§ 23. Разложение многочлена на множители

с помощью комбинаций различных приемов

№ 640

а) 5x

2

– 5 = 5(x

2

– 1) = 5(x – 1)(x + 1);

б) 10x

2

– 10y

2

= 10(x

2

– y

2

) = 10(x – y)(x + y);

в) 3a2

– 12 = 3(a2

– 4) = 3(a – 2)(a + 2);

г) 9b3

– b = b(9b2

– 1) = b(3b – 1)(3b + 1).

№ 641

а) 9x

2

– 81x = 9x(x – 9);

б) y

3

– 100y = y(y

2

– 100) = y(y – 10)(y + 10);

в) 64a – a3

= a(64 – a2

) = a(8 – a)(8 + a);

г) b3

– 144b = b(b2

– 122

) = b(b – 12)(b + 12).

№ 642

а) c

3

– 25c = c(c

2

– 25) = c(c – 5)(c + 5);

б) 50m – 2n2

m = 2m(25 – n2

) = 2m(5 – n)(5 + n);

в) 0,04s – sa2

= s(0,04 – a2

) = s(0,2 – a)(0,2 + a);

г)

23 22 16 16 4 4

49 49 7 7

pqqq pq qpq pq

⎛⎞⎛⎞⎛⎞

−= − = − + ⎜⎟⎜⎟⎜⎟

⎝⎠⎝⎠⎝⎠

.

№ 643

а) 5a2

+ 10ab + 5b2

= 5(a2

+ 2ab + b2

) = 5(a + b)

2

;

б) 2x

2

+ 4xy + 2y

2

= 2(x

2

+ 2xy + y

2

) = 2(x + y)

2

;

в) 3m2

+ 3n2

– 6mn = 3(m2

– 2mn + n2

) = 3(m – n)

2

;

г) 8n2

– 16n + 8 = 8(n2

– 2n + 1) = 8(n – 1)

2

.

№ 644

а) –3x

2

+ 12x – 12 = –3(x

2

– 4x + 4) = –3(x – 2)

2

;

б) –2a2

+ 20ab – 50b2

= –2(a2

– 10ab + 25b2

) = –2(a – 5b)

2

;

в) –5p2

– 10pq – 5q2

= –5(p2

+ 2pq + q2

) = –5(p + q)

2

;

г) –12z

3

– 12z

2

– 3z = –3z(4z

2

+ 4z + 1) = –3z(2z + 1)

2

.

№ 645

а) a4

– 16 = (a2

)

2

– 42

= (a2

– 4)(a2

+ 4) = (a – 2)(a + 2)(a2

+ 4);

б) b4

– 81 = (b2

– 9)(b2

+ 9) = (b – 3)(b + 3)(b2

+ 81); 107

в) y

8

–1=(y

4

– 1)(y

4

+ 1) = (y

2

– 1)(y

2

+ 1)(y

4

+ 1)=(y–1)(y+1)(y

2

+ 1)(y

4

+ 1);

г) x

4

– z

4

= (x

2

– z

2

)(x

2

+ z

2

) = (x – z)(x + z)(x

2

+ z

2

).

№ 646

а) 4m3

– 4n3

= 4(m3

– n3

) = 4(m – n)(m2

+ mn + n2

);

б) 13a3

+ 13b3

= 13(a3

+ b3

) = 13(a + b)(a2

– ab + b2

);

в) 15c

3

+ 15d3

= 15(c

3

+ d3

) = 15(c + d)(c

2

– cd + d2

);

г) 21s

3

– 21t

3

= 21(s

3

– t

3

) = 21(s – t)(s

2

+ st + t

2

).

№ 647

а) 6x

5

y – 24xy

3

= 6xy(x

4

– 4y

2

) = 6xy(x

2

– 2y)(x

2

+ 2y);

б) 3a4

b2

+ 24ab5

= 3ab2

(a3

+ 8b3

) = 3ab(a + 2b)(a2

– 2ab + 4b2

;

в) 0,3y

2

– 2,7y

6

= 0,3y

2

(1 – 9y

4

) = 0,3y

2

(1 – 3y

2

)(1 + 3y

2

);

г) 0,1x

4

y – 2,7xy

4

= 0,1xy(x

3

– 27y

3

) = 0,1xy(x – 3y)(x

2

+ 3xy + 9y

2

).

№ 648

а) (m + 3)

3

– 8 = (m + 3 – 2)((m + 3)

2

+ 2(m + 3) + 4) =

= (m + 1)(m2

+ 6m + 9 + 2m + 6 + 4) = (m + 1)(m2

+ 8m + 19);

б) (c–1)

3

+27 = (c – 1 + 3)(c

2

– 2c + 1 – 3c + 3 + 9) = (c + 2)(c

2

– 5c + 12);

в) (a–12)

3

–125 = (a–12–5)(a2

–24a+144+5a–60+25)=(a – 17)(a2

–19a+109);

г) (b+4)

3

+64=(b+4+4)(b2

+8b+16+4b+16+16) = (b + 8)(b2

+ 12b + 48).

№ 649

а) (x

2

+ 1)

2

– 4x

2

= (x

2

+ 1 – 2x)(x

2

+ 1 + 2x) = (x – 1)

2

(x + 1)

2

;

б) (y

2

+ 2y)

2

– 1 = (y

2

+ 2y – 1)(y

2

+ 2t + 1) = (y

2

+ 2y – 1)(y + 1)

2

;

в) 81 – (c

2

+ 6c)

2

= (9 – c

2

– 6c)(9 + c

2

+ 6c) = (9 – c

2

– 6c)(c + 3)

2

;

г) 16m2

– (m – n)

2

= (4m – m + n)(4m + m – n) = (3m + n)(5m – n).

№ 650

а) (a2

+ 2ab + b2

) – c

2

= (a + b)

2

– c

2

= (a + b – c)(a + b +c);

б) 16 – (x

2

– 2xy + y

2

) = 16 – (x – y)

2

= (4 – x + y)(4 + x – y);

в) 1 – m2

– 2mn – n2

= 1 – (m + n)

2

= (1 – m – n)( 1 + m + n);

г) 4 – p2

– 2pq – q2

= 4 – (p + q)

2

= (2 – p – q)(2 + p + q).

№ 651

а) x

2

– 2xc + c

2

– d2

= (x – c)

2

– d2

= (x – c – d)(x – c + d);

б) a2

+ 2a – b2

+ 1 = (a + 1)

2

– b2

= (a + 1 – b)(a + 1 + b);

в) c

2

– d2

+ 6c + 9 = (c + 3)

2

– d2

= (c + 3 – d)(c + 3 + d);

г) r

2

– s

2

– 10s – 25 = r

2

– (s + 5)

2

= (r – s – 5)(r + s + 5).

№ 652

а) x

2

+ 2xy – m2

+ y

2

= (x + y)

2

– m2

= (x = y – m)(x = y + m);

б) c

2

– a2

+ 2ab – b2

= c

2

– (a – b)

2

= (c – a + b)(c + a – b);

в) m2

– n2

– 8m + 16 = (m – 4)

2

– n2

= (m – 4 – n)(m – 4 + n);

г) 9 – p2

+ q2

– 6q = (q – 3)

2

– p2

= (q – 3 – p)(q – 3 + p). 108

№ 653

а) x

3

– x

2

y – xy

2

+ y

3

= x

3

+ y

3

– x

2

y – xy

2

= (x + y)(x

2

– xy + y

2

) – xy(x + y) =

= (x + y)(x

2

– xy + y

2

– xy) = (x + y)(x

2

– 2xy + y

2

) = (x + y)(x – y)

2

;

б) a3

+ a2

b – ab2

– b3

= (a – b)(a2

+ ab + b2

) + ab(a – b) =

= (a – b)(a2

+ 2ab + b2

) = (a – b)(a + b)

2

;

в) c

2

+2c–d2

+2d=c

2

–d2

+2 (c + d) = (c – d)(c + d) + 2 (c + d)=(c+d)(c – d + 2);

г) m2

– 2n – m – 4n2

= m2

– 4n2

– (2n + m) =

= (m – 2n)(m + 2n) – (2n + m) = (2n + m)(m – 2n – 1).

№ 654

а) x

2

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

2

– 2x + 1) = (x – 3)(x – 1)

2

;

б) (1 – a)

2

– 4a(1 – a)

2

+ 4a(1 – a)

2

= (1 – a)

2

(1 – 4a + 4a) = (1 – a)

2

.

№ 655

а) a3

+ 8b3

+ a2

– 2ab + 4b2

= (a + 2b)(a2

– 2ab + 4b2

) + a2

– 2ab + 4b2

=

= (a + 2b + 1)(a2

– 2ab + 4b2

);

б) 8c

3

– d3

+ 4c

2

+ 2cd + d2

= (2c – d)(4c

2

+ 2cd + d2

) + 4c

2

+ 2cd + d2

=

= (2c – d + 1)(4c

2

+ 2cd + d2

).

№ 656

а) x

3

+ 8y

3

+ x

2

+ 4xy + 4y

2

= (x + 2y)(x

2

– 2xy + 4y

2

) + (x + 2y)

2

=

= (x + 2y)(x

2

– 2xy + 4y

2

+ x + 2y);

б) 8p3

– q3

+ 4p2

– 4pq + q2

= (2p – q)(4p2

+ 2pq + q2

) + (2p – q)

2

=

= (2p – q)(q2

+ 2pq + 4p2

+ 2p – q).

№ 657

а) a3

– a2

– 2a + 8 = a3

+ 8 – a(a + 2) = (a + 2)(a2

– 2a + 4) – a(a + 2) =

= (a + 2)(a2

– 2a + 4 – a) = (a + 2)(a2

– 3a + 4);

б) b3

– 6b2

– 6b + 1 = b3

+ 1 – 6b(b + 1) = (b + 1)(b2

– b + 1) – 6b(b + 1) =

= (b + 1)(b2

– b + 1 – 6b) = (b + 1)(b2

– 7b + 1).

№ 658

а) x

2

– 10x + 24 = x

2

– 10x + 25 – 1 = (x – 5)

2

– 1 = (x – 6)(x – 4);

б) y

2

– 14y + 40 = y

2

– 14y + 49 – 9 = (y – 7)

2

– 9 = (y – 10)(y – 4);

в) b4

+ 4b2

– 5 = b4

+ 4b2

+ 4 – 9 = (b2

+ 2)

2

– 9 =

= (b2

+ 2 – 3)(b2

+ 2 + 3) = (b2

– 1)(b2

+ 5) = (b – 1)(b + 1)(b2

+ 5);

г) a2

– 6a + 5 = a2

– 6a + 9 – 4 = (a – 3)

2

– 4 = (a – 5)(a – 1).

№ 659

а) 4a2

– 12ab + 5b2

= 4a2

– 12ab + 9b2

– 4b2

= (2a – 3b)

2

– 4b2

=

= (2a – 5b)(2a – b);

б) 9c

2

– 24cd + 7d2

= 9c

2

– 24cd + 16d2

– 9d2

= (3c – 4d)

2

– 9d2

=

= (3c – 7d)(3c – d);

в) 25a2

– 20ab – 12b2

= 25a2

– 20ab + 4b2

– 16b2

= (5a + 2b)

2

– 16b2

=

= (5a – 2b)(5a + 6b);

г) 9m2

– 30mk + 16k

2

= 9m2

– 30mk + 25k

2

– 9k

2

= (3m – 5k

2

) – 9k

2

=

= (3m – 8k)(3m – 2k). 109

№ 660

а) a2

+ 7a + 10 = a2

+ 5a + 2a + 10 = a(a + 5) + 2(a + 5) = (a + 2)(a + 5);

б) x

4

+7x

2

+12 = x

4

+ 3x

2

+ 4x

2

+ 12 = x

2

(x

2

+ 3) + 4(x

2

+ 3) = (x

2

+ 4)(x

2

+ 3);

в) b2

– 3b – 4 = b2

– 1 – 3b – 3 = (b – 1)(b + 1) – 3(b + 1) = (b – 4)(b + 1) г)

y

4

– 5y

2

+ 4 = y

4

– 4y

2

– y

2

+ 4 = y

2

(y

2

– 4) – y

2

– 4 =

= (y

2

– 1)(y

2

– 4) = (y – 1)(y + 1)(y – 2)(y + 2).

№ 661

а) x

2

+5xy+6y

2

=x

2

+ 2xy + 3xy + 6y

2

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

б) 4m2

–5mn+n2

=4m2

–4mn – mn + n2

= 4m(m – n) + n(n – m)=(m–n)(4m – n);

в) p2

–pq–2q2

=p2

+pq–2q2

– 2pq = p(p + q) – 2q(p + q) = (p + q)(p – 2q);

г) a2

+7ab+6b2

=a2

+ ab + 6ab + 6b2

= a(a + b) + 6b(a + b) = (a + b)(a + 6b).

№ 662

а) x

3

– x = 0; б) 16y – y

3

= 0;

x(x

2

– 1) = 0; y(16 – y

2

) = 0;

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

x = 0, x = 1, x = –1. y = 0, y = 4, y = –4.

Ответ: 0; 1; –1. Ответ: 0; 4; –4.

в) c

3

+ c

2

= 0; г) d3

+ d = 0;

c

2

(c + 1) = 0; d(d2

+ 1) = 0;

c = 0, c = –1. d = 0, d2

+ 1 ≠ 0 не при каких d.

Ответ: 0; –1. Ответ: 0.

№ 663

а) x

3

+ x

2

– 4x – 4 = 0; б) y

3

+ 2y

2

– 4y – 8 = 0;

x

2

(x + 1) – 4(x + 1) = 0; y

2

(y + 2) – 4(y + 2);

(x

2

– 4)(x + 1) = 0; (y

2

– 4)(y + 2) = 0;

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

2

= 0;

x = 2, x = –2, x = –1. y = 2, y = – 2.

Ответ: 2; –2; –1. Ответ: 2; –2.

в) 9z + 9 – z

3

– z

2

= 0; г) p3

– p2

– 4p + 4 = 0;

9(z + 1) – z

2

(z + 1) = 0; p2

(p – 1) – 4(p – 1) = 0;

(9 – z

2

)(z + 1) = 0; (p2

– 4)(p – 1) = 0;

(3 – z)(3 + z)(z + 1) = 0; (p – 2)(p + 2)(p – 1) = 0;

z = 3, z = –3, z = –1. p = 2, p = –2, p = 1.

Ответ: 3; –3; –1. Ответ: 2; –2; 1.

№ 664

x1 + x2 = 7; x1 · x2 = 2;

а) x1x2

2

+ x1

2

x2 = x1x2(x1 + x2) = 2 · 7 = 14;

б) (x1 = x2)

2

= 72

= 49;

в) x1

2

+ x2

2

= x1

2

+ x2

2

+ 2x1x2 – 2x1x2 = (x1 + x2)

2

– 2x1x2 = 49 – 4 = 45;

г) (x1

3

+ x2

3

) = (x1 + x2)(x1

2

– x1x2 + x2

2

) =

= (x1 + x2)((x1 + x2)

2

– 3x1x2) = 7(49 – 6) = 7 · 43 = 301. 110

№ 665

x1 + x2 = 5; x1 · x2 = –3

а) x1

4

+ x2

4

= x1

4

+ 2x1

2

x2

2

+ x1

4

– 2x1

2

x2

2

= (x1

2

+ x2

2

)

2

– 2x1

2

x2

2

=

= ((x1 + x2)

2

– 2x1x2)

2

– 2x1

2

x2

2

= (25 + 6)

2

– 18 = 312

– 18 = 961 – 18 = 943;

б) (x1 – x2)

2

= x1

2

– 2x1x2 + x2

2

= (x1 + x2)

2

– 4x1x2 = 25 + 12 = 37;

в) x1

3

x2

2

+ x1

2

x2

3

= x1

2

x2

2

(x1 + x2) = 9 · 5 = 45;

г) x1

2

x2

4

+ x1

4

x2

2

= x1

2

x2

2

(x1

2

+ x2

2

) = x1

2

x2

2

(x1

2

+ 2x1x2 + x2

2

– 2x1x2) =

= x1

2

x2

2

((x1 + x2)

2

– 2x1x2) = 9 · (25 + 6) = 279.