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– расстояние между ними.Ответ: 20 км.Пусть V = (Vв – Vп)№ 434км.– разность скорости велосипедиста и пешехода.ч.км.33Тогда V = 7 – 1;V = 6; V = 8– разность скоростей.44ч.1,5 · 8 = 12 км.Ответ: 12 км.Пусть V = (Vв – Vп)§ 16. Умножение многочлена на многочлен№ 435а) (x +1)(x + 2) = x(x + 1) + 2(x+1) = x2 + x + 2x + 1 = x2 + 3x + 1;б) (a – 3)(a + 8) = a(a + 8) – 3(a + 8) = a2 + 8a – 3a – 24 = a2 + 5a – 24;в) (b+10)(b–4) = b(b – 4) + 10(b – 4) = b2 – 4b + 10b – 40 = b2 + 6b – 40;г) (y – 5)(y – 9) = y(y – 5) – 9(y – 5) = y2 – 5y – 9y + 45 = y2 – 14y + 45.№ 436а) (x – 5)(9 – x) = 9(x – 5) – x(x – 5) = 9x – 45 – x2 + 5x = –x2 + 14x – 45;б) (y–10)(–y–6) = –y(y – 10) – 6(y – 10) = –y2 + 10y – 6y + 60=–y2 + 4y – 60;69в) (b + 10)(b – 4) = b(b – 4) + 10(b – 4) = b2 – 4b + 10b – 40 = b2 + 6b – 40;г) (–7 – b)(a – 4) = a( –7 – b) – 4( –7 – b) = –7a – ab + 28 + 4b.№ 437а) (2a + 4)(5a + 6) = 2a(5a + 6) + 4(5a + 6) = 10a2 +32a + 24;б) (7b – 3)(8b + 4) = 7b(8b + 4) – 3(8b + 4) = 56b2 + 4b – 12;в) (8c + 12)(3c – 1) = 8c(3c – 1) + 12(3c – 1) = 24c2 + 28c + 12;г) (15d + 27)(–5d – 9) = –75d2 – 135d – 135d – 243 = –75d2 – 270d – 243.№ 438а) (m2 + n)(m + n) = m2 (m + n) + n(m + n) = m3 + m2n + mn + n;б) (2x2 – 1)(x + 3) = 2x2 (x + 3) – (x + 3) = 2x3 + 6x2 – x + 3;в) (3y2 + 5)(y – 6) = 3y2 (y – 6) + 5(y – 6) = 3y3 + 5y – 18y2 – 30;г) (7c2 – 1)(c – 3) = 7c2 (c – 3) – (c – 3) = 7c3 – 21c2 – c + 3.№ 439а) (3a + 5)(3a – 6) + 30 = 9a2 + 15a – 18a – 30 + 30 = 9a2 – 3a;б) (8 – y)(8 + y) – (y2 + 4) = 64 – y2 –y2 – 4 = 60 – 2y2;в) x(x – 3) + (x + 1)(x + 4) = x2 – 3x + x2 + 5x + 4 = 2x2 + 2x + 4;г) (c + 2)c – (c + 3)(c – 3) = c2 + 2c – c2 + 9 = 2c + 9.№ 440а) 0,3a(4a2 – 3)(2a2 + 5 = 0,3a(8a4 – 6a2 + 20a2 – 15) == 0,3(8a4 + 14a2 – 15) = 2,4a4 + 4,2a2 – 4,5;б) 1,5x(3x2 – 5)(2x2 + 3) = 1,5x(6x4 – x2 – 15 = 9x5 – 1,5x3 – 22,5x;в) 3p(2p + 4) · 2p(2p – 3) = 6p2 (4p2 + 2p – 12) = 24p4 + 12p3 – 72p2;г) –0,5y(4 – 2y2)(y2 + 3) = –0,5y(4y2 – 2y4 + 12 – 6y2) = y3 + y5 – 6y.№ 441а) (3m3+5)(3m2–10)=3m3 (3m2–10) + 5(3m2 – 10)=9m5 + 15m2 – 30m3 – 50;б) (4n5 – 1)(2n3 + 3) = 4n5 (2n3 + 3) – (2n3 + 3) = 8n8 – 2n3 + 12n5 – 3;в) (5k4 + 2)(6k2 – 1) = 5k4 (6k2 – 1) + 2(6k2 – 1) = 30k6 + 12k2 – 5k4 – 2;г) (6p8 – 4)(2p2 + 5) = 6p8(2p2 + 5) – 4(2p2 + 5) = 12p10 – 8p2 + 30p8 –20.№ 442а) (a + 2)(a2 – a – 3) = a3 + 2a2 – a2 – 2a – 3a – 6 = a3 + a2 – 5a – 6;б) (5b – 1)(b2 – 5b + 1) 5b3 – b2 – 25b2 + 5b + 5b – 1 = 5b3 – 26b2 + 10b – 1;в) (m – n + 1)(m + n) = m2 + mn – mn – n2 + m + n = m2 – n2 + m + n;г) (c – 2d)(c + 2d – 1) = (c – 2d)(c + 2d) – c + 2d = c2 – 4d2 – c + 2d.№ 443а) (x2 – xy + y2)(x + y) =x(x2 – xy + y2) + y(x2 – xy + y2) == x3 – x2y + y2x + yx2 – x2y + y3 = x3 + y3;б) (n2 + np + p2)(n – p) = n(n2 + np + p2) – p(n2 + np + p2) == n3 + n2p + np2 – pn2 – np2 – p3 = n3 – p3;в) (a + x)(a2 + ax + x2) = a(a2 + ax + x2) + x(a2 + ax + x2) == a3 + a2x + ax2 + xa2 + ax2 + x3 = a3 + x3 + 2ax2 + 2a2x;70г) (c2 + cd + d2)(c – d) = c(c2 + cd + d2) – d(c2 + cd + d2) == c3 + c2d + cd2 – dc2 – d2c – d3 = c3 – d3.№ 444а) (2a+ 3b)(4a2 – 6ab + 9b2) = 2a(4a2 – 6ab + 9b2) + 3b(4a2 – 6ab + b2) == 8a3 – 12a2b + 18ab2 + 12a2b – 18ab2 + 27b2 = 8a3 + 27b3;б) (5x + 2y)(25x2 + 10xy + 4y2)=5x(25x2+10xy+4y2)+2y(25x2 + 10xy + 4y2) == 125x3 + 50x2y + 20xy2 + 50yx2 + 20xy2 + 8y3;в) (5 – 2a + a2)(4a2 – 3a – 1) = 20a2 – 8a3 + 4a4 – 15a + 6a2 – 3a3 – 5 ++ 2a – a2 = 4a4 – 11a3 + 25a2 – 13a – 5;г) (m2 – m + 2)(3m2 + m – 2) = 3m4 – 3m3 + 6m2 + m3 – m2 + 2m – 2m2 ++ 2m – 4 = 3m4 – 2m3 + 3m2 + 4m – 4.№ 445а) (a–1)(a–2)–(a–5)(a + 3) = a2 – 3a + 2 – a2 + 2a + 15 = 17 – a=17+8 = 27;б) (a–3)(a+4)–(a+2)(a+5)=a2 + a – 12 – a2 – 7a – 10= –6a–24=1–24 = – 23;в) (a–7)(a+4)–(a+3)(a–10)=a2 – 3a – 28 – a2 + 7a + 30=4a+2=–0,6+2 = 1,4;г) (a + 2)(a + 5) – (a + 3)(a + 4) = a2 + 7a + 10 – a2 – 7a – 12=10 – 12 = –2.№ 446а) 12x2 – (4x – 3)(3x + 1) = –2; 12x2 – 12x2 + 5x + 3 = –2;5x + 5 = 0; x = – 1.Ответ: –1.б) (x + 1)(x + 2) – (x + 3)(x + 4) = 0; x2 + 3x + 2 – x2 – 7x – 12 = 0;–4x – 10 = 0; x = – 2,5.Ответ: – 2,5.в) 10x2 – (2x – 3)(5x – 1) = 31; 10x2 – 10x2 + 17x – 3 = 31;17x = 34; x = 2.Ответ: 2.г) 16x2 – (4x – 1)(4x – 3) =13; 16x2 – 16x2 + 16x – 3 = 13;16x = 16; x = 1.Ответ: 1.№ 447а) (3x + 5)(4x – 1) = (6x – 3)(2x + 7); 12x2 + 17x – 5 = 12x2 + 36x – 21;16.19x = 16; x =1916Ответ:.19б) (5x – 1)(2 – x) = (x – 3)(2 – 5x); 11x – 2 – 5x2 = 17x – 6 – 5x2;26x =4; x = .32Ответ: .371в) (5x + 1)(2x – 3) = (10x – 3)(x + 1); 10x2 – 13x – 3 = 10x2 + 7x – 3;20x = 0; x = 0.Ответ: 0.г) (7x – 1)(x + 5) = (3 + 7x)(x + 3); 7x2 + 34x – 5 = 7x2 + 24x + 9;10x = 14; x = 1Ответ: 12.52.5№ 448Пусть х м.

– длина прямоугольника.Тогда (х – 20) м. – его ширина.(x – 10)(x – 20 + 6 = x(x – 20) + 12;x2 – 10x – 20x + 200 + 6x = x2 – 20x + 12;–4x + 200 = 12; 4x = 188; x = 47;47 – 20 = 27 м. – ширина прямоугольника.Ответ: 47; 27.№ 449Пусть х – наименьшее натуральное число из этих чисел.Тогда (x + 1) – второе число.(х + 2) – третее число.(х + 3) – четвертое число.(x + 2)(x + 3) – x(x + 1) = 58; x2 + 5x + 6 – x2 – x = 58; 4x = 52; x = 13;13 + 1 = 14 – второе число.13 + 2 = 15 – третее число.13 + 3 = 16 – третее число.Ответ: 13; 14; 15; 16.№ 450Пусть х см.

– длина прямоугольника.⎛ 60⎞Тогда ⎜ − x ⎟ см. – ширина прямоугольника.2⎝⎠(x + 10)(30 – x – 6) = x(30 – x) – 32;(x + 10)(24 – x) = 30x – x2 – 32; 14x + 240 – x2 = 30x – x2 – 32; 16x = 272;x = 17 см. – длина прямоугольника.Из того что площадь прямоугольника равна произведению его длины наширину следует, что S = 17 · (30 – 17) = 17 · 13 = 221 см2.Ответ: 221 см2.№ 451Пусть х – меньшее число.Тогда (х + 1) – второе число.(х + 2) – третее число.x2 + 65 = (a + 1)(a +2); x2 + 65 = x2 + 3a + 2; 3x = 63;72x = 21 – первое число.21 + 1 = 22 – второе число.21 + 2 = 23 – третее число.Ответ: 21; 22; 23.№ 452а) a(3a2–4)(3a2+4)=(3a3 – 4a)(3a2 + 4) = 9a5 – 12a3 + 12a3 – 16a=9a5–16a;б) (a – 5)(a + 5)(a2 + 25) = (a2 – 5a + 5a – 25)(a2 + 25) == (a2 – 25)(a2 + 25) = a4 – 25a2 + 25a2 – 625 = a4 – 625;в) (a2 + 9)(a +3)(a – 3) = (a2 + 9)(a2 – 3a + 3a – 9) = (a2 + 9)(a2 – 9) == a4 – 9a2 + 9a2 – 81 = a4 – 81;г) (a2 + 16)(a – 4)(a + 4) = (a2 + 16)(a2 – 4a + 4a – 16) == (a2 + 16)(a2 – 16) = a4 – 16a2 + 16a2 – 256 = a4 – 256.№ 453а) (3,5p–1,2k)(3,5p + 1,2k) = 12,25p2 – 4,2p + 4,2p – 1,44 = 12,25p2 – 1,44;б) (1,7s+0,3t2)(0,3t2–1,7s)=0,09t4 – 0,51st2 + 0,51st2 – 2,89s2=0,09t4–2,89s2;в) (2,4m2–0,8n2)(0,8n2+2,4m2)=5,76m4–1,92m2n2+1,92m2n2–0,64n4 == 5,76m4 – 0,64n4;г) (1,3x3–1,8y2)(1,8y2+1,3x3)=1,69x6–2,34y2x3+2,34y2x3–3,24y4=1,69x6–3,24y4.№ 454а) (a2 + a – 1)(a2 – a + 1) = a4 – a3 + a2 + a3 – a2 + a – a2 + a – 1 == a4 – a2 + 2a – 1;б) (m2 + 2m – 1)(m2 – 2m + 1) = m4 + 2m3 – m2 – 2m3 – 4m2 + 2m + m2 ++ 2m – 1 = m4 – 4m2 + 4m – 1;в) (2x2 + 3x + 2)(–2x2 + 3x – 2) == – 4x4 – 6x3 – 4x2 + 6x3 + 9x2 + 6x – 4x2 – 6x – 4 = –4x4 + x2 – 4;г) (b3 + 5b + 3)(– b3 – 5b + 3) = –b6 – 5b4 –3b3 – 5b4 – 25b2 – 15b ++ 3b3 + 15b + 9 = 9 – b6 – 10b4 – 25b2.№ 455а) (m – 1)(m3 + m2 + m +1) =m4 – m3 + m3 – m2 + m2 – m + m – 1=m4 – 1;б) (2 – s)(16 + 8s + 4s2 + 2s3 + s4) == 32 – 16s + 16s – 8s2 + 8s2 – 4s3 + 4s3 – 2s4 + 2s4 – s5 = 32 – s5;в) (x + y)(x3 – x2y + xy2 – y3) == x4 + x3y – x3y – x2y2 +x2y2 + xy3 – xy3 – y4 = x4 – y4;г) (a + 3)(81 – 27a + 9a2 – 3a3 + a4) = 81a + 243 – 27a2 – 81a + 9a3 ++ 21a2 – 3a4 – 9a3 + a5 + 3a4 = 243 – 6a2 + a5.№ 456Пусть х см.

– длина первого прямоугольника.Тогда (61 – х) – ширина первого прямоугольника;(х – 5) см. – длина второго прямоугольника;(61 – (х – 5) см. – ширина второго прямоугольника;(x – 5)(66 – x) – 120 = x(61 – x); 66x – 330 + 5x – x2 – 120 = 61x – x2;739x = 450; x = 50 см.Из того что площадь прямоугольника равна произведению его сторонследует: 50 · (61 – 50) = 50 · 11 = 550 см2 – площадь первого;550 + 120 = 670 см2 – площадь второго прямоугольника.Ответ: 550 см2; 670 см2.№ 457Пусть х см.

– длина прямоугольника.Тогда (120 – х) см. – ширина прямоугольника.(x – 14)(130 – x) – 4 = x(120 – x);130x – 1820 + 14x – x2 – 4 = 120x – x2; 24x = 1824;x = 76 см. – длина прямоугольника;120 – 76 = 44 см. – ширина прямоугольника.Ответ: 44 см.; 76 см.№ 458Пусть х – первое число.Тогда (х + 3) – второе число; (х + 6) – третее число.x(x + 6) + 54 = (x + 3)(x + 6); x2 + 6x + 54 = x2 + 9x + 18; 3x = 36;x = 12 – первое число; 12 + 3 = 15 – второе число;12 + 6 = 18 – третее число.Ответ: 12; 15; 18.№ 459Пусть х – первое число.Тогда (х + 12) – второе число; (х + 24) – третее число.x(x + 12) + 432 = (x + 12)(x + 24); x2 + 12x + 432 = x2 + 36x + 288;24x = 144; x = 6 – первое число; 6 + 12 = 18 – второе число;6 + 24 = 18 – третее число.Ответ: 6; 12; 18.№ 460Пусть х – первое число.Тогда (х + 3) – второе число; (х + 8) – третее число;(х + х + 3) – четвертое число.x(x+3)=(x + 8)2 – (2x + 3) – 74,2; x2 + 3x = x2 + 8x + 8x + 64 – 2x – 3 – 74,2;8x + 8x – 2x – 3x = –64 + 3 + 74,2; 11x = 13,2;x = 1,2 – первое число; 1,2 + 3 = 4,2 – второе число;1,2 + 8 = 9,2 – третее число; 2 · 1,2 + 3 = 5,4 – четвертое число.Ответ: 1,2; 4,1; 9,2; 5,4.§ 17.

Формулы сокращенного умножения№ 461а) (a + x) 2 = a2 + 2ax + x2; б) (b – y) 2 = b2 – 2by + y2;в) (c + d) 2 = c2 + 2cd + d2; г) (m – n) 2 = m2 – 2mn + n2.74№ 462а) (x + 1) 2 = x2 + 2x + 1;в) (a – 5) 2 = a2 – 10a + 25;б) (y – 2) 2 = y2 – 4y + 4;г) (c + 8) 2 = c2 + 16c + 64.№ 463а) (7 – a) 2 = 49 – 14a + a2;в) (4 + n) 2 = 16 + 8n + n2;б) (9 + b) 2 = 81 + 18b + b2;г) (12 – p) 2 = 144 – 24p + p2.№ 464а) (–x + 1) 2 = 1 – 2x + x2;б) (–z – 3) 2 = z2 + 6z + 9;22в) (–n + 8) = 64 – 16n + n ; г) (–m – 10) 2 = m2 – 20m + 100.№ 465а) (2a + 1) 2 = 4a2 + 4a + 1; б) (3c – 2) 2 = 9c2 – 12c + 4;в) (6x – 3) 2 = 36x2 – 36x + 9; г) (7y + 6) 2 = 49y2 + 84y + 36.№ 466а) (8x + 3y) 2 = 64x2 + 48xy + 9y2;в) (9p – 2q) 2 = 81p2 – 36pq + 4q2;б) (6m – 4n) 2 = 36m2 – 48mn + 16n2;г) (10z + 3t) 2 = 100z2 + 60zt + 9t2.№ 467а) ( –3a + 5x)2 = 25x2 – 30ax + 9a2;б) (– 6y – 2z)2 = 36y2 + 24yz +4z2;222в) (– 3m + 4n) = 16n – 24mn + 9m ; г) (–12z – 3t)2 =9t2 + 72tz + 144z2.№ 468а) (0,2x – 0,5a) 2 = 0,04x – 0,2xa + 0,25a2;221⎞311⎛1⎞⎛; в) ⎜ 6a − ⎟ = 36a2 – 2a +;б) ⎜ + 3n ⎟ = 9n2 + mn +216366⎠⎝4⎠⎝г) (10c + 0,1) 2 = 100c + 2cy + 0,01y2.№ 469а) (x2 + 1) 2 = x4 + 2x2 + 1;в) (p2 – 10) 2 = p4 – 20p2 + 100;б) (y2 – 6) 2 = y4 – 12y2 + 36;г) (q2 + 8) 2 = q4 + 16q2 + 64.№ 470а) (a2 + 3x) 2 = a4 + 6a2x + 9x2;в) (m2 – 6n) 2 = m4 – 12m2n + 36n2;б) (b2 – 5y) 2 = b4 – 10b2y + 25y2;г) (r2 + 4s) 2 = r4 + 8r2s + 16s2.№ 471а) (c2 + d2) 2 = c4 + 2c2d2 + d4;в) (p2 – q2) 2 = p4 – 2p2q2 + q4;б) (m2 – n3) 2 = m4 – 2m2n3 + n6;г) (z2 + t3) 2 = z4 + 2z2t3 + t6.№ 472а) (a3 + 3b) 2 = a4 + 6a2b + 9b2;б) (4x2 – 3c) 2 = 16x4 – 24x2x + 9c2;в) (5m2 + 3n2) 2 = 25m4 + 30m2n2 + 9n4;г) (6p2 – 8q3) 2 = 36p4 – 96p2q3 + 64q6.75№ 473221 ⎞15 ⎞49 2225 2⎛7⎛ 1а) ⎜ 2 a − 1 b ⎟ = ⎜ a − b ⎟ =a – 5ab +b =3143149196⎝⎠⎝⎠429= 5 a2 – 5ab 1;91962213 ⎞ ⎛40 ⎞21600 2⎛y =б) ⎜ 0,9 x + 1 y ⎟ = ⎜ 0,9 x + 1 y ⎟ = 0,81x2 + 2 xy +27 ⎠ ⎝27 ⎠3729⎝2142= 0,81x2 + 2 xy + 2;3729221 ⎞25 ⎞625 2⎛⎛y ⎟ =1,44x2 + 10xy +y =в) ⎜ − 1,2 x − 4 y ⎟ = ⎜1,2 x +6 ⎠6 ⎠36⎝⎝13= 1,44x + 10xy + 17;36222 ⎞23 ⎞625 2⎛ 25⎛y⎟ =b – 5ab + 5,29a2 =г) ⎜ − 2,3a − 1 b ⎟ = ⎜ b −529232310⎝⎠⎝⎠96 22=1b – 5ab + 5,29a .529№ 474а) 792 = (80 – 1) 2 = 6400 – 160 + 1 = 6241;б) 392 = (40 – 1) 2 = 1600 – 80 + 1 = 1521;в) 592 = (60 – 1) 2 = 3600 – 120 + 1 = 3481;г) 692 = (70 – 1) 2 = 4900 – 140 + 1 = 4761.№ 475а) 212 = (20 + 1) 2 = 400 + 40 + 1 = 441;б) 312 = (30 + 1) 2 = 900 + 60 + 1 = 961;в) 612 = (60 + 1) 2 = 3600 + 120 + 1 = 3721;г) 912 = (90 + 1) 2 = 8100 + 180 + 1 = 8281.№ 476а) 422 = (40 + 2) 2 = 1600 + 160 + 4 = 1764;б) 622 = (60 + 2) 2 = 3600 + 240 + 4 = 3844;в) 822 = (80 + 2) 2 = 6400 + 320 + 4 = 6724;г) 322 = (30 + 2) 2 = 900 + 120 + 4 = 1024.№ 477а) 982 = (100 – 2) 2 = 10000 – 400 – 4 = 9596;б) 282 = (30 – 2) 2 = 900 – 120 – 4 = 776;в) 882 = (90 – 2) = 8100 – 360 + 4 = 7736;г) 582 = (60 – 2) 2 = 3600 – 240 – 4 = 3356.76№ 47822221⎞11⎛⎛ 1⎞а) ⎜12 ⎟ = ⎜12 + ⎟ = 144 + 2 += 146;1212144144⎝⎠⎝⎠2⎞2⎞44⎛⎛;б) ⎜ − 7 ⎟ = ⎜ − 7 − ⎟ = 49 + 4 += 53774949⎝⎠⎝⎠223⎞3⎞99⎛⎛в) ⎜ − 13 ⎟ = ⎜ − 13 − ⎟ = 169 + 6 +;= 17513 ⎠13 ⎠169169⎝⎝223⎞99⎛ 3⎞⎛г) ⎜ 7 ⎟ = ⎜ 7 + ⎟ = 49 + 3 +.= 5214 ⎠196196⎝ 14 ⎠⎝№ 47922221⎞11⎛ 12 ⎞⎛;а) ⎜12 ⎟ = ⎜13 − ⎟ = 169 − 2 += 1691313169169⎝⎠⎝⎠2⎞44⎛ 13 ⎞⎛б) ⎜14 ⎟ = ⎜15 − ⎟ = 225 − 4 +;= 22115 ⎠225225⎝ 15 ⎠⎝221 ⎞11⎛ 39 ⎞⎛в) ⎜ 39 ⎟ = ⎜ 40 − ⎟ = 1600 − 2 +;= 159840 ⎠16001600⎝ 40 ⎠⎝2293⎞9⎛ 13 ⎞⎛г) ⎜15 ⎟ = ⎜16 − ⎟ = 256 − 6 +.= 25016 ⎠256256⎝ 16 ⎠⎝№ 480а) (a – b)(a + b) = a2 – b2;в) (m – n)(m + n) = m2 – n2;б) (c – d)(c + d) = c2 – d2;г) (p – q)(p + q) = p2 – q2.№ 481а) (x – 1)(x + 1) = x2 – 1;в) (10m – 4)(10m + 4) = 100m2 – 16;б) (c – 2)(c + 2) = c2 – 4;г) (8a – 1)(8a + 1) = 64a2 – 1.№ 482а) (3b – 1)(3b + 1) = 9b2 – 1;в) (10m – 4)(10m + 4) = 100m2 – 16;б) (6x – 2)(6x + 2) = 36x2 – 4;г) (8a – 1)(8a + 1) = 64a2 – 1.№ 483а) (4a – b)(4a + b) = 16a2 – b2;в) (4b + 1)(4b – 1) = 16b2 – 1;б) (x + 7)(7 – x) = 49 – x2;г) (5m + 2)(5m – 2) = 25m2 – 4.№ 484а) (3x – 5y)(3x + 5y) = 9x2 – 25y2;б) (7a – 8b)(7a + 8b) = 49a2 – 64b22в) (13c – 11d)(13c + 11d) = 169c – 121d2;г) (8m – 9n)(8m + 9n) = 64m2 – 81n2.77№ 485а) (5x – 2y2)(5x + 2y2) = 25x2 – 4y4;б) (2c – 3a2)(2c + 3a2 = 4c2 – 9a4;3362в) (10p –7q)(10p +7q)=100p – 49q ; г) (8d + 6c3)(6c3 – 8d) = 36c6 – 64d2.№ 486а) (4x2–2y2)(4x2+2y2)=16x4–4y4; б) (10a3+5b2)(10a3–5b2)=100a6–25b4;в) (3n4–m4)(3n4 + m4) = 9n8 – m8; г) (10m8+8n8)(10m8–8n8)=100m16–64n16.№ 487а) 69 · 71 = (70 – 1)(70 + 1) = 4900 – 1 = 4899;б) 31 · 29 = (30 + 1)(30 – 1) = 900 – 1 = 8999;в) 89 · 91 = (90 – 1)(90 + 1) = 8100 – 1 = 8099;г) 99 · 101 = (100 – 1)(100 + 1) = 10000 – 1 = 9999.№ 488а) 58 · 62 = (60 – 2)(60 + 2) = 3600 – 4 = 3596;б) 82 · 78 = (80 + 2)(80 – 2) = 6400 – 4 = 6396;в) 42 · 38 = (40 + 2)(40 – 2) = 1600 – 4 = 1596;г) 18 · 22 = (20 – 2)(20 + 2) = 400 – 4 = 396.№ 489а) 0,49 · 0,51 = (0,5 – 0,01)(0,5 + 0,01) = 0,25 – 0,0001 = 0,2499;б) 0,78 · 0,82 = (0,8 – 0,02)(0,8 + 0,02) = 0,64 – 0,0004 = 0,6396;в) 0,67 · 0,73 = (0,7 – 0,03)(0,7 + 0,03) = 0,49 – 0,0009 = 0,4891;г) 1,21 · 1,19 = (1,2 + 0,01)(1,2 – 0,01) = 1,44 – 0,01 = 1,4399.№ 49021⎞91 5 ⎛ 13 3 ⎞⎛ 13 3 ⎞ ⎛а) 10 ⋅ 9 = ⎜ 9 + ⎟⎜ 9 − ⎟ = ⎜10 − ⎟ −=14 ⎠ 1967 7 ⎝ 14 14 ⎠⎝ 14 14 ⎠ ⎝1019208280 + 8= 100 − +−= 100 −−= 100 −=7 196 19614 1961969210426;= 100 − 1= 98= 9819619649452 5 ⎛2 ⎞⎛2⎞4= 80 ;б) 10 ⋅ 9 = ⎜ 9 + ⎟⎜ 9 − ⎟ = 81 −7 7 ⎝7 ⎠⎝7⎠4949821 ⎛1 ⎞⎛1⎞1в) 99 ⋅100 = ⎜100 + ⎟⎜100 − ⎟ = 10000 − = 9999 ;33 ⎝3 ⎠⎝3⎠9944 1 ⎛1 ⎞⎛1⎞124= 63,96.г) 7 ⋅ 8,2 = 7 ⋅ 8 = ⎜ 8 − ⎟⎜ 8 + ⎟ = 64 −= 6355 5 ⎝5 ⎠⎝5⎠2525№ 491а) (x – 1)(x2 + x + 1) = x3 – 1;в) (x + 3)(x2 – 3x + 9) = x3 – 27;78б) (x – 2)(x2 + 2x + 4) = x3 – 8;г) (x + 4)(x2 – 4x + 16) = x3 – 64.№ 492а) (5m + 3n)(25m2 – 15mn + 9n2) = 125m3 + 27n3;б) (2a – 3x)(4a2 + 6ax + 9x2) = 8a3 – 27x3;в) (4x – 5y)(16x2 + 20xy + 25y2) = 64x3 – 125y3;г) (3x + 5y)(9x2 – 15xy + 25y2) = 27x3 + 125y3.№ 493а) 3(x – y) 2 = 3x2 – 6xy + 3y2;б) b(1 +2b) 2 = b(1 + 4b + 4b2 = b + 4b2 + 4b3;в) –6(5m – n) 2 = –6(25m2 – 10mn + n2) = – 150m2 + 60mn – 6n2;г) –c(3a + c) 2 = –c(9a2 + 6ac + c2) = –9a2c – 6ac2 – c3.№ 494а) a2 + (3a – b) 2 = a2 + 9a2 – 6ab + b2 = 10a2 – 6ab + b2;б) 9p2 – (q – 3p) 2 = 9p2 – q2 + 6qp – p2 = 8p2 + 6qp – q2;в) (5c + 7d) 2 – 70cd = 25c2 + 70cd + 49d2 – 70cd = 25c2 + 49d2;г) (8m – n) 2 – 64m2 = 64m2 – 16mn + n2 – 65m2 = n2 – 16mn.№ 495а) (a – 4) 2 + a(a + 8) = a2 – 8a + 16 + a2 + 8a = 2a2 + 16;б) (x – 7)x + (x +3) 2 = x2 – 7x + x2 + 6x + 9 = 2x2 – x + 9;в) (y – 5) 2 – (y – 2) = y2 – 10y + 25 – y + 2 = y2 – 11y + 27;г) b(b +4) – (b + 2) 2 = b2 + 4b – b2 – 4b – 4 = –4.№ 496а) (3a – b)(3a + b) + b2 = 9a2 – b2 + b2 = 9a2;б) 9x2 – (y +4x)(y – 4x) = 9x2 – y2 + 16x2 = 25x2 – y2;в) (5c – 6d)(5c + 6d) – 25c2 = 25c2 – 36d2 – 25c2 = – 36d2;г) (7m – 10n)(7m + 10n) – 100n2 = 49m2 – 100n2 – 100n2 = 49m2 – 200n2.№ 497а) 2(a – 2)(a + 2) = 2(a2 – 4) = 2a2 – 8;б) x(x + 4)(x – 4) = x(x2 – 16) = x3 – 16x;в) 5c(c + 3)(c – 3) = 5c(c2 – 9) = 5c3 – 45c;г) 7d2 (d – 1)(d + 1) = 7d2 (d2 – 1) = 7d4 – 7d2.№ 498а) (a – c)(a + c) – (a – 2c) 2 = a2 – c2 – a2 + 4ac – 4c2 = –5c2 + 4ac;б) (x – 4)(x + 4) – (x + 8)(x – 8) = x2 – 16 – x2 + 64 = 48;в) (3b – 1)(3b + 1) – (b – 5)(b + 5) = 9b2 – 1 – b2 + 25 = 8b2 + 24;г) (m+3n) 2 + (m + 3n)(m – 3n) = m2 + 6mn + 9n2 + m2 – 9n2 = 2m2 + 6mn.№ 499а) (b – 5)(b + 5)(b2 + 5) = (b2 – 25)(b2 + 25) = b4 – 625;б) (3 – y)(3 + y)(9 + y2) = (9 – y2)(9 + y2) = 81 – y4;в) (a – 2)(a + 2)(a2 – 4) = (a2 – 4)(a2 + 4) = a4 – 16;г) (c2 – 1)(c2 + 1)(c4 + 1) = (c4 – 1)(c4 + 1) = c8.79№ 500(2a – b)(2a + b) + (b – c)(b + c) + (c – 2a)(c + 2a) == 4a2 – b2 + b2 – c2 + c2 – 4a2 = 0.№ 501а) (a+3) 2–(a – 2)(a + 2) = a2 + 6a + 9 – a2 + 4 = 6a + 13=–3,5 · 6 + 13 = –8;б) (x–3) 2–(x +3)(x – 3) = x2 – 6x + 9 – x2 + 9 = 18 – 6x = 18 + 0,1 · 6 = 18,6;в) (m+3) 2–(m – 9)(m + 9) = m2 + 6m + 9 – m2 + 81 = 6m + 100=100–3 = 97;г) (c + 2) 2 – (c +4)(c – 4) = c2 +4c + 4 – c2 + 16 = 4c + 20 = 1 +20 = 21.№ 502а) (5a – 10) 2 – (3a – 8) 2 + 132a = 25a2 – 100a + 100 – 9a2 + 48a – 64 ++ 132a = 16a2 + 80a + 36 = 16a2 + 48a + 36 + 32a = (4a + 6) 2 + 32a == 182 – 192 = 324 – 192 = 132;б) (3p – 8) 2 + (4p + 6) 2 + 100p = 9p2 – 48p + 64 + 16p2 + 48p + 36 ++ 100p = 25p2 + 100p + 100 = (5p + 10) 2 = 0;в) (5b – 3) 2 + (12b – 4) 2 – 4b = 25b2 – 30b + 9 + 144b2 – 96b + 16 –– 4b = 169b2 – 130b + 25 = (13b – ) 2 = 182 = 324;г) (13 – 5m) 2 – (12 – 4m) 2 + 4m = 169 – 130m + 25m2 – 144 + 96m –– 16m2 + 4m = 9m2 – 30m + 25 = (3m – 5) 2 = 72 = 49.№ 503а) 8x(1 + 2x – (4x + 3)(4x – 3) = 2x; 8x + 16x2 – 16x2 + 9 = 2x;36x = – 9; x = – .23Ответ: – .2б) x – 3x(1 – 12x) = 11 – (5 – 6x)(5 + 6x); x – 3x + 36x2 = 11 – 25 + 36x2;–2x = –14; x = 7.Ответ: 7.в) (6x – 1)(6x + 1) – 4x(9x + 2) = – 1; 36x2 – 1 – 36x2 – 8x = –1;–8x = 0; x = 0.Ответ: 0.г) (8 – 9x)x = –40 + (6 – 3x)(6 + 3x) ; 8x – 9x2 = – 40 + 36 – 9x2;8x = –4; x = –Ответ: –1.21.2№ 504а) (x – 6) 2 – x(x + 8) =2; x2 – 12x + 36 – x2 – 8x = 2;20x = 34; x = 1Ответ: 1,7.807.10б) 9x(x + 6) – (3x + 1) 2 = 1; 9x2 + 54x – 9x2 – 6x – 1 = 1;148x = 2; x =.241.Ответ:24в) x(x – 1) – (x – 5) 2; x2 – x – x2 + 10x – 25 = 0;25.9x = 25; x =97Ответ: 2 .9г) 16x(2 – x) + (4x – 5) 2 = 1; 32x – 16x2 + 16x2 – 40x + 25 = 1;–8x = –24; x = 3.Ответ: 3.№ 505а) 9x2 – 1 – (3x – 2) 2 = 0; 9x2 – 1 – 9x2 + 12x – 4 = 0;5.12x = 5; x =125Ответ:.12б) x + (5x +2) 2 = 25(1+ x2); x + 25x2 + 20x + 4 = 25 + 25x2;21x = 21; x = 1.Ответ: 1.в) (2x – 3) 2 – 2x(4 + 2x) = 11; 4x2 – 12x + 9 – 8x – 4x2 = 11;–20x = 2; x = –2.20Ответ: –0,1.г) (4x – 3)(3 + 4x) – 2x(8x – 1) = 0; 16x2 – 9 – 16x2 = 2x = 0; 2x=9; x =9.2Ответ: 4,5.№ 506а) (x – 1)(x +1) = 2(x – 3) 2 – x2; x2 – 1 = 2x2 – 24x + 18 – x2;24x = 19 30x = –71; x =Ответ:19.2419.24б) 3(x +5) 2 – 4x2 = (2 – x)(2 + x); 3x2 + 30x + 75 – 4x2 = 4 – x2; x = –Ответ: – 271.301.3081в) (2x + 3) 2 – 4(x – 1)(x + 1) = 49; 4x2 + 12x + 9 – 4x2 + 4 = 49;1312x = 26; x =.61Ответ: 2 .6г) (3x + 1) 2 – (3x – 2)(2 + 3x) = 17; 9x2 + 6x + 1 – 9x2 + 4 = 17;6x = 12; x = 2.Ответ: 2.№ 507а) (10x2 – 4xy3) 2 = 100x4 – 80x3y3 + 16x2y6;б) (8p3 + 5p2q) 2 = 64p6 + 80p5 + 25p4q2;в) (0,6b3 – 5b2c4) 2 = 0,36b6 – 6b5c4 + 25b4c8;г) (3z7 + 0,5z3t) 2 = 9z14 + 3z10t + 0,25z6t2.№ 508а) 20x3z + 0,03z2) 2 = 400x6z2 + 1,2x3z3 + 0,0009z4;⎛3⎞29б) ⎜ n3 + 4mn 2 ⎟ = n6 + 3mn516m2n 4 ;64⎝8⎠в) (0,15r4n3 – 10n4) 2 = 0,625r8n6 – 3r4n7 + 100n8;⎛1⎞21г) ⎜ 6a 2 − ab ⎟ = 36a4 – 4a3b + a2b2.3 ⎠9⎝№ 509а) (xn – 23)(xn + 23) = x2n – 26;б) (a2n + bn)(a2n – bn) = a4n – b2n;в) (cn – d3n)(cn + d3n) = c2n – d6n);г) (an + 1 – bn – 1)(an + 1 + bn – 1) = a2n + 2 – b2n – 2.№ 510а) (3x2 – 2)(9x4 + 6x2 + 4) = 27x6 – 8;б) (5x2 + 3)(25x4 – 15x2 + 9) = 152x6 + 27;в) (8b2 + 3)(64b4 – 24b2 + 9) = 512b6 + 27;г) (7a2 – 1)(49a4 + 7a2 + 1) = 343a3 – 1.№ 511а) (x – 2) 2 (x + 2) 2 = (x2 – 4) 2 = x4 – 8x2 + 4;б) (y – 4) 2 (y + 4) = (y2 – 4) (y – 4) = y3 – 4y – 4y2 + 16;в) (m – 6) 2 (m + 6) 2 = (m2 – 36) = m4 – 72m2 + 1296;г) (n – 7) 2 (7 + n) = (n2 – 7)(n – 7) = n3 – 7n – 7n2 + 49.№ 512а) (x – y)(x + y)(x2 + y2) = (x2 – y2)(x2 + y2) = x4 – y4; б) (3a – b)(3a + b);б) (9a2 + b2) = (9a2 – b2)(9a2 + b2) = 81a4 – b4;82в) (p3 + q)(p3 – q)(p6 + q2) = (p6 – q2)(p6 + q2) = p12 – q4;г) (s4 + r4)(s – r)(s + r)(s2 + r2) = (s4 – r4)(s4 + r4) = s8 – r8.№ 513а) (3x2 + 4) 2 + (3x2 – 4) 2 – 2(5 – 3x2)(5 + 3x2) = 9x4 + 24x2 + 16 ++ 19x4 – 24x2 + 16 + 9x4 – 24x2 + 16 – 50 + 18x4 = 36x4 – 18;б) (4a3+5) 2+(4a3–1) 2–2(4a3+5)(4a3 – 1) = (4a3 + 5 – 4a3 + 1) 2 = 62 = 36;в) p(p – 2c)(p +2c) – (p – c)(p2 + pc + c2) = p3 – p · 4c2 – p3 + c3 = c3 – 4pc2;г) m(2m–1) 2–2(m+1)(m2 –m + 1) = 4m3–4m2+m–2m3–2=2m3 – 4m2 + m – 2.№ 514а) (a – b)(a + b)(a2 + b2)(a4 + b4)(a8 + b8) = (a8 – b8)(a8 + b8) = a16 – b16;б) x32 – (x – 1)(x + 1)(x2 + 1)(x4 +1)(x8 + 1) (x16 + 1) = x32 – x32 + 1 = 1.№ 515а) (6a5 + ∗1) 2 = ∗2 + ∗3 + 25x2; ∗12 = 25x2, значит ∗1 = 5x;∗2 = (6a5) 2 = 36a10; ∗3 = 6a5 · ∗1 · 2 = 6a5 · 5x · 2 = 60a5x;б) (10m5 + ∗1) 2 = ∗2 + ∗3 + 36m4n6; ∗12 = 36m4n6, значит ∗1 = 6m2n3;∗2 = (10m5) 2 = 100m10; ∗3 = 2 · ∗1 · 10m5 = 2 · 6m2n3 · 10m5 = 120m7n3;в) (∗1 – 4x7) 2 = 25x4y2 – ∗2 + ∗3; (∗1) 2 = 25x4y2, значит ∗1 = 5x2y;∗2 = 2 · ∗1 · 4x7 = 2 · 5x2y · 4x7 = 40x9y; ∗3 = (4x7) 2 = 16x14;г) (8a3 – ∗1) 2 = ∗2 – ∗3 + 49a8b6; ∗12 = 49a8b6, значит ∗1 = 7a4b3;∗2 = (8a3) 2 = 64a6; ∗3 = 2 · 8a3 · ∗1 = 16a3 · 7a4b3 = 112a7b3.№ 516а) (∗1 + 4d4) 2 = ∗2 + 24c2d5 + ∗3; ∗3 = (4d4) 2 = 16d8;2 · ∗1 · 4d4 = 24c2d5, значит ∗1 = 3c2d;∗2 = (∗1) 2 = (3c2d) 2 = 9c4d2;4 26 22б) (∗1 – 8a ) = 81a b – ∗2 + ∗3; (∗1) = 81a6b2, значит ∗1 = 9a3b;∗2 = 2 · ∗1 · 8a4 = 2 · 9a3b · 8a4 = 144a7b; ∗3 =(8a4) 2 = 64a8;в) (4p2q2 + ∗1) 2 = ∗2 + ∗3 + 0,01q8; (∗1) 2 = 0,01q8, значит ∗1 = 0,1q4;∗2 = (4p2q2) 2 = 16p4q4; ∗3 = 2 · ∗1 · 4p2q2 = 8p2q2 · 0,1q4 = 0,8 p2q6;г) (8q4t 3 – ∗1) 2 = ∗2 – ∗3 + 0,16t4;(∗1) 2 = 0,16t4, значит ∗1 = 0,4t2;4 3 28 64 3∗2 = (8q t ) = 64q t ; ∗3 = 2 · 8q t · ∗1 = 16q4t3 · 0,4t2 = 6,4q4t5.№ 517а) (5b3 – 7c) 2 = 25b6 – 70b3c + 49c2;б) (9x – 10x2y3) 2 = 81x2 – 180x3y3 + 100x4y6;в) (7x3 +5y2) 2 = 49x6 + 70x3y2 + 25y4; г) (6c5 – 4d3) 2 = 36c10 – 48c5 + 16d6.№ 518а) (2c – 15a)(2c + 15a) = 4c2 – 225a2;б) (9a + 11c)(9a – 11c) = 81a2 – 121c2;⎛⎝3⎞⎛⎠⎝3⎞⎠9в) ⎜ 0,5 y 2 − x3 ⎟⎜ 0,5 y 2 + x3 ⎟ = 0, 25 y 4 − x6 ;4416г) (10m3 – 0,4n2)(10m3 + 0,4n2) =100m6 – 0,16n4.83№ 519а) (0,7x3 – 10z2)(0,7x3 +10z2) = 0,49x6 – 100z4;⎛4⎞⎛⎞416q4;б) ⎜ 7 p 6 + q 2 ⎟⎜ 7 p 6 − q 2 ⎟ = 49p12 –11 ⎠⎝11 ⎠121⎝⎛ 3⎝ 4⎞⎛ 3⎠⎝ 4⎞⎠в) ⎜1 x7 − 8 y 2 z 5 ⎟⎜1 x 7 + 8 y 2 z 5 ⎟ = 31 14x – 64y4z10;16г) (5a4 – 6x2) 2 25a8 – 60a4x2 + 36x4.№ 52064= –64;27б) 25 – (2 – 3a)(4 + 6a + 9a2) = 25 – 8 + 27a3 =17 + 27a3 = 17 – 1 = 16;в) 127+(5c–3)(25c2+15c+9)=127+125c–27=125c+100=100–125 · 1,2= – 50;г) 64 – (4 – 3a)(16 + 12a + 9a2) = 64 – 64 + 27a3 = – 8.а) 125 – (5 – 3x)(25 + 15x + 9x2) = 125 – 125 + 27x3=27x3=– 27 ·№ 521а) (2 – 1)(2 + 1)(22 + 1)(24 + 1)(28 +1) – 216 = 216 – 216 = 0;б) 3(22 + 1)(24 + 1)(28 +1)(216 +1) – 232 == (22 – 1)(22 + 1)(24 + 1)(28 +1)(216 +1) – 232 = 232 – 232 = 0.№ 522(32 + 22)(34 + 24)(38 +28)(316 + 216) ==(32 − 22 )(32 + 22 )(34 + 24 )(38 + 28 )(316 + 216 )23 −22=332 − 232= 0,2(332 – 232).5§ 18.

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