makarytchev-gdz-7-1-1289-2003 (542427), страница 12
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Вынесение общего множителя за скобки№702.а) mx + my = m(x + y);б) kx – px = x(k – p);в) –ab + ac = a(c – b);г) –ma – na = –a(m + n).Проверим:а) m(x + y) = mx + my;б) x(k – p) = kx – px;в) a(c – b) = –ab + ac;г) –a(m + n) = –ma – na.№703.а) 5x + 5y = 5(x + y); б) 4a – 4b = 4(a – b);в) 3c + 15d = 3(c + 5d);г) –6m – 9n = –3(2m + 3n); д) ax + ay = a(x + y); е) bc – bd = b(c – d);ж) ab + a = a(b + 1); з) cy – c = c(y – 1); и) –ma – a= –a(m + 1).83№704.а) 7a – 7y = 7(a – y); б) –8b + 8c = 8(c – b); в) 12x + 48y = 12(x + 4y);г) –9m–27n=–9(m+3n); д) 12a+12 = 12(a + 1); е) –10 – 10c = –10(1 + c).№705.а) 7ax+7bx = 7x(a + b); б) 3by – 6b = 3b(y – 2); в) –5mn + 5n = 5n(1 – m);г) 3a + 9ab = 3a(1 + 3b); д) 5y2 – 15y = 5y(y – 3); е) 3x + 6x2 = 3x(1 + 2x);ж) a2–ab=a(a–b); з) 8mn–4m2 = 4m(2n – m); и) –6ab2 + 9b2 = 3b(3b – 2a);к) x2y–xy2 = xy(x – y); л) ab – a2b = ab(1 – a); м) –p2q2 – pq = –pq(pq + 1).№706.а) a2 + a = a(a + 1); б) x3 – x2 = x2(x – 1); в) c5 + c7 = c5(1 + c2);г) a3–a7 = a3(1 – a4); д) 3m2 + 9m3 = 3m2(1 + 3m); е) 9p3 – 8p = p(9p2 – 8);ж) 4c2–12c4=4c2(1–3c2); з) 5x5–15x3=5x3(x2–3); и) –12y4–16y=–4y(3y3+4).№707.а) 14x+21y=7(2x+3y); б) 15a+10b=5(3a + 2b); в) 8ab – 6ac = 2a(4b – 3c);г) 9xa+9xb=9x(a + b); д) 6ab – 3a = 3a(2b – 1); е) 4x – 12x2 = 4x(1 – 3x);ж) m4 – m2 = m2(m2 – 1); з) c3 + c4 = c3(1 + c); и) 7x – 14x3 = 7x(1 – 2x2);к) 16y3 + 12y2 = 2y2(4y + 3); л) 18ab3 – 9b4 = 9b3(2a – b);м) 4x3y2 – 6x2y3 = 2x2y2(2x – 3y).№708.а) 3,28x–x2=x(3,28–x), при x=2,28, x(3,28 – x) = 2,28(3,28 – 2,28) = 2,38;б) a2y+a3=a2(y+a), при a=–1,5; y=–8,5, a2(y+a)=(–1,5)2 (–1,5+(–8,5))=–22,5;в) ay2–y3=y2(a–y), при a=8,8; y=–1,2, y2(a–y)=(–1,2)2(8,8 – (–1,2)) = 14,4;г) –mb–m2=–m(b+m), при m=3,48; b=96,52, –m(b+m)=–3,48(96,52+3,48)= –348.№709.
а) x2 + 8x = 0, x(x + 8) = 0, x1 = 0; x2 = –8;б) 5x2 – x = 0, x(5x – 1) = 0, x1 =0; x2 = 0,2;в) 6y2 – 30y = 0, 6y(y – 5) = 0, y1 = 0; y2 = 5;г) 3x2 – 1,2x = 0, 3x(x – 0,4) = 0, x1 = 0; x2 = 0,4;д) 6x2 – 0,5x = 0, x(6x – 0,5) = 0, x1 = 0; x2 =1;121 2⎛1⎞y + y = 0, y ⋅ ⎜ y + 1⎟ = 0, y1 = 0; y2 = –4;4⎝4⎠ж) x – 10x2 = 0, x(1 – 10x) = 0, x1 = 0; x2 = 0,1;з) 6x – 0,2x2 = 0, 2x(3 – 0,1x) = 0, x1 = 0; x2 = 30;2⎞22⎛и) y2 + y = 0, y ⋅ ⎜ y + ⎟ = 0, y1 = 0; y2 = – .3⎠33⎝№710. а) 5x2 + 3x = 0, x(5x + 3) = 0, x1 = 0; x2 = –0,6;б) x2 – 11x = 0, x(x – 11) = 0, x1 = 0; x2 = 11;в) 6x2 – 3,6x = 0, 6x(x – 0,6) = 0, x1 = 0; x2 = 0,6;г) 0,3x2 – 3x = 0, 3x(0,1x – 1) = 0, x1 = 0; x2 = 10;д) 5x2 – 0,8x = 0, x(5x – 0,8) = 0, x1 = 0; x2 = 0,16;е) 7x2 – 0,28x = 0, 7x(x – 0,04) = 0, x1 = 0; x2 = 0,04.№711.
а) 165 + 164 = 164(16 + 1) = 164 ⋅ 17 делится на 17;б) 389 – 388 = 388(38 – 1) = 388 ⋅ 37 делится на 37;5в) 36 – 69 = (62)5 – 69 = 610 – 69 = 69 ⋅ 5 = 68 ⋅ 6 ⋅ 5 = 68 ⋅ 30 кратно 30;г) 518 – 258 = 518 – (52)8 = 518 – 516 = 518(52 – 1) = 516 ⋅ 24 == 516 ⋅ 5 ⋅ 24 = 516 ⋅ 120 кратно 120.е)84№712.а) x5 + x4 – x3 = x3(x2 + x – 1); б) y7 – y5 – y2 = y2(y5 – y3 – 1);в) a4 + a5 – a8 = a4(1 + a – a4); г) –b10 – b15 – b20 = –b10(1 + b5 + b10).№713.а) 78 – 77 + 76 = 76(49 – 7 + 1) = 76 ⋅ 43 кратно 43;б) 213 – 210 – 29 = 29(24 – 2 – 1) = 29(16 – 3) = 29 ⋅ 13 кратно 13;в) 274 – 95 + 39 = (33)4 – (32)5 + 39 = 312 – 310 + 39 = 39(33 – 3 + 1) == 39(27 – 2) = 39 ⋅ 25 кратно 25;г) 164 – 213 – 45 = (24)4 – 213 – (22)5 = 216 – 213 – 210 == 210(26 – 23 + 1) = 210(64 – 8 – 1) = 210 ⋅ 55 = 210 ⋅ 5 ⋅ 11 кратно 11.№714.а) x3 – 3x2 + x = x(x2 – 3x + 1); б)m2 – 2m3 – m4 = m2(1 – 2m – m2);в) 4a5 – 2a3 + a = a(4a4 – 2a2 + 1); г) 6x2 – 4x3 + 10x4 = 2x2(3 – 2x + 5x2);д)15a3–9a2+6a=3a(5a2–3a+2); е) –3m2–6m3 + 12m5 = –3m2(1 + 2m – 4m3).№715.а) c3 – c4 + 2c5 = c3(1 – c + 2c2); б) 5m4 – m3 + 2m2 = m2 (5m2 – m + 2);в) 4x4 + 8x3 – 2x2 = 2x2(2x2 + 4x – 1); г) 5a – 5a2 – 10a4 = 5a(1 – a – 2a3).№716.а) 3a3–15a2b+5ab2=a(3a2–15ab+5b2); б) 20x4–25x2y2–10x3=5x2(4x2–5y2–2x);в) –6am2+9m3–12m4=6ab(2a–3m+4m2); г) 12a2b–18ab2–30ab3=6ab(2a–3b–5b2);д) 4ax3+8a2x2–12a2x=4ax(x2–2ax–3a); е) –3x4y2–6x2y2+9x2y4=–3x2y2(x2+2–3y2).№717.а) 4c4 – 6x2c2 + 8c = 2c(2c3 – 3x2c + 4);б) 10a2x – 15a3 – 20a4x = 5a2(2x – 3a – 4a2x);в) 3ax – 6ax2 – 9a2x = 3ax(1 – 2x – 3a);г) 8a4b3 – 12a2b4 + 16a3b2 = 4a2b3(2a2b – 3b2 + 4a).№718.а) 2a(x + y) +b(x + y) = (x + y)(2a + b);б) y(a – b) – (a – b) = (a – b)(y – 1);в) (c + 3) –x(c + 3) = (c + 3)(1 – x);г) 9(p – 1) + (p – 1)2 = (p – 1)(9 + p – 1) = (p – 1)(8 + p);д) (a + 3)2 – a(a + 3) = (a + 3)(a + 3 – a) = 3(a + 3);е) –3b(b – 2) + 7(b–2)2=(b–2)(–3b + 7b – 14)=(b – 2)(4b – 14) = 2(b – 2)(2b – 7).№719.а) a(b – c) +d(c – b) = a(b – c) – d(b – c) = (b – c)(a – d);б) x(y – 5) – y(5 – y) = x(y – 5) + y(y – 5)= (y – 5)(x + y);в) 3a(2x – 7) + 5b(7 – 2x) = 3a(2x – 7) – 5b(2x – 7) = (2x – 7)(3a – 5b);г) (x – y)2 – a(y – x) = (y – x)2 – a(y – x) = (y – x)(y – x – a);д) 3(a – 2)2 – (2 – a) = 3(2 – a)2 – (2 – a) = (2 – a)(6 – 3a – 1) = (2 – a)(5 – 3a);е) 2(3–b) + 5(b–3)2=2(3 – b) + 5(3 – b)2 = (3 – b)(2 +15 – 5b) = (3 – b)(17 – 5b).№720.а) 8m(a – 3) + n(a – 3) = (a – 3)(8m + n);б) (p2 – 5) – q(p2 – 5) = (p2 – 5)(1 – q);в) x(y – 9) + y(9 – y) = x(y – 9) – y(y – 9) = (y – 9)(x – y);г) 7(c + 2) + (c + 2)2 = (c + 2)(7 + c + 2) = (c + 2)(9 + c);д) (a – b)2 – 3(b – a) = (b – a)2 – 3(b – a) = (b – a)(b – a – 3);е) –(x + 2y) – 4(x + 2y)2 = –(x + 2y)(1 + 4x + 8y).85№721.из A в Bv12 км/ чиз B в A18 км/чtxч12xч18Sx кмx кмна1ч меньше4xx 1– = , 3x – 2x = 9, x = 9 км — расстояние от A до B.12 18 4Ответ: 9 км.№722.3x − 5 8 x − 12а)+= 9 , 7(3x – 5) + 2(8x – 12) = 126,27185, x = 5;21x – 35 + 16x – 24 = 126, 37x = 126 + 59, x =3721 − 4 x 8 x + 15б)−= 2 , 21 – 4x – 3(8x + 15) = 18,9321 – 4x – 24x – 45 = 18, –28x = 18 – 21 + 45, 28x = –42, x = –1,5.№723.11== −2 ;Т.к.
a – b =0,5, то а) b – a = –(a – b) = –0,5;б)b − a −0 ,5г) (b – a)2 = (a – b)2 =0,25;в) (a – b)2 = 0 ,52 =0,25;33е) (b – a)3 = –(a – b)3 = –0,125.д) ( a – b) = 0,5 =0,125;№724.а) (a – b)(a + b); б) a2 + b2; в) (a + b)2; г) b2 – c2; д) (b – c)3;е) b3 + c3.§ 11. Произведение многочленов28. Умножение многочлена на многочлен№725.а) (x + m)(y + n) = xy + xn + my + mn;б) (a – b)(x + y) = ax + ay – bx – by;в) (a – x)(b – y) = ab – ay – bx + by;г) (x + 8)(y – 1) = xy – x + 8y – 8;д) (b – 3)(a – 2) = ab – 2b – 3a + 6;е) (–a + y)(–1 – y) = a + ay – y – y2.№726.а) (x + 6)(x + 5) = x2 + 6x + 5x + 30 = x2 + 11x + 30;б) (a – 4)(a + 1) = a2 – 4a + a – 4 = a2 – 3a – 4;в) (2 – y)(y – 8) = 2y – 16 – y2 + 8y = –y2 + 10y – 16;г) (a – 4)(2a + 1) = 2a2 – 8a + a – 4 = 2a2 – 7a – 4;д) (2y – 1)(3y + 2) = 6y2 + 4y – 3y – 2 = 6y2 + y – 2;е) (5x – 3)(4 – 3x) = 20x – 15x2 – 12 + 9x = –15x2 + 29x – 12.№727.а) (m – n)(x + c) = mx + mc – nx – nc;б) (k – p)(k – n) = k2 – kn – kp + pn;22в) (a + 3)(a – 2) = a + 3a – 2a – 6 = a + a – 6;86г) (5 – x)(4 – x) = 20 – 4x – 5x + x2 = x2 – 9x + 20;д) (1 – 2a)(3a + 1) = 3a + 1 – 6a2 – 2a = – 6a2 + a + 1;е) (6m – 3)(2 – 5m) = 12m – 30m2 – 6 + 15a = –30m2 + 27m – 6.№728.MBCaKAObcLdND(a + b)(c + d) = ac + bd + ad + bd.SABCD = (a + b)(c + d) — площадь прямоугольника ABCDSKBMO = ab; SOMCL = AD; SAKON = bc; SOLDN = bd.SABCD = SKBMO + SOMCL + SAKON + SOLDNСумма площадей прямоугольников КМВО, АКОN, OLDN, OMCL равнаплощади ABCD.№729.а) (x2 + y)(x + y2) = x3 + x2y2 + xy + y3;б) (m2 – n)(m2 + 2n2) = m4 + 2m2n2 – m2n – 2n3;в) (4a2 + b2)(3a2 – b2) = 12a4 –4a2b2 + 3a2 b2 – b4 = 12a4 – a2b2 – b4;г)(5x2 – 4x)(x + 1) = 5x3 + 5x2 – 4x2 – 4x = 5x3 + x2 – 4x;д) (a – 2)(4a3 – 3a2) = 4a4 – 3a3 – 8a3 + 6a2 = 4a4 – 11a3 + 6a2;е) (7p2 – 2p)(8p – 5) = 56p3 – 35p2 – 16p2 + 10p = 56p3 – 51p2 + 10p.№730.а) (2x2 – y)(x2 + y) = 2x4 + 2x2y – x2y – y2 = 2x4 + x2y – y2;б) (7x2 + a2)(x2 – 3a2) = 7x4 – 21a2x2 + a2x2 – 3a4 = 7x4 – 20a2x2 – 3a4;в) (11y2 – 9)(3y – 2) = 33y3 – 22y2 – 27y + 18;г) (5a – 3a3)(4a – 1) = 20a2 – 5a – 12a4 + 3a3.№731.а) (x + 10)2 = (x + 10)(x + 10) = х2 + 10х + 10х + 100 = x2 + 20x + 100;б) (1 – y)2 = (1 – y)(1 – y) = 1 –у – у + у2 = 1 – 2y + y2;в) (3a – 1)2 = (3a – 1)(3a – 1) = 9а2 – 3а – 3а + 1 = 9a2 – 6a + 1;г) (5 – 6b)2 = (5 – 6b)(5 – 6b) = 25 – 30b – 30b + 36b2 = 25 – 60b + 36b2.№732.а) (x2 + xy – y2)(x + y) = x3 + x2y + x2y + xy2 – xy2 – y3 = x3 + 2x2y – y3;б) (n2 – np + p2)(n – p) = n3 – n2p – n2p + np2 + np2 – p3 = n3 – 2n2p + 2np2 – p3;в) (a + x)(a2 – ax – x2) = a3 – a2x – ax2 + a2x – ax2 – x3 = a3 – 2ax2 – x3;г) (b – c)(b2 – bc – c2) = b3 – b2c – bc2 – cb2 + bc2 + c3 = b3 – 2b2c + c3;д) (a2 – 2a + 3)(a – 4) = a3 – 4a2 – 2a2 + 8a + 3a – 12 = a3 – 6a2 + 11a – 12;е) (5x – 2)(x2 – x – 1) = 5x3 – 5x2 – 5x – 2x2 + 2x + 2 = 5x3 – 7x2 – 3x + 2;ж) (2 – 2x + x2)(x + 5) = 2x + 10 – 2x2 – 10x + x3 + 5x2 = x3 + 3x2 – 8x + 10;з) (3y – 4)(y2 – y + 1) = 3y3 – 3y2 + 3y – 4y2 + 4y – 4 = 3y3 – 7y2 + 7y – 4.87№733.а) (c2 – cd – d2)(c + d) = c3 + c2d – c2d – d3 = c3 – 2cd2 – d3;б) (x – y)(x2 – xy – y2) = x3 – x2y – xy2 + xy2 + y3 = x3 – 2x2y + y3;в) (4a2 + a + 3)(a – 1) = 4a3 + a2 – a + 3a – 3 = 4a3 – 3a2 + 2a – 3;г) (3 – x)(3x2 + x – 4) = 9x2 + 3x – 12 –3x3 – x2 + 4x = – 3x3 + 8x2 + 7x – 12.№734.а) (4n2– 6np + 9p2)(2n + 3p) = 8n3 + 12n2p – 12n2 p – 18np2 + 18np2 + 27p3;б) (25x2+10xy+4y2)(5x–2y)=125x3–50x2y + 50x2y – 20xy2+20xy2–8y3=125x3–8y3;в) (–2a2+3a+1)(3a–2)=–6a3 + 4a2 + 9a2 – 6a + 3a – 2 = –6a3 + 13a2 – 3a – 2;г) (7–2a)(4a2 + 4a + 3) = 28a2 + 28a + 21 – 8a2 – 6a–8a3=–8a3+20a2 + 22a + 21;д) (x2–x+2)(3x2+x–2)=3x4+x3–2x2–3x3–x2+2x+6x2 + 2x–4=3x4–2x3 + 3x2 + 4x – 4;е) (5 – 2a + a2)(4a2 – 3a – 1)=20a2–15a–5 – 8a3 + 6a2 + 2a + 4a2 – 3a3 – a2 == 4a4 – 11a3 + 25a2 – 13a – 5.№735.
а) y2(y + 5)(y – 3) = y2(y2 + 2y – 15) = y4 + 2y3 – 15y2;б) 2a2(a – 1)(3 – a) = 2a2(–a2 + 4a – 3) = –2a4 + 8a3 – 6a2;в) –3b3(b + 2)(1 – b) = –3b3(–b2 – b + 2) = 3b5 + 3b4 – 6b3;г) –0,5c2(2c – 3)(4 – c2) = –0,5c2(8c – 2c3 – 12 + 3c) = c5 – 1,5c4 – 4c3 + 6c2.№736.а)(x+1)(x+2)(x+3)=(x2+3x+2)(x+3)=x3+3x2+3x2+9x+2x+6=x3 + 6x2 + 11x + 6;б) (a – 1)(a – 4)(a + 5) = (a2 – 5a + 4)(a + 5) == a3 + 5a2 – 5a2 – 25a + 4a + 20 = a3 – 21a + 20.№737.а) (3b – 2)(5 – 2b) + 6b2 = 15b – 6b2 – 10 + 4b + 6b2 = 19b – 10;б) (7y – 4)(2y + 3) – 13y = 14y2 + 21y – 8y – 12 – 13y = 14y2 – 12;в) x3–(x2–3x)(x + 3) = x3 – (x3 + 3x2 – 3x2 – 9x) = x3 – x3 – 3x2 + 3x2 + 9x = 9x;г) 5b3 + (a2+5b)(ab – b2) = 5b3 + a3b – a2b2 + 2ab2 – 5b3 = a3b – a2b2 + 5ab2;д) (a–b)(a+2)–(a+b)(a – 2) = a2 + 2a – ab – 2b – a2 + 2a – ab + 2b=4a – 2ab;е) (x + y)(x – y) – (x – 1)(x – 2) = x2 + xy – xy – y2 – (x2 – 3x + 2) == x2 – y2 – x2 + 3x – 2 = –y2 + 3x – 2.№738.а) (2x – y)(y + 4x) + 2x(y – 3x) = 2xy + 8x2 – y2 – 4xy + 2xy – 6x2 = 2x2 – y2;б) (3a–2b)(2a–3b)–6a(a–b)=6a2 – 9ab – 4ab + 6b2 – 6a2 + 6ab = 6b2 – 7ab;в) 5a(2x–a)–(8a–x)(2x–a)=10ax–5a2–16ax + 8a2 + 2x2 – ax=3a2 – 7ax + 2x2;г) 2c(b+15c)+(b–6c)(5c+2b)=2bc+30c2 + 5bc + 2b2 – 30c2 – 12bc=2b2 – 5bc.№739.а) (8a – b)(a + 7b) – 55ab = 8a2 + 56ab – ab – 7b2 – 55ab = 8a2 – 7b2;б) (3x + 2y)(4x – y) + 2y2 = 12x2 – 3xy + 8xy – 2y2 + 2y2 = 12x2 + 5xy;в) (3p – 1)(2p + 5) – 6p(p – 2) = 6p2 + 15p – 2p – 5 – 6p2 + 12p = 25p – 5;г) (7m+3)(2m–1)–2m(7m – 1) = 14m2 – 7m + 6m – 3 – 14m2 + 2m = m – 3.№740.а) (x – 3)(x + 7) – (x + 5)(x – 1) = x2 + 4x – 21 – (x2 + 4x – 5) == x2 + 4x – 21 – x2 – 4x + 5 = –16 верно;б) x4 – (x2 – 7)(x2 + 7) = x4 – (x4 + 7x2 – 7x2 – 49) = x4 – x4 + 49 = 49 верно.№741.а) (x – 5)(x + 8) – (x + 4)(x – 1) = x2 + 3x – 40 – (x2 + 3x – 4) == x2 + 3x – 40 – x2 – 3x + 4 = – 36, не зависит от x;б) x4 – (x2 – 2)(x2 + 1) = x4 – (x4 – 1) = x4 – x4 + 1= 1, не зависит от х.88№742.(y – 6)(y + 8) – 2(y – 25) = y2 + 2y – 48 – 2y + 50 = y2 + 2 > 0,для любого y, т.к.
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