§ 22. Разложение многочлена на множители

с помощью формул сокращенного умножения

№ 594

а) 4z

2

= 2z · 2z = (2z)

2

; 9b4

= 3b2

· 3b2

= (3b2

)

2

;

25m2

= 5m · 5m = (5m)

2

; 64p2

= 8p · 8p = (8p)

2

;

б) 16a2

b4

= (4ab2

)

2

; 81x

6

y

4

= (9x

3

y

2

)

2

;

49s

2

t

8

= (7st

4

)

2

; 25k

2

t

10

= (5kt

5

)

2

;

в)

16

25

p2

s

4

t

2

=

2

2 4

5

ps t

⎛⎞

⎜⎟

⎝⎠

;

9

16

m4

n12

=

2

26 3

4

mn

⎛⎞

⎜⎟

⎝⎠

;

4

49

a2

b12

=

2

6 2

9

ab

⎛⎞

⎜⎟

⎝⎠

;

25

81

x

4

y

8

z

16

=

2

248 5

9

x yz

⎛⎞

⎜⎟

⎝⎠

;

г) 0,01a4

b8

= (0,1a2

b4

)

2

; 0,04x

6

y

6

= (0,2x

3

y

3

)

2

;

0,49k

2

L10

= (0,7k

4

L5

)

2

; 1,21m6

n4

= (1,1m3

n2

)

2

.

№ 595

а) 4 – 36a2

= 22

– (6a)

2

= (2 – 6a)(2 + 6a);

б) 100 – 49b2

= 102

– (7b)

2

= (10 – 7b)(10 + 7b);

в) 400 – 121c

2

= 202

– (11c)

2

= (20 – 11c)(20 + 11c);

г) 225 – 144d2

= 152

– (12d)

2

=(15 – 12d)(15 + 12d).

№ 596

а) a2

– 9b2

= (a – 3b)(a + 3b); б) c

2

– 16d2

= (c – 4d)(c + 4d);

в) m2

– 64n2

= (m – 8n)(m + 8n); г) p2

– 100q2

= (p – 10q)(p + 10q).

№ 597

а) 59x

2

–121a2

=(7x – 11a)(7x + 11a); б) 64p2

– 81q2

= (8p – 9q)(8p + 9q);

в) 9m2

– 16n2

= (3m – 4n)(3m + 4n); г) 144y

2

–81r

2

= (12y – 9r)(12y + 9r).

№ 598

а) x

2

y

2

–1=(xy)

2

–12

= (xy – 1)(xy + 1); б) c

2

d2

–4=(cd)

2

–22

=(cd – 2)(cd + 2);

в) m2

n2

– 25 = (mn)

2

– 52

= (mn – 5)(mn + 5);

г) p2

q2

– 100 = (pq)

2

– 102

= (pq – 10)(pq + 10).

№ 599

а) 25 – 36p2

c

2

= 52

– (6pc)

2

= (5 – 6pc)(5 + 6pc);

б) 100m2

n2

– 81 = (10mn)

2

– 92

= (10mn – 9)(10mn + 9);

в) 49x

2

y

2

– 400 = (7xy)

2

– 202

= (7xy – 20)(7xy + 20);

г) 225 – 144c

2

d2

= 152

– (12cd)

2

= (15 – 12cd)(15 + 12cd).

№ 600

а) c

2

d2

– m2

= (cd)

2

– m2

= (cd – m)(cd + m);

б) a2

x

2

– 0,25p2

q2

=(ax)

2

– (0,5pq)

2

= (ax – 0,5pq)(ax + 0,5pq);

в) 16y

2

z

2

– 4a2

n2

= (4yz)

2

– (2an)

2

= (4yz – 2an)(4yz + 2an);

г) x

2

y

2

– 0,25p2

q2

= (xy)

2

– (0,5pq)

2

= (xy – 0,5pq)(xy + 0,5pq). 98

№ 601

а) 144a4

– 625c

2

= (12a2

)

2

– (25c)

2

= (12a2

– 25c)(12a2

+ 25c);

б) 25p10

–

1

9

q12

= (5p5

)

2

–

2

65656 111

55

333

qpqpq

⎛⎞⎛ ⎞⎛ ⎞

=− + ⎜⎟⎜ ⎟⎜ ⎟

⎝⎠⎝ ⎠⎝ ⎠

;

в) 169x

8

– 400y

16

= (13x

4

)

2

– (20y

8

)

2

= (13x

4

– 20y

8

)(13x

4

+ 20y

8

);

г) 4b16

–

1

16

d4

= (2b8

)

2

–

2

28282 111

22

444

d bdbd

⎛⎞⎛ ⎞⎛ ⎞

=− − ⎜⎟⎜ ⎟⎜ ⎟

⎝⎠⎝ ⎠⎝ ⎠

.

№ 602

а) x

2

– 49 = 0; б) y

2

– 100 = 0;

(x – 7)(x + 7) = 0; (y – 10)(y + 10) = 0;

x – 7 = 0; x + 7 = 0; y – 10 = 0; y + 10 = 0;

x = 7; x = – 7. y = 10; y = – 10.

Ответ: 7; – 7. Ответ: 10; – 10.

в) z

2

– 625 = 0; г) t

2

– 1 = 0;

(z – 25)(z + 25) = 0; ( t – 1)(t + 1) = 0;

z – 25 = 0; z + 25 = 0; t – 1 = 0; t + 1 = 0;

z = 25; z = – 25. t = 1; t = – 1.

Ответ: 25; – 25. Ответ: 1; – 1.

№ 603

а) 4x

2

– 1 = 0; б) 25y

2

– 49 = 0;

(2x – 1)(2x + 1) = 0; (5y – 7)(5y + 7) = 0;

2x – 1 = 0; 2x + 1 = 0; 5y – 7 = 0; 5y + 7 = 0;

x =

1

2

; x = –

1

2

. y =

5

7

; y = –

5

7

.

Ответ:

1

2

;–

1

2

. Ответ:

5

7

; –

5

7

.

в) 26a2

– 25 = 0; г) 144z

2

– 1 = 0;

(6a – 5)(6a + 5) = 0; (12z – 1)(12z + 1) = 0;

6a = 5; 6a = – 5; 12z = 1; 12z = – 1;

a = 1,2; a = – 1,2. z =

1

12

; z = –

1

12

.

Ответ: 1,2; – 1,2. Ответ:

1

12

; –

1

12

.

№ 604

а) a + b; a2

– ab + b2

; б) m2

+ 2n2

; m4

– 2m2

n2

+ 4n4

в) 2c + 3d; 2c

2

– 6cd + 9d2

; г) 3p + 4q2

; 9p2

– 12pq2

+ 16q4

.

№ 605

а) k –L; k

2

+ kL + L2

; б) 5a2

– b2

; 25a4

+ 5a2

b2

+ b4

;

в) 3p – 2m; 9p2

+ 6pm + 4m2

; г) 4s – 3t

2

; 16s

2

+ 125t

2

+ 9t

4

. 99

№ 606

а) a3

b3

= (ab)

3

, x

6

y

9

= (x

2

y

3

)

3

, 8m3

n9

= (2mn3

)

3

, 125k

9

t

27

= (5k

3

t

9

)

3

;

б)

1

64

p9

=

3

1

4

p

⎛⎞

⎜⎟

⎝⎠

,

27

125

s

18

=

3

6 3

5

s

⎛⎞

⎜⎟

⎝⎠

;

1

343

m12

=

4 1

7

m ⎛⎞

⎜⎟

⎝⎠

,

24 125

216

a =

3

8 5

6

a

⎛⎞

⎜⎟

⎝⎠

в) 0,064a3

b3

= (0,4ab)

3

, 0,125x

9

y

3

= (0,5x

3

y)

3

,

0,216m3

n18

= (0,6mn6

)

3

, 0,008p9

q12

= (0,2p3

q4

)

3

;

г) 125x

3

y

6

z

9

=(5xy

2

z

3

)

3

, 216a12

b36

c

24

= (6a4

b12

c

8

)

3

,

8m6

n3

p12

= (2m2

np4

)

3

, 0,343k

9

L18

p15

= (0,7k

3

L6

p5

)

3

.

№ 607

а) a3

+ 8 = a3

+ 23

= (a + 2)(a2

– 2a + 4);

б) b3

– 27 = b3

– 33

= (b – 3)(b2

+ 3b + 9);

в) c

3

– 64 = c

3

– 43

= (c – 4)(c

2

+ 4c + 16);

г) d3

+ 125 = d3

+ 53

= (d + 5)(d2

– 5d + 25).

№ 608

а) 216 – m3

= 63

– m3

= (6 – m)(36 + 6m + m2

);

б) 1000 + m3

= 103

+ m3

= (10 + m)(100 – 10m + m2

);

в) 729 + p3

= 93

+ p3

= (9 + p)(81 – 9p + p2

);

г) 343 – q3

= 73

– q3

= (7 – q)(49 + 7q + q2

).

№ 609

а) 64a3

+ 1 = (4a + 1)(16a2

– 4a + 1);

б) 27d3

– 8=(3d – 2)(9d2

+ 6d + 4);

в) 512b3

– 125 = (8b – 5)(64b2

+ 40b + 25);

г) 216c

3

+ 1000 = (6c + 10)(36c

2

– 60c + 100).

№ 610

а) a3

b3

–1=(ab – 1)(a2

b2

+ ab + 1); б) 8 + c

3

d3

= (2 + cd)(4 – 2cd + c

2

d2

);

в) m3

n3

–27=(mn–3)(m2

n2

+3mn+9); г) p3

q3

+64=(pq + 64)(p2

q2

– 4pq + 16).

№ 611

а) a2

– 2ab + b2

= (a – b)

2

; б) x

2

+ 2xy + y

2

= (x + y)

2

;

в) z

2

+ 2zt + t

2

= (z + t)

2

; г) m2

– 2mn + n2

= (m – n)

2

.

№ 612

а) m2

+ 4m + 4 = m2

+ 2 · 2m + 22

= (m + 2)

2

;

б) a2

– 12a + 36 = a2

– 2 · 6a + 62

= (a – 6)

2

;

в) 1 – 2b + b2

= (1 – b)

2

;

г) y

2

+ 18y + 81 = y + 2 · 9y + 92

= (y + 9)

2

.

№ 613

а) p2

+ 10p + 25 = p2

+ 2 · 5p + 52

= (p + 5)

2

;

б) x

2

– 14x + 49 = x

2

– 2 · 7x + 72

= (x – 7)

2

;

в) 225 – 30y + y

2

= 152

– 2 · 15y + y

2

= (15 – y)

2

;

г) 64 + 16z + z

2

= 82

+ 2 · 8z + z

2

= (z + 8)

2

. 100

№ 614

а) 4y

2

– 12y + 9 = (2y)

2

– 2 · 3 · 2y + 32

= (2y – 3)

2

;

б) 9p2

+ 48pq + 64q2

= (3p)

2

+ 2 · 3p · 8q + (8q)

2

= (3p + 8q)

2

;

в) 9m2

+ 24m + 16 = (3m)

2

+ 2 · 4 · 3m + 42

= (3m + 4)

2

;

г) 9a2

– 30a + 25 = (3a)

2

– 2 · 5 · 3a + 52

= (3a – 5)

2

.

№ 615

а) 9x

2

+ 24xy + 26y

2

= (3x)

2

+ 2 · 3x · 4y + (4y)

2

= (3x + 4y)

2

;

б) 36a2

– 84ab + 49b2

= (6a)

2

– 2 · 6a · 7b + (7b)

2

= (6a – 7b)

2

;

в) 4m2

– 28mn + 49n2

= (2m)

2

– 2 · 2m · 7n + (7n)

2

= (2m – 7n)

2

;

г) 0,25x

2

+ 3xy + 9y

2

= (0,5x)

2

+ 2 · 0,5x · 3y + (3y)

2

.

№ 616

а) a2

– 10a + 25 = (a – 5)

2

– неотрицательное;

б) 49 + 14a + a2

= (a + 7)

2

– неотрицательное;

в) – a2

– 4a – 4 = – 1 · (a2

+ 4a + 4) = – (a + 2)

2

– неположительное;

г) – a2

– 12a – 36 = – 1 · (a2

+ 12a + 36) = – (a + 6)

2

– неположительное.

№ 617

а) 342

+ 2 · 34 · 36 + 362

= (34 + 36)

2

= 702

= 4900;

б) 272

– 2 · 27 · 13 + 132

= (27 – 13)

2

= 142

= 196;

в) 982

– 2 · 98 · 8 + 82

= (98 – 8)

2

= 902

= 8100;

г) 76,4 + 13,6 + 2 · 76,4 · 13,6 = (76,4 + 13,6)

2

= 902

= 8100.

№ 618

а) 2572

– 1432

= (257 – 143)(257 + 143) = 114 · 400 = 45600;

б) 73,62

– 26,42

= (73,6 – 26,4)(73,6 + 26,4) = 37,2 · 100 = 3720;

в) 1652

– 652

= (165 – 65)(165 + 65) = 100 · 230 = 23000;

г) 72,52

– 47,52

= (72,5 – 47,5)(72,5 + 47,5) = 25 · 120 = 3000.

№ 619

а) (x + 1)

2

– 25 = (x + 1 – 5)(x + 1 + 5) = (x – 4)(x + 6);

б) (y – 2)

2

– 4 = (y – 2 – 2)(y – 2 + 2) = y(y – 4);

в) (z + 10)

2

– 100 = (z + 10 – 10)(z + 10 + 10) = z(z + 20);

г) (t – 7)

2

– 100 = (t – 7 – 10)(t – 7 + 10)= (t – 17)(t + 3).

№ 620

а) 49 – (m – 3)

2

= (7 – m + 3)(7 + m – 3) = (m + 4)(10 – m);

б) 400 – (a + 9)

2

= (20 – a – 9)(20 + a + 9) = (11 – a)(29 + a);

в) 625 – (n + 12)

2

= (25 – n – 12)(25 + n + 12) = (13 – n)(37 + n);

г) 121 – (b – 13)

2

= (11 – b + 13)(11 + b – 13) = (24 – b)(b – 2).

№ 621

а) (a+4)

2

–(b+2)

2

=(a + 4 – b – 2)(a + 4 + b + 2) = (a – b + 2)(a + b + 6);

б) (x–5)

2

–(y + 8)

2

= (x – 5 – y – 8)(x – 5 + y + 8) = (x – y – 13)(x + y +3);

в) (m+10)

2

–(n–12)

2

= (m+10–n+12)(m+10+n – 12)=(m – n + 22)(m + n – 2);

г) (c–1)

2

–(d – 23)

2

= (c – 1 – d + 23)(c – 1+ d – 23) =(c – d + 22)(c + d – 24). 101

№ 622

а) (3x+1)

2

–(4x + 3)

2

= (3x + 1 – 4x – 3)(3x + 1 + 4x + 3) = – (x + 2)(7x + 4);

б) (6y – 7)

2

–(9y + 4)

2

=(6y – 7 – 9y – 4)(6y – 7 + 9y +4)=(– 3y – 11)(15y – 3);

в) (15z + 4)

2

– (3z – 2)

2

= (15z + 4 – 3z + 2)(15z + 4 + 3z – 2) =

= (12z + 6)(18z + 2) = 12(2z + 1)(3z + 2);

г) (3t – 9)

2

– (8t – 7)

2

= (13t – 9 – 8t + 7)(13t – 9 +8t – 7)=(5t – 2)(21t – 16).

№ 623

а)

2 11

16 25

a − = 0;

1111

4545

aa

⎛⎞⎛⎞

−+ ⎜⎟⎜⎟

⎝⎠⎝⎠

= 0;

11

45

a = ;

11

45

a = − ;

a =

4

5

; a =

4

5

− . Ответ:

4

5

;

4

5

− .

б)

2 416

49 121

b − = 0;

2424

711711

bb

⎛⎞⎛⎞

−+ ⎜⎟⎜⎟

⎝⎠⎝⎠

= 0;

24

711

b = ;

24

711

b = −

b = 1

3

11

; b = – 1

3

11

. Ответ: 1

3

11

; – 1

3

11

.

в)

2 981

16 100

c − = 0;

3939

410410

cc

⎛⎞⎛⎞

−+ ⎜⎟⎜⎟

⎝⎠⎝⎠

= 0;

39

410

c = ;

39

410

c = − ;

c = 1

1

5

; c = – 1

1

5

. Ответ: 1

1

5

; – 1

1

5

.

г)

36

1225

d2

–

64

441

= 0;

6868

35 21 35 21

dd

⎛⎞⎛⎞

−+ ⎜⎟⎜⎟

⎝⎠⎝⎠

= 0;

68

35 21

d = ;

68

35 21

d =− ; d = 2

2

9

; d = – 2

2

9

. Ответ: 2

2

9

; – 2

2

9

.

№ 624

а) (2x – 5)

2

– 36 =0; б) (4 – 11y)

2

– 1= 0;

(2x – 41)(2x + 31) = 0; (3 – 11y)(5 – 11y) = 0;

2x = 41; 2x = – 31; 11y = 3; 11y = 5;

x = 20

2

1

; x = – 15

2

1

. y =

11

3

; y =

11

5

.

Ответ: 20

2

1

; – 15

2

1

. Ответ:

11

3

;

11

5

.

в) (5z – 3)

2

– 9z

2

= 0; г) (4t – 3)

2

– 25t

2

= 0;

(2z – 3)(14z – 3) = 0; (– t – 3)(9t – 3) = 0;

2z = 3; 14z = 3; – t = 3; 9t = 3;

z = 1

2

1

; z =

14

3

. t = – 3; t =

3

1

.

Ответ: 1;

14

3

. Ответ: – 3;

3

1

. 102

№ 625

а) (a + 1)

2

– (2a + 3)

2

= 0; б) (3b – 2)

2

– (b + 1)

2

= 0;

(– a – 2)(3a + 4) = 0; (2b – 3)(4b – 1) = 0;

a = – 2; 2b = 3;

3a = – 4; 4b = 1;

a = – 1

3

1

. b = 1

2

1

;

b =

4

1

.

Ответ: – 2; – 1

3

1

. Ответ: 1

2

1

;

4

1

.

в) (5c + 8)

2

– (c – 10)

2

= 0; г) (7d – 13)

2

– (9d – 25)

2

= 0;

(4c + 18)(6c – 2) = 0; (– 2d + 12)(16d – 38) = 0;

4c = – 18; 2d = 12;

6c = 2; 16d = 38;

c = – 4

2

1

; d = 6;

c = 3. d =2

8

3

.

Ответ: – 4

2

1

; 3. Ответ: 6; 2

8

3

.

№ 626

а)

33 2 2 1 8 12114

827 23439

ababaabb

⎛⎞⎛ ⎞

−=− ++ ⎜⎟⎜ ⎟

⎝⎠⎝ ⎠

;

б)

33 2 2 64 729 8 9 64 36 81

343 1000 7 10 49 35 100

cdcdccdd

⎛⎞⎛ ⎞

+=+ −+ ⎜⎟⎜ ⎟

⎝⎠⎝ ⎠

;

в)

33 2 2 125 216 5 6 25 15 36

512 343 8 7 64 28 49

x yxyxxyy

⎛⎞⎛ ⎞

−=− ++ ⎜⎟⎜ ⎟

⎝⎠⎝ ⎠

;

г)

33 2 2 1 125 1 5 1 5 25

729 216 9 6 81 54 36

mnmnmmnn

⎛⎞⎛ ⎞

+=+ −+ ⎜⎟⎜ ⎟

⎝⎠⎝ ⎠

.

№ 627

а) a6

– 8 = (a2

)

3

– 23

= (a2

– 2)(a4

+ 2a2

+ 4);

б) 27 + b9

= 33

+ (b3

)

3

= (3 + b3

)(9 – 3b3

+ b6

);

в) ()

2 3 62224 11 111

82 242

x xxxx

⎛⎞ ⎛ ⎞⎛ ⎞

−= − = − + + ⎜⎟ ⎜ ⎟⎜ ⎟

⎝⎠ ⎝ ⎠⎝ ⎠

;

г) –

64

1

– x

6

=

6224 1111

64 4 16 4

yyyy

⎛⎞⎛ ⎞⎛ ⎞

−+=−− − + ⎜⎟⎜ ⎟⎜ ⎟

⎝⎠⎝ ⎠⎝ ⎠

. 103

№ 628

а) x

3

y

3

– c

3

= (xy)

3

– c

3

= (xy – c)(x

2

y

2

+ xyc + c

2

);

б) a3

+ m3

n9

= a3

+ (mn3

)

3

= (a + mn3

)(a2

– amn3

+ m2

n6

);

в) m6

n3

– p12

= (m2

n)

3

– (p4

)

3

= (m2

n – p4

)(m4

n2

+ m2

np4

+ p8

);

г) q3

+ c

15

d18

= q3

+ (c

5

d6

)

3

= (q + c

5

d6

)(q2

– qc

5

d6

+ c

10

d12

).

№ 629

а) ()

3 3 69 2 3 23 4 236 11 111

82 242

abab abaabb

⎛⎞ ⎛ ⎞⎛ ⎞

−= − = − + + ⎜⎟ ⎜ ⎟⎜ ⎟

⎝⎠ ⎝ ⎠⎝ ⎠

;

б)

27

8

a3

+ x

9

=

3

2

3

a

⎛⎞

⎜⎟

⎝⎠

+ (x

3

)

3

=

32 36 242

393

ax a ax x

⎛⎞⎛ ⎞

+−+ ⎜⎟⎜ ⎟

⎝⎠⎝ ⎠

;

в) ()

3 3 36 2 2 2 24 11 111

125 5 5 25 5

x yxy xy xxyy

⎛⎞ ⎛ ⎞⎛ ⎞

+= + = + − + ⎜⎟ ⎜ ⎟⎜ ⎟

⎝⎠ ⎝ ⎠⎝ ⎠

;

г)

36

36 64 343 4 7

729 1000 9 10

mnmn

⎛⎞⎛ ⎞

−=−= ⎜⎟⎜ ⎟

⎝⎠⎝ ⎠

22 2 4 4 7 16 14 49

9 10 81 45 100

mn m mn n

⎛⎞⎛ ⎞

−++ ⎜⎟⎜ ⎟

⎝⎠⎝ ⎠

.

№ 630

а)(2с + 1)

3

– 64 = (2c + 1)

3

– 43

=(2c + 1 – 4)((2c + 1)

2

+ 4(2c + 1)+42

) =

= (2c – 3)(4c

2

+ 4c + 1 + 8c + 4 + 16) = (2c – 3)(4c

2

+ 12c + 21);

б) (3p – 4)

3

+ 1 = (3p – 4 + 1)((3p – 4)

2

– 3p + 4 + 1) =

= (3p – 3)(9p2

– 24p + 16 – 3p + 4 + 1) = 3(p – 1)(9p2

– 27p + 21) =

= 9(p – 1)(3p2

– 9p + 7);

в) 8 – (3 – k)

3

= (k – 1)(4 + 6 – 2k + 9 – 6k + k

2

) = (k – 1)(k

2

– 8k + 19);

г) (5a + 4)

3

– 27 = (5a + 1)(25a2

+ 40a + 16 + 15a + 12 + 9) =

= (5a + 1)(25a2

+ 55a + 37).

№ 631

а) (6b + 8)

3

– 125 = (6b + 8 – 5)((6b + 8)

2

+ 5(6b + 8) + 25) =

= 9(2b + 1)(12b2

+ 42b + 43);

б) 1000 + (3q + 12)

3

= (3q + 12)(100 – 10(3q + 12) + (3q + 12)

2

) =

= (3q + 12)(100 – 30q – 120 + 9q2

+ 72q + 144) =3(q + 4)(9q2

+ 42q + 124);

в) 8x

3

–(5x–3)

3

=(2x–5x+3)((2x)

2

+ 2x(5x – 3) + (5x–3)

2

)=– 3(x–1)(4x

2

+10x

2

–

– 6x + 25x

2

– 30x + 9) = – 9(x – 1)(13x

2

– 12x + 3);

г)(3x + 2y)

3

+ 729y

3

= (3x + 2y + 9y)((3x + 2y)

2

– 9y(3x + 2y) + 81y

2

) =

= (3x+11y)(9x

2

+12xy+4y

3

–27xy–18y

2

+81y

2

)= (3x + 11y)(9x

2

– 15xy + 67y

2

).

№ 632

а)

222

22 9163 434

22

16 9 4 3 4 3

aabb a ab b ab

⎛⎞ ⎛⎞⎛ ⎞

−+ = −+ = − ⎜⎟ ⎜⎟⎜ ⎟

⎝⎠ ⎝⎠⎝ ⎠

;

б)

22

62 44 26 22 2 22 2 9253 5

25 36 5 6

ab ab ab ab a ab b

⎛⎞ ⎛⎞ ⎛⎞

⎜⎟ ++ = ++ ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ⎝⎠

=a2

b2

2

22 35

56

ab

⎛⎞

+ ⎜⎟

⎝⎠

; 104

в) b8

+ a2

b4

+

4

1

a4

=(b4

)

2

+ a2

b4

+

22

242 11

22

aba

⎛⎞⎛ ⎞

=+ ⎜⎟⎜ ⎟

⎝⎠⎝ ⎠

;

г) 0,01x

4

+ y

2

– 0,2x

2

y = (0,1x

2

)

2

– 0,2x

2

y + y

2

= (0,1x

2

– y)

2

.

№ 633

а) 513

– 263

= (51 – 26)(512

+ 51 · 26 + 262

) = 25 · (512

+ 51 · 26 + 262

) 25 –

это сомножитель, значит, выражение делится на 25;

б) 433

+ 173

= (43 + 17)(432

– 43 · 17 + 172

) = 60(432

– 43 · 17 + 172

) 60 –

это сомножитель, значит, выражение делится на 60;

в) 543

– 143

= (54 – 14)(542

– 54 · 14 + 142

) = 40(542

– 54 · 14 + 142

) 40 –

это сомножитель, значит, выражение делится на 40;

г) 383

+ 373

= (38 + 37)(382

– 38 · 37 + 372

) = 75(382

– 38 · 37 + 372

) 75 –

это сомножитель, значит, выражение делится на 75.

№ 634

а) (532

+222

–472

–162

) : (652

– 2 · 65 · 59 + 592

) = (532

– 472

+ 222

– 162

) :

: (65 – 59)

2

= ((53 – 47)(53 + 47) + (22 – 16)(22 + 16)) : 62

=

= (100 · 6 + 38 · 6) : 36 = 6 · (100 + 38) : 36 = 138 : 6 = 23;

б)

33

59 41

18

−

+ 59 · 41 = (59 – 41)(592

+ 59 · 41 + 412

) : 18 + 59 · 41 =

592

+ 2 · 59 · 41 + 412

= (59 + 41)

2

= 1002

= 10000;

в) (1092

– 2 · 109 · 61 + 612

) : (792

+ 732

– 492

– 552

) =

= (109 – 61)

2

: ((79 – 49)(79 + 49) + (73 – 55)(73 + 55)) =

= 482

: (30 · 128 + 18 · 128) = 482

: (128· (30 + 18)) = 48 : 128 =

8

3

;

г)

33

67 52

119

+

– 67 · 52 = (67 + 52)(672

– 67 · 52 + 522

) : 119 – 67 · 52 =

= 672

– 2 · 67 · 52 + 522

= (67 – 52)

2

= 152

= 225.

№ 635

а)

33

97 53

97 53

44

⎛⎞ −

+⋅ ⎜⎟

⎜⎟

⎝⎠

: (152,52

– 27,52

) = =

()( )

22

97 53 97 97 53 53

44

⎛⎞ −+⋅+

⎜⎟

⎜⎟

⎜⎟

⎝⎠

:

: ((152,5–27,5)(152,5+27,5)) = (972

+2 · 97 · 53 + 532

) : (125 · 180) =

= (97 + 53)

2

: 22500 = 1502

: 22500 = 1;

б) (36,52

– 27,52

) :

33

57 33

57 33

90

⎛⎞ +

−⋅ ⎜⎟

⎜⎟

⎝⎠

=

= ((36,5–27,5)(36,5+27,5))

()( )

22

57 33 57 57 33 33

57 33

90

⎛⎞ +−⋅+

⎜⎟

−⋅

⎜⎟

⎜⎟

⎝⎠

=

= (9 · 64) : (572

– 2 · 57 · 33 + 332

) = 576 : (57 – 33)

2

= 1; 105

в)

33

79 41

79 41

38

⎛⎞ −

+⋅ ⎜⎟

⎜⎟

⎝⎠

: (133,52

– 58,52

) =

()( )

22

79 41 79 79 41 41

79 41

38

⎛⎞ −+⋅+

⎜⎟

+ ⋅

⎜⎟

⎜⎟

⎝⎠

· (133,5–58,5)(133,5+ 58,5) = (79 + 41)

2

: (75 · 192) = 1202

: 14400 = 1;

г) (94,52

– 30,52

) :

33

69 29

69 29

98

⎛⎞ +

−⋅ ⎜⎟

⎜⎟

⎝⎠

=

= (94,5 – 30,5)(94,5 + 30,5)

22

(69 29)(69 69 · 29 29 )

69 · 29

98

⎛⎞ +−+

⎜⎟

⎜⎟

⎝⎠

=

= 64 · 125 : (69 – 29)

2

= 8000 : 1600 = 5.

№ 636

а) a2

+ ∗ + b2

= (a + b)

2

; (a + b)

2

= a2

+ 2ab + b2

; ∗ = 2ab;

б) b2

+ 20ab + ∗ = (b + 10)

2

; (b + 10)

2

= b2

+ 20ab + 100; ∗ = 100;

в) ∗ – 56a + 49 = (4a – 7)

2

; (4a – 7)

2

= 16a2

– 56a + 49; ∗ = 16a;

г) ∗1 – 12c + ∗2 = (3c – 2)

2

; (3c – 2)

2

= 9c

2

– 12c + 4; ∗1 = 9c

2

, ∗2 = 4.

№ 637

а) b2

– 20b + ∗1 = (∗2 – 10)

2

; (∗2–10)

2

=∗2

2

– 2 · ∗2 · 10 + 100; ∗1=100, ∗2=b;

б) ∗1 – 42pq+49q2

= (3p – ∗2)

2

; (3p–∗2)

2

=9p2

–2 · ∗2 · 3p+∗2

2

; ∗1=9p2

, ∗2=7q;

в) 25a2

+ ∗1 +

4

1

b2

= (∗2 +

2

1

b)

2

;

(∗2 +

2

1

b)

2

= ∗2

2

+ ∗2b +

4

1

b2

; ∗2 = 5a, ∗1 = ∗2 · b = 5ab;

г) 0,01b2

+ ∗1 + 100c

2

= (0,1b + ∗2)

2

; (0,1b + ∗2)

2

= 0,01b2

– 0,2b · ∗2 + ∗2

2

;

∗2 = 10c, ∗1 = ∗2 · 0,2b = 10c · 0,2b = 2bc.

№ 638

а) ∗1 + 56ab + 49b2

= (4a + ∗2)

2

;

(4a + ∗2)

2

= 16a2

+ 2 · ∗2 · 4a + ∗2

2

; ∗1 = 16a2

, ∗2 = 7b;

б) 225x

2

– ∗1 + 64y

2

= (15x – ∗2)

2

;

(15x – ∗2)

2

= 225x

2

– 2 · 15x · ∗2 + ∗2

2

;

∗2 = 8y, ∗1 = 30x · ∗2 = 30x · 8y = 240xy;

в) ∗1 + 96xy + 36y

2

= (8x + ∗2)

2

;

(8x + ∗2)

2

= 64x

2

+ 16x · ∗2 + ∗2

2

; ∗1 = 64x

2

, ∗2 = 6y;

г) 100a2

+ ∗1 + 49b2

= (10a + ∗2)

2

;

(10a + ∗2)

2

= 100a2

+ 20a · ∗2 + ∗2

2

; ∗2 = 7b, ∗1 = 140ab.

№ 639

а) m2

+ 40m + ∗1 = (∗2 + 20)

2

;

(∗2 + 20)

2

= ∗2

2

+ 2 · 20 · ∗2 + 400; ∗1 = 400, ∗2 = m;

б) ∗1 – 70pq + ∗2 = (7p – ∗3)

2

;

(7p – ∗3)

2

= 49p2

– 2 · 7p · ∗3 + ∗3

2

; 106

∗1 = 49p2

, ∗3 =

p

pq

14

70

= 5q, ∗2 = 25q2

;

в) ∗1 + 42ac + 49c

2

= (∗2 + ∗3)

2

;

∗3 = 7c, 2 · ∗2 · ∗3 = 42ac, ∗2 =

c

ac

7

21

= 3a, ∗1 = ∗2

2

= 9a2

;

г) 25z

2

– ∗1 + ∗2 = (∗3 – 8t)

2

;

∗2 = (8t)

2

= 64t

2

, ∗3 = 5z, ∗1 = 5z · 8t · 2 = 80zt.