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Деление многочлена на одночлен№ 523а) (12 + 8) : 4 = 20 : 4 = 5;в) (44y + 22) : 11 = 4y + 2;б) (54d + 36) : (– 18) = – 3d – 2;г) (– 15y – 5y) : (– 5) = – 20y : (– 5) = 4y.№ 524а) (a – ab) : a = 1 – b;в) (– m – mn) : m = – 1 – n;б) (x – xy) : (– x) = y – 1;г) (– c + cd) : (– c) = 1 – d.№ 525а) (a2 + 3ab) : a = a + 3b;в) ( c2 – 2cd) : c = c – 2d;б) (m3 – m2n) : m2 = m – n;г) (p4 – p3q) : p3 = p – q.№ 526а) (4ab2 + 3ab) : ab = ab + 3; б) (1,2cd3 – 0,7cd) : cd = 1,2d2 – 0,7;в) (– 3,5m2n – 0,2mn) : mn = – 3,5m – 0,2;111⎛ 1⎞г) ⎜ − xy + x 3 y ⎟ : mn = –+ x2.323⎝ 2⎠84№ 52734x – y = 0,6x – 0,8y;55б) (4x + 12y – 16) : (– 4) = – x – 3y + 4;в) (2ab + 6a2b2 – 4b2) : (– 2b) = – a – 3a2b + 2b;г) (– a5b3 + 3a6b2) : 4a2b2 = – 0,25a3b + 0,75a4.№ 528а) (18a4 – 27a3) : 9a2 – 10a3 : 5a = 2a2 – 3a – 2a2 = – 3a = 24;1б) (36x2y – 4xy2) : 4y + yx = 9x2 – xy + xy = 9x2 = .9№ 529а) x, x2, 1; б) x, 2x, 7x; в) ab, 5ab, a2b2; г) mn, m2n2, 5mn.№ 530а) нет; б) нет;в) (2x3y2 + 3x2y – 5x4y4) : xy = 2x2y + 3x – 5x3y3;г) (2x3y2 + 3x2y – 5x4y4) : (– x2y) = – 2xy – 3 + 5x2y3.а) (3x2y – 4xy2) : 5xy =№ 531а)12a8b6 + 60a 6b85 54a b= 3a3b + 15ab3 ; б)15a 7 x9 − 45a9 x75a 6 x 6в)132n3 p 2 − 44n 2 p3 + 110n 2 p 4= 6n2p – 2np2 + 5np3;22npг)108k 4 n 2 − 144k 3n3 − 180k 2 n 4= 3k3n – 4k2n2 – 5kn3.36kn= 3ax3 – 9a3x;№ 532а) (7a2 + 10a3b) : a4 =7 + 10aba2б) (4x2 – 3x) : (– x2) = – 4 +=7a2b+ 10 ;a313; в) (27a3 – 81b3) : 9a3b3 = 3 – 9 · 3 ;xbaг) (42x3y – 63xy3 + 14xy) : 7xy = 6x2 – 9y2 + 2.№ 533а) kL, kL; б) pq, p2q2; в) cd, c2d2; г) xy, x2y2.№ 534а) bc, b2c2, b2c, b2c3, b2c4;в) z, m, zm, z2m, z3m;б) xy, x2y2, xy2, x2y, x;г) k, L, kL, kL2, kL3.№ 535а) xy;в) y2, 3, 142xyz, 15x;б) xy2z, 5z, 6xyz, 20xy;г) 4xy2, y2z, 8, 7xyz, 2xy2z.85№ 536а)15a 4b − * + 24a 2b35 a 2bОтвет:б)= 3a 2 −*5a 2 b15a 4b − 35a3b 2 + 24a 2b325a b*1 − 100a 2b 4 + 75ab 2∗1 = 3a2 · 25ab3 = 75a3b3; ∗2 =в)24 223 2b ; ∗ = 5a · 7ab = 35a b .52= 3a – 7ab +24 2b.5= 3a2 – ∗2 + ∗3;25ab3Ответ:+100a 2b 425ab75a3b3 − 100a 2b 4 + 75ab 225ab33= 4ab; ∗3 == 3а2 – 4ab +75ab 225ab3=3.b3.b*1 − 24a3 x 424a3 x 4= 7a2 – 8ax3; ∗2 == 3a2x;3*28ax∗1 = ∗2 · 7a2 = 7a2 · 3a2x = 21a4x.Ответ:г)3a 2 x − 24a3 x 421a 4 x= 7a2 – 8ax3.57c 4 d 3 − 38c 3 d 257c 4 d 3= 3cd2 – ∗2; ∗1 == 19c3d;*13cd 2∗2 =38c3d 2 38c3d 2== 2d.*119c3dОтвет:57c 4 d 3 38c3d 219c3d= 3cd2 – 2d.№ 537а)4a 2 x 4 21a3 x3 + 72a 4 x 2= ∗2 – ∗3 + 12a2x;*1∗1 =∗3 =б)72a 4 x 2272a x= 6a2x;∗2 =4a 2 x 46a 2 x2 3x;321a3 x321a3 x37== ax2;*126a 2 x30k 3 p3175k 2 p 4 − *1= 3k2 – ∗3 – 14p2;*2∗1 = 14p2 · 10kp3 = 240kp5; ∗3 =86=∗2 =30k 3 p33k 2=10kp3;175k 2 p 4175k 2==17,5kp;*210kp3в)45c10 d 3 + 54c n + 2 d 7 − *1=∗3 + 3,6cnd5 – 2c6d8;*2∗2 =∗3 =г)54c n + 2 d 73,6c n d 5= 15c2d2;∗1 = 2c6d8 · ∗2 = 2c6d8 · 15c2d2 = 30c8d10;45c10 d 345c10 d 3== 3c8d;*215c 2 d 2*1 − *2 + 63a n x5= 2a5x3 – 3a6x2 + 4,5an – 3x;*3∗3 =63a n x54,5a n −3 x= 14a3x4;∗1 = ∗3 · 2a5x3 = 14a3x4 · 2a5x3 = 28a8x7;∗2 = ∗3 · 3a6x2 = 14a3x4 · 3a6x2 42a9x6.№ 538а) 3a3 – 1, 2ab – частное; 10b3 – делитель;30a4b – 12ab2 – частное; b2 делитель; б) нет; в) нет;г) 15a4b3 – 6a2b4 – частное; 2 – делитель;3a2 – 1,2b – частное; 10a2b3 – делитель.№ 539а) (6x4y3 + 8x3y) · 7xy = 42x5y4 + 56x4y2;б) (21x2y3 + 28xy) · 2x3y = 42x5y4 + 56 x4y2;в) (4,2x4y2 + 5,6x3) · 10xy2 = 42x5y2 + 56x4y2;г) (10,5x2y3 + 14xy) · 4x3y = 42x5y4 + 56x4y2.ГЛАВА 5.

Разложение многочленов на множители§ 19. Что такое разложение многочленовна множители и зачем оно нужно№ 540а) x(x + 2) = 0;б) y( y – 8) = 0;в) z(z + 1,6) = 0;⎛ 1⎞г) t ⎜ t − ⎟ = 0;⎝ 2⎠x = 0; x = – 2;y = 0; y = 8;z = 0; z = – 1,6;1t = 0; t = .2№ 541а) (x + 1)(x + 4) = 0; x = – 1; x = – 4. Ответ: – 1; – 4.в) (y + 2)(y – 6) = 0; y = – 2; y = 6. Ответ: – 2; 6.б) (c – 12)(c + 25) = 0; c = 12; c = – 25. Ответ: 12; – 25.г) (d – 7)(d + 1,4) = 0.

d = 7; d = – 1,4. Ответ: 7; – 1,4.87№ 542а) (2x + 3)(3x – 6) = 0;2x + 3 = 0; 3x – 6 = 0;2x = – 3; 3x = 6;x = – 1,5; x = 2.Ответ: – 1,5; 2.в) (4a – 8)(6a – 12) = 0;4a – 8 =0; 6a – 12 = 0;4a = 8; 6a = 12;a = 2; a = 2.Ответ: 2.б) (11b + 44)(13b + 26) = 0;11b + 44 = 0; 13b + 26 = 0;11b = – 44; 13b = – 26;b = – 4; b = – 2.Ответ: – 4; – 2.г) (25c – 125)(54c – 108) =0;25c – 125 = 0; 54c – 108 = 0;25c = 125; 54c = 108;c = 5; c = 2.Ответ: 5; 2.№ 543а) m(m + 1)(m + 2) = 0;б) n(n – 3)(n – 8) = 0;m = 0; m + 1 =0; m + 2 =0;n = 0; n – 3 =0; n – 8 =0;m = – 1; m = – 2.n = 3; n = 8.Ответ: – 1; – 2.Ответ: 3; 8.в) p(p + 13)(p – 17) = 0;г) q(q – 21)(q – 105) = 0;p =0; p + 13 = 0; p – 17 = 0;q = 0; q – 21 = 0; q – 105 = 0;p = – 13; p = 17.q = 21; q = 105.Ответ: – 13; 17.Ответ: 21; 105.№ 544а) (4x + 12)(15x + 30)(17x + 34) = 0;4x + 12 = 0; 15x + 30 = 0; 17x + 34 = 0;4x = – 12; 15x = – 30; 17x = – 34; x = – 3; x = – 2; x = – 2.Ответ: – 3; – 2.б) (9y + 18)(12y – 48)(36y – 72) = 0; 9y + 18 = 0; 12y – 48 =0; 36y – 72 = 0;y = – 2; y = 4; y =2.Ответ: – 2; 4; 2.в) (23z – 46)(45z + 90)(3z + 24) = 0; 23z – 46 = 0; 45z + 90 = 0; 3z + 24 = 0;z = 2; z = – 2; z = – 8.Ответ: 2; – 2; – 8.г) (4t – 1)(8t – 3)(12t – 17) = 0; 4t – 1 = 0; 8t – 3 = 0; 12t – 17 = 0;13175;t= ;t==1 .4812121 35Ответ: ; ; 1 .4 812t=№ 545а) x2 – x = 0; x(x – 1) =0; x = 0; x = 1.

Ответ: 0; 1.б) 2x2 + 4x = 0; 2x(x + 2) = 0; x = 0; x = – 2. Ответ: 0; – 2.1313в) 3x2 – 7x = 0; x(3x – 7) = 0; x = 0; x = 2 . Ответ: 0; 2 .г) x2 =4x; x(x – 4) = 0; x = 0; x= 4. Ответ: 0; 4.88№ 546а) x2 – 16 = 0;б) y2 – 25 = 0;в) z2 – 36 = 0;г) t2 – 100 = 0;(x – 4)(x + 4) = 0; x = 4; x = – 4.y – 5)(y + 5) = 0; y = 5; y = – 5.(z – 6)(z + 6) = 0; z = 6; z = – 6.(t – 10)(t + 10) = 0; t = 10; t = – 10.Ответ: 4; – 4.Ответ: 5; – 5.Ответ: 6; – 6.Ответ: 10; – 10.№ 547а) 532 – 432 = (53 – 43)(53 + 43) = 10 · 96 = 960;б) 1082 – 982 = (108 – 98)(108 + 98) = 10 · 206 = 2060;⎛ 1⎞2⎛ 1⎞2⎛ 1⎞2⎛ 11 ⎞⎛ 11⎞⎛ 11 ⎞⎛ 11⎞22в) ⎜ 6 ⎟ − ⎜ 5 ⎟ = ⎜ 6 − 5 ⎟⎜ 6 + 5 ⎟ = 1 ⋅ 11 = 11 ;3 ⎠⎝ 33⎠33⎝ 3⎠ ⎝ 3⎠⎝ 3⎛ 1⎞2г) ⎜ 7 ⎟ − ⎜ 3 ⎟ = ⎜ 7 − 3 ⎟⎜ 7 + 3 ⎟ = 4 · 11 = 44.2 ⎠⎝ 22⎠⎝ 2⎠ ⎝ 2⎠⎝ 2№ 548а) 0,25a2 – 9 = 0;(0,5a – 3)(0,5 + 3) = 0;0,5a – 3 = 0; 0,5a + 3 = 0;a = 6; a = – 6.Ответ: 6; – 6.в) 4x2 – 144 = 0;(2x – 12)(2x + 12) = 0;2x – 12 = 0; 2x + 12 = 0;x = 6; x = – 6.Ответ: 6; – 6.б) 0,04b2 – 4 = 0;(0,2b – 2)(0,2b + 2) = 0;0,2b – 2 = 0; 0,2b + 2 = 0;b = 10; b = – 10.Ответ: 10, – 10.г) 0,25y2 – 25 = 0;(0,5y – 5)(0,5y + 5) = 0;0,5y – 5 = 0; 0,5y + 5 =0;y = 10; y = – 10.Ответ: 10; – 10.№ 549910910910131а)==== ;137 2 − 1232 (137 − 123)(137 + 123) 14 ⋅ 260 14 ⋅ 26 4б)2742 − 342 ( 274 − 34 )( 274 + 34 ) 240 ⋅ 308 308==== 77 ;9609609604в)3242 − 362 ( 324 − 36 )( 324 + 36 ) 288 ⋅ 360 288==== 72 ;1440144014404г)3522 − 522 ( 352 − 52 )( 352 + 52 ) 300 ⋅ 404 300==== 150 .8088088082№ 550(144 − 18)(144 + 18) = 126 ⋅ 162 = 2 ⋅ 2 ⋅ 34 = 4 = 1 1 ;331532 − 902 (153 − 90 )(153 + 90 ) 63 ⋅ 24335532 − 27 2 ( 53 − 27 )( 53 + 27 ) 24 ⋅ 806 ⋅ 8 48б);====792 − 512 ( 79 − 51)( 79 + 51) 28 ⋅ 130 7 ⋅ 13 91а)1442 − 182=89в)г)782 − 302662 − 621012 − 3121392 − 292( 78 − 30 )( 78 + 30 ) = 48 ⋅ 108 = 4 ⋅ 108 = 108 = 6 = 1 1 ;( 66 − 6 )( 66 + 6 ) 60 ⋅ 72 5 ⋅ 72 5 ⋅ 18 5 5(101 − 31)(101 + 31) = 70 ⋅ 132 = 7 ⋅ 132 = 7 ⋅ 12 = 1=(139 − 29 )(139 + 29 ) 110 ⋅ 168 11 ⋅ 168 168 2=№ 551а)б)в)г)7,8 ⋅ 8,7 + 7,8 ⋅ 1,3 7,8 ( 8,7 + 1,3) 7,8 ⋅ 10=== 0,78;10010010013, 2 ⋅ 9,8 + 13, 2 ⋅ 2, 2 13, 2 ( 9,8 + 2, 2 ) 13, 2 ⋅ 12= 6,6;==2424242,6 ⋅ 5, 4 + 5, 4 ⋅ 0,6 5, 4 ( 2,6 + 0,6 ) 5, 4 ⋅ 3, 2=== 1,6;10,810,810,84,5 ⋅ 3,1 + 4,5 ⋅ 2,1 4,5 ( 3,1 + 2,1) 4,5 ⋅ 5, 2=== 4,5 · 52 = 234.0,10,10,1§ 20.

Вынесение общего множителя за скобки№ 552а) 1, 2, 2m; б) x, 2x, 4x; в) a, b, ab; г) z, 2z, 7z.№ 553а) (3x + 3y) = 3(x + y);б) (5a – 5b) = 5(a – b);в) 7a + 7y = 7(a + y);г) 8x – 8a = 8(x – a).№ 554а) 3x + 6y = 3(x + 2y);б) 5a – 15b = 5(a – 3b);г) 8x – 32a = 8(x – 4a).в) 7a + 14y = 7(a + 2y);№ 555а) 8x + 28y = 4(x + 3y);б) 15a – 25 b = 5(3a – 5b);в) 21a + 28y = 7(3a + 4y); г) 24x – 32a = 8(3x – 4a).№ 556а) 2,4x + 7,2y = 2,4(x + 3y);б) 1,8a – 2,4b = 0,6(3a – 4b);в) 0,01a + 0,03y = 0,01(a + 3y); г) 1,25x – 1,75a =0,25(5x – 7a).№ 55714181682а) x + y = (x + 4y);б) a –b = (a – b);33392793в)1812a+y=25356⎛32 ⎞⎜ a + y⎟ ;5⎝57 ⎠г)3⎛ 41 ⎞123x–y = ⎜ x − y⎟ .7⎝74 ⎠4928№ 5581525а) 3 x + 3 y =9016171x + y = (16x + 17y);55527б) 4 a − 11301515(4a – b).b= a− b=1471414№ 559а) 3b2 – 3b = 3b(b – 1);в) 4c2 – 12c5 = 4c2(1 – 3c3);б) a4 + 2a2 = a2(a2 + 2);г) 8d4 – 32d2 = 8d2(d2 – 4).№ 560а) x3 – 3x2 – x = x(x2 – 3x – 1);б) y5 – 2y4 + y2 = y2(y3 – 2y2 + 1);№ 561а) ab – a2b = ab(1 – a);в) x2y – xy2 = xy(x – y);в) 2m6–4m3 + 6m = 2m(m5 – 2m3 + 3);г) 9p4 – 18p2 – 27p = 9p(p3 – 2p – 3).б) – p2q2 – pq = – pq(pq + 1);г) m3n2 – n3m2 = n2m2(m – n).№ 562а) 2z5q2 – 4z3q + 6z2q3 = 2z2q(z3q – 2z + 3q3);б) xy3 + 5x2y2 – 3x2y = xy(y2 + 5xy – 3x);в) 7a4b3 – 14a3b4 + 21a2b5 = 7a2b3(a2 – 2ab + 3b2);г) 8x3y2 + 88x2y3 – 16x3y4 = 8x2y2(x + 11y – 2xy2).№ 563а) 15x3y2 + 10x2y – 20x2y3 = 5x2y(3xy + 2 – 4y2);б) 12a2b4 – 36a2b + 44abc = 4ab(3ab3 – 9a + 11c);в) 195c6p5 – 91c5p6 + 221c3p10 = 13c3p5(15c3 – 7c2p + 17p5);г) 42a4b – 48a3b2 – 78a2b3 = 6a2b(7a2 – 8ab – 13b2).№ 564а) 3x(a+b)+y(a+b)=(a +b)(3x + y); б) m(x – y) – (x – y) = (x – y)(m – 1);в) (c + 2) – d(c + 2) = (c + 2)(1 – d); г) 5p(r–s) + 6q(r – s) = (r – s)(5p +6q).№ 565а) 15c(a+b)+8(b+a)=(15c+8)(a + b); б) n(2a+1)+m(1+a) = (n +m)(2a + 1);в) 4a(x+y)–9b(y+x)=(4a–9b)(x+);г) 11p(c+8d)–9(8d+c)=(11p–9)(8d+c).№ 566а) a(b–c)+3(c–b)=(a – 3)(b – c);в) 4(p–q)–a(q–p)=(4+a)(p – q);б) 6(m – n) + s(n – m) = (6 – s)(m – n);г) 7z(x – y) – 5(y – x) = (7z + 5)(x – y).№ 567а) (x – y)2 – a(x – y) = (x –y)(x – y –a);б) 5(a + 3)2 – (a +3) = (a +3)(5(a +3) – 1) = (a +3)(5a + 14);в) 8c(m+n)2 + 9d(m + n) = (m + n)(8c(m + n) + 9d)=(m + n)(8mc + 8cn + 8);г) (p2 – 6) – q(p2 – 6) = (p2 – 6)(1 – q(p2 – 6)) = (p2 – 6)(1 – qp2 + 6q).№ 568а) x2 – 3x = 0; x(x – 3) = 0; x = 0; x = 3.

Ответ: 0; 3.б) y2 – 5y = 0; y(y – 5) = 0; y = 0; y = 5. Ответ: 0; 5.91в) a2 + 10a = 0; a(a + 10) = 0; a = 0; a = – 10. Ответ: 0; – 10.г) b2 + 20b = 0; b(b + 20) = 0; b = 0; b = – 20. Ответ: 0; – 20.№ 569а) 0,45p2 + 18p = 0;9p(p + 40) = 0;209p = 0; p + 40 = 0;20p =0; p = – 40.Ответ: 0; – 40.в) 4q2 + 3q = 0;q(4q + 3) = 0;3.43Ответ: 0; – .4q = 0; q = –б) 9m2 + 0,27m = 0;⎛ 100⎞m + 1⎟ = 0;⎝ 3⎠0,27m ⎜100m + 1 = 0;3m = 0; m = – 0,03.Ответ: 0; – 0,03.г) 7x2 + 2x = 0;x(7x + 2) = 0;0,27m = 0;2.72Ответ: 0; – .7x = 0; x = –№ 570а) 1542 + 154 · 46 = 154(154 + 46) = 154 · 200 = 308 · 100 = 30800;б) 0,23 + 0,22 · 0,8 = 0,22(0,2 + 0,8) = 0,22 = 0,04;в) 1672 – 167 · 67 = 167(167 – 67) = 167 · 100 = 16700;г) 0,93–0,81 · 0,29=0,81 · 0,9–0,81 · 2,9=0,81(0,9–2,9)=0,81 · (– 2) = – 1,62.№ 571а) a(2a–b)(a+b)–3a(a + b)2 = a(a + b)(2a – b – 3a – 3b) = – a(a + b)(a + 4b);б) 5x2(3x–8) + 10x(3x – 8)2 = 5x(3x – 8)(x + 6x – 16) = 5x(3x – 8)(7x – 16);в) 4c(4c – 1) – 3(4c – 1)2 = (4c – 1)(4c – 12c + 3) = (4c – 1)(3 – 8c);г) 6d2(2d–5)2–12d2(2d–5)(d+5)=6d2(2d–5)(2d–5 – 2d – 10) = – 90d2(2d – 5).№ 572а) 0,7562 – 0,241 · 0,756 – 0,415 · 0,756 == 0,756(0,756 – 0,241 – 0,415) = 0,756 · 0,1 = 0,0756;б) 2,49 · 1,63–2,12 · 1,63 + 1,632 = 1,63(2,49 – 2,12 + 1,63)=1,63 · 2 = 3,26;в) 0,252 · 2,4 + 0,25 · 2,42 – 0,25 · 2,4 · 0,65 == 0,25 · 2,4(0,25 + 2,4 – 0,65) = 0,25 · 2,4 · 2 = 0,5 · 2,4 = 1,2;г) 0,16 · 6,41 · 1,25 – 0,16 · 1,252 – 0,162 · 1,25 == 0,16 · 1,25(6,41 – 1,25 – 0,16) = 0,16 · 1,25 · 5 = 1.№ 573а) 176 + 175 = 175(17 + 1) = 175 · 18; кратно 18;б) 428 + 427 = 427(42 + 1) = 427 · 43; кратно 43;в) 317 + 315 = 315(32 +1) = 315 · 10 = 314 · 30; кратно 30;г) 223 + 220 = 220(23 + 1) = 220 · 9 = 217 · 8 · 9 = 217 · 72; кратно 72.92№ 574а) 87 – 218 = 87 – 86 = 86(8 – 1) = 4 · 2 · 85 · 7; кратно 4;б) 106 – 57 = 56 · 26 – 57 = 56(64 – 5) = 56 · 59; кратно 59;в) 97 + 312 = 97 + 96 = 96(9 + 1) = 95 · 90; кратно 90;г) 810 – 227 = 810 – 89 = 89 (8 – 1) = 88 · 4 · 14.

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