = (x – 5)(x – 3);

в) x

2

+ 3x + 2 = x

2

+ x + 2x + 2 = x(x + 1) + 2(x + 1) = (x + 2)(x + 1);

г) x

2

– 5x + 6 = x

2

– 2x – 3x + 6 = x(x – 2) – 3(x – 2) = (x – 2)(x – 3).

№ 590

а) a2

– 7a + 6 = a2

– a – 6a + 6 = a(a – 1) – 6(a – 1) = (a – 6)(a – 1);

б) b2

+ 9b – 10 = b2

– b + 10b – 10 = b(b – 1) + 10(b – 1) = (b – 1)(b + 10); 96

в) y

2

– 10y + 24 = y

2

– 4y – 6y + 24 = y(y – 4) – 6(y – 4) = (y – 6)(y – 4);

г) z

2

– 18z – 40 = z

2

+ 2z – 20z – 40 = z(z + 2) – 20(z + 2) = (z – 20)(z + 2).

№ 591

а) a2

+ 8ab – 9b2

= a2

+ ab – 9ab – 9b2

= a(a + b) – 9b(a+b)=(a – 9b)(a + b);

б) a2

+ 16ab + 55b2

= a2

+ 5ab + 11ab + 55b2

=

= a(a + 5b) + 11b(a + 5b) = (a +11b)(a + 5b);

в) x

2

+ 4xy – 12y

2

= x

2

– 2xy + 6xy – 12y

2

=

= x(x – 2y) + 6y(x – 2y) = (x + 6y)(x – 6y);

г) x

2

+ 16xy + 39y

2

= x

2

+ 13xy + 3xy + 39xy

2

=

= x(x + 3y) + 13y(x + 3y) = (x + 13y)(x + 3y).

№ 592

а) x

2

– 3x + 2 = 0; б) x

2

+ 8x + 15 = 0;

x

2

– x – 2x + 2 =0; x

2

+ 3x + 5x + 15 = 0;

x(x – 2) – (x – 2) =0; x(x + 3) + 5(x + 3) = 0;

(x – 1)(x – 2) =0; (x + 3)(x + 5) = 0;

x = 1; x = – 3;

x = 2. x = – 5.

Ответ: 1; 2. Ответ: – 3; – 5.

в) x

2

– 6x + 8 =0; г) x

2

– 3x – 4 = 0;

x

2

– 2x – 4x + 8 =0; x

2

+ x – 4x – 4 = 0;

(x – 2)(x – 4) = 0; (x – 4)(x + 1) = 0;

x = 2; x = 4;

x = 4. x = – 1.

Ответ: 2; 4. Ответ: 4; – 1.

№ 593

а) 2x

2

– 5x + 2 = 0; б) 3x

2

+ 10x + 3 = 0;

2x

2

– 4x – x + 2 = 0; 3x

2

+ 9x + x + 3 = 0;

2x(x – 2) – (x – 2) = 0; 3x(x + 3) + (x + 3) = 0;

(2x – 1)(x – 2) = 0; (3x + 1)(x + 3) = 0;

x =

1

2

; x = 2. x = –

1

3

; x = – 3.

Ответ:

1

2

; 2. Ответ: –

1

3

; – 3.

в) 4x

2

+ 5x – 6 =0; г) 3x

2

– x – 2 = 0;

4x

2

+ 8x – 3x – 6 = 0; 3x

2

– 3x + 2x – 2 = 0;

4x(x + 2) – 3(x + 2) = 0; 3x(x – 1) + 2(x – 1) = 0;

(4x – 3)(x + 2) = 0; (3x + 2)(x – 1) = 0;

x =

3

4

; x = – 2. x = –

2

3

; x = 1.

Ответ:

3

4

; – 2. Ответ: –

2

3

; 1. 97