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Значит, для трехзначных этосвойство не выполнено.№833.а) 2n + 2n+1+2n+2 = 2n ⋅ (1 + 2 + 22) = 2n ⋅ 7 = 2n–1 ⋅ 2 ⋅ 7=(2n–1 ⋅ 14) кратно 14;б) 5k + 5k+1 = 5k ⋅ (1 + 5) = 5k ⋅ 6 = 5k–1 ⋅ 5 ⋅ 6 = (5k–1 ⋅ 30) кратно 30.К параграфу 11№834.а) (x – 2)(5 + x) = x2 + 3x – 10; б) (y + 7)(y – 11) = y2 – 4y – 77;в) (10 – z)(z – 4) = –z2 + 14z – 40;г) (3a + 4)(8 – a) = 24a – 3a2 + 32 – 4a = – 3a2 + 20a + 32;д) (5c + 2)(2c – 1) = 10c2 – 5c + 4c – 2 = 10c2 – c – 2;е)(3n – 2)(1 – 4n) = 3n – 12n2 – 2 + 8n = –12n2 + 11 – 2.№835.а) (x – 2)(x + 3) + (x + 2)(x – 3) = x2 + x – 6 + x2 – x + 6 = 2x2 + 12;б) (y – 1)(y + 2) + (y + 1)(y – 2) = y2 + y – 2 + y2 – y – 2 = 2y2 – 4;в) (a + 1)(a + 2) + (a + 3)(a+4)=a2 + 3a + 2 + a2 + 7a + 12 = 2a2 + 10a + 14;г) (c – 1)(c – 2) + (c – 3)(c – 4) = c2 – 3c + 2 + c2 – 7c + 12 = 2c2 – 10c + 14.№836.а) (x2 – x – 4)(x – 5) = x3 – x2 – 4x – 5x2 + 5x + 20= x3 – 6x2 + x + 20;б) (2y – 1)(y2 + 5y – 2) = 2y3 + 10y2 – 4y – y2 – 5y + 2 = 2y3 + 9y2 – 9y + 2;101в) (2–3a)(–a2+4a–8)=–2a2+8a–16+3a3–12a2+24a = 3a3 – 14a2 + 32a – 16;г) (3 – 4c)(2c2 – c – 1) = 6c2 – 3c – 3 – 8c3 + 4c2 + 4c = –8c3 + 10c2 + c – 3;д) (x2–x+1)(2x2–x+4)=2x4–x3+4x2–2x3+x2–4x+2x2–x+4=2x4–3x3+7x2 – 5x + 4;е) (–5a2 + 2a + 3)(4a2 – 2a + 1) = – 20a4 + 10a3 – 5a2 + 8a3 – 4a2 + 2a ++ 12a2 – 6a + 3 = –20a4 + 18a3 + 3a2 – 4a + 3;ж) y(y – 3)(y + 2) = y(y2 – y – 6) = y3 – y2 – 6y;з) (c–4)(c+2)(c+3)=(c2–2c–8)(c+3)=c3 + 3c2 – 2c2 – 6c – 8c–24=c3+c2–14c – 24.№837.а) (x + y)(x2 – xy + y2) = x3 – x2y + xy2 + x2y – xy2 + y3 = x3 + y3;б) (x – y)(x2 + xy + y2) = x3 + x2y + xy2 – x2y – xy2 – y3 = x3 – y3;в) (a+b)(a3–a2b+ab2–b3)=a4–a3b+a2b2–ab3 + a3b – a2b2 + ab3 – b4 = a4 – b4;г) (a–b)(a3+a2b+ab2+b3)=a4 + a3b + a2b2 – ab3 – a3b – a2b2 – ab3–b4=a4 – b4.№838.а) (a2–7)(a+2)–(2a–1)(a–14)=a3+2a2–7a – 14 – (2a2–28a–a+14) = a3 + 22a – 28;б) (2 – b )(1 + 2b) + (1 + b)(b3 – 3b) = 2+ 4b – b – 2b2 + b3 – 3b + b4 – 3b2 == b4 + b3– 5b2 + 2;в) 2x2 – (x – 2y)(2x + y) = 2x2 – 2x2 – xy + 4xy + 2y2 = 2y2 + 3xy;г) (m – 3n)(m + 2n) – m(m – n) = m2 + 2mn – 3mn – 6n2 – m2 + mn = –6n2.№839.(y + 8)(y – 7) – y(y + 1) = y2 + y – 56 – y2 – y = – 56 < 0.№840.а) (35 – 34)(33 + 32) = 34(3 – 1) ⋅ 32(3 + 1) = 36 ⋅ 2 ⋅ 4 = 36 ⋅ 8 == 35 ⋅ 3 ⋅ 8 = (35 ⋅ 24) кратно 24;б) (210 + 28) ⋅ (25 – 23) = 28(22 + 1) ⋅ 23(22 – 1) = 210 ⋅ 5 ⋅ 3 = 210 ⋅ 15 == 28 ⋅ 22 ⋅ 15 = (28 ⋅ 60) кратно 60;в) (163 – 83)43 + 23) = (212 – 29)(26 + 22) = 29 ⋅ (23 – 1) ⋅ 23 ⋅ (23 + 1) == 212 ⋅ 7 ⋅ 9 = (212 ⋅ 63) кратно 63;г) (1252 + 252)(52 – 1) = (56 + 54) ⋅ 24 = 54(52 + 1) ⋅ 24 = 54 ⋅ 26 ⋅ 24 == 54 ⋅ 13 ⋅ 2 ⋅ 3 ⋅ 8 = (54 ⋅ 16 ⋅ 39) кратно 39.№841.а) (126y3 + 9x – 5y)(x2 + 25y2 + 5xy) = 126y3 + x3 – 125y3 = y3 + x3,при x = –3, y =–2, y3 + x3 = (–2)3 + (–3)3 = –8 – 27 = –35;б) m3 + n3 – (m2 – 2mn – n2)(m – n) = m3 + n2 – m3 + m2n + 2m2n – 2mn2 ++ n2m – n3 = 3m2n – n2m = nm(3m – n),при m = –3, n = 4, nm(3m – n) = 4 ⋅ (–3) ⋅ (3 ⋅ (–3) – 4) = (–12) ⋅ (–13) = 156.№842.а) (a – 3)(a2 – 8a + 5) – (a – 8)(a2 – 3a + 5) == a3 – 8a2 + 5a – 3a2 + 24a – 15 – a3 + 3a2 – 5a + 8a2 – 24a + 40 = 25б) (x2 – 3x + 2)(2x + 5) – (2x2 + 7x + 17)(x – 4) == 2x3 + 5x2 – 6x2 – 15x + 4x + 10 – 2x3 + 8x2 – 7x2 + 28x – 17x + 68 = 78.№843.а) x + (x + 1) + (x + 2) + (x + 3) + (x + 4) = 5x + 10 = [5 ⋅ (x + 2)] кратно 5;б) x + (x + 2) + (x + 4) + (x + 6) = 4x + 12 = 4 ⋅ (x + 3), т.к.

x — нечетное число,то (x + 3) — четное число, поэтому (x + 3) делится на 2, значит 4(x + 3)кратно 8.102№844.Если I число — x; II число — (x + 1); III число — (x + 2); IV — (x + 3), тоx(x + 1) + 38 = (x + 2)(x + 3), x2 + x + 38 = x2 + 5x + 6, 4x = 32,x = 8 — I число; II число — 9; III число — 10; IV число — 11.Ответ: 8; 9; 10; 11.№845.а) Пусть a; a + 1; a + 2; a + 3 — последовательные натуральные числа.

Тогда (a + 1)(a + 2) – a(a + 3) = a2 + 3a + 2 – a2 – 3a = 2 — верно;б) Пусть b, b + 2, b + 4 — последовательные нечетные числа,(b + 2)2 – b(b + 4) = b2 + 4b + 4 – b2 – 4b = 4 — верно.№846.Если x см — сторона квадрата, то (x – 2) см — ширина прямоугольника, (x +5) см — длина прямоугольника. Тогдаx2 + 50 = (x – 2)(x + 5), x2 + 50 = x2 + 3x – 10, 3x = 60,x = 20 см — сторона квадрата, а S = 400 см2 — площадь квадрата.Ответ: 400 см2.№847.ДлинаШиринаПлощадьПрямоугольник(x + 4) см(x – 5) см(x + 4)(x – 5)Квадратx смx смx2222x – (x + 4) ⋅ (x – 5) = 40, x – x – 4x + 5x + 20 = 40, x = 20,24 см — длина прямоугольника; 15 см — ширина прямоугольника.Тогда S = 24 ⋅ 15 = 360 см2.Ответ: 360 см2.№848.P=18м, значит если ширина прямоугольника—x м, то его длина—(18–x) м.2ДлинаШиринаПлощадьI прямоуг.xм(18 – x) мx(18 – x) м2II прямоуг.(x + 2) м(18 – x + 1) м(x + 2)(19 – x) м2(x + 2) ⋅ (19 – x) – x ⋅ (18 – x) = 30, 19x – x2 + 38 – 2x – 18x + x2 = 30,x = 8 м — ширина первого прямоугольника.10 м — длина первого прямоугольника.S = 10 ⋅ 8 = 80 м2.Ответ: 80 м2.№849.Пусть a см — ширина прямоугольника, (15 – a) см — длина прямоугольника, (12 – a) см — новая длина прямоугольника, (a + 5) см — новая ширинапрямоугольника.

Тогда по условию(12 – a)(a + 5) + 8 = a(15 – a), 12a + 60 – a2 – 5a + 8 = 15a – a2,7a – 15a = – 68, 8a = 68, a= 8,5 см — ширина первого прямоугольника.6,5 см — длина первого прямоугольника.S = 8,5 ⋅ 6,5 = 55,25 см2 — площадь первого прямоугольника.Ответ: 55,25 см2.103№850.а) a2+ ab – 7a – 7b = a(a + b) – 7(a + b) = (a + b)(a – 7),при a = 6,6, b = 0,4, (a + b)(a – 7) = (6,6 + 0,4) ⋅ (6,6 – 7) = –2,8;б) x2 – xy – 4x + 4y = x(x – y) – 4(x – y) = (x – y)(x – 4),при x = 0,5, y = 2,5, (x – y)(x – 4) = (0,5 – 2,5)(0,5 – 4) = 7;в) 5a2 – 5ax – 7a + 7x = 5a(a – x) – 7(a – x) = (a – x)(5a – 7),при a = 4, x = –3, (a – x)(5a – 7)= (4 – (–3))(5 ⋅ 4 – 7) = 91;г) xb – xc + 3c – 3b = a(b – c) – 3(b – c) = (b – c)(x – 3),при x = 2, b = 12,5, c = 8,3, (b – c)(x – 3) = (12,5 – 8,3) ⋅ (2–3) = –4,2;д) ay – ax – 2x + 2y = a(y – x) + 2(y – x) = (y – x)(a + 2),при a = –2, x = 9,1, y = –6,4, (y – x)(a + 2) = (–6,4 – 9,1)(–2 + 2) = 0;е) 3ax – 4by – 4ay + 3bx = 3x(a +b) – 4y(a + b) = (a + b)(3x – 4y),при a=3, b=–13, x=–1, y=–2, (a+b)(3x–4y)=(3+(–13))(3⋅(–1)–4⋅(–2))=–50.№851.а) a3 – 2a2 + 2a – 4 = a2(a – 2) + 2(a – 2) = (a – 2)(a2 + 2);б) x3 – 12 + 6x2 – 2x = x2 (x + 6) – 2(x + 6) = (x + 6)(x2 – 2);в) c4 – 2c2 + c3 – 2c = c2(c2 – 2) + c(c2 – 2) = (c2 – 2)(c2 + c)=c(c + 1)(c 2 – 2);г) –y6–y5 + y4 + y3 = –y5(y + 1) + y3(y + 1) = (y + 1)(y3 – y5)=y3(1 – y2)(y + 1);д) a2b – b2c + a2c – bc2 = a2(b + c) – bc(b + c) = (b + c)(a2 – bc);е) 2x3 + xy2 – 2x2y – y3 = 2x2(x – y) + y2(x – y) = (x – y)(2x2 + y2);ж) 16ab2–10c2+32ac2–5b2c=16a(b2+2c2) – 5c(2c2 + b2)=(2c2 + b2)(16a – 5c);з) 6a3 – 21a2b + 2ab2 – 7b3=2a(3a2 + b2) – 7b(3a2 + b2) = (3a2 + b2)(2a – 7b).№852.а) ma–mb+na–nb+pa–pb = m(a – b) + n(a – b) + p(a – b)=(a – b)(m + n + p);б) ax – bx – cx + ay – by – cy = x(a – b – c) + y(a – b – c) = (a – b – c)(x + y);в) x2+ax2 – y – ay + cx2 – cy = x2(1 + a + c) – y(1 + a + c)=(1 + a + c)(x2 – y);г) ax2–2y – bx2 + ay + 2x2 – by = x2(a – b + 2) + y(a – b + 2)=(a–b+2)(x2 + y).№853.а) x2 – 10x + 24 = x2 – 4x – 6x + 24 = x(x – 4) – 6(x – 4) = (x – 4)(x – 6);б) x2 – 13x + 40 = x2 – 8x – 5x + 40 = x(x – 8) – 5(x – 8) = (x – 8)(x – 5);в) x2 + 8x + 7 = x2 + x + 7x + 7 = x(x + 1) + 7(x + 1) = (x + 1)(x + 7);г) x2 + 15x + 54 = x2 + 6x + 9x + 54 = x(x + 6) + 9(x + 6) = (x + 6)(x + 9);д) x2 + x – 12 = x2 + 4x – 3x – 12 = x(x + 4) – 7(x + 4) = (x + 4)(x – 3);е) x2 – 2x – 35 = x2 – 7x + 5x – 35 = x(x – 7) + 5(x – 7) = (x – 7)(x +5).№854.а) (x + a)(x + b) = x2 + ax + bx + ab = x2 + (a + b)x + ab;б) (x – a)(x – b) = x2 – ax – bx + ab = x2 – (a + b)x + ab.№855.а) (x4 + x3)(x2 + x) = x3(x + 1) ⋅ x(x + 1) = x4(x + 1)2;б) (y4 + y2)(y2 – у) = y2(y2 + 1)y(y – 1) = y3(y2 + 1)(y – 1);в) (a2 + ab + b2)(a2 – ab + b2) == a4 – a3b + a2b2 + a3b – a2b2 + ab3 + a2b2 – ab3 + b4 = a4 + a2b2 + b4;г) (c4–c2+1)(c4 + c2 + 1) = c8 + c6 + c4 – c6 – c4 – c2 – c2 + c2 + 1 = c8 + c4 + 1.№856.(10a + b)(10a + c) = 100a2 + 10ac + 10ba + bc = 100a2 + 10a(b + c) + bc == 100a2 + 10a + bc = 100a(a + 1) + bc;104а) 23 ⋅ 27 = (20 + 3)(20 + 7) = 200 ⋅ 2(2 + 1) + 3 ⋅ 7 = 600 + 21 = 621;б) 42 ⋅ 48 = (40 + 2)(40 + 8) = 100 ⋅ 4(4 + 1) + 2 ⋅ 8 = 2000 + 16 = 2016;в) 59 ⋅ 51 = (50 + 9)(50 + 1) = 100 ⋅ 5(5 + 1) + 9 ⋅ 1 = 3000 + 9 = 3009;г) 84 ⋅ 86 = (80 + 4)(80 + 6) = 100 ⋅ 8(8 + 1) + 4 ⋅ 6 = 7200 + 24 = 7224.№857.

(a + c)(b + c) + (a – c)(b – c) = ab + ac + c2 + ab – ac – bc + c2 == 2ab + 2c2 = 2(ab + c2) = 2 ⋅ 0 = 0.№858.(a + 1)(b + 1) – (a – 1)(b – 1) = ab + a + b + 1 – ab + a + b – 1 == 2a + 2b = 2(a + b) = 2 ⋅ 9 = 18.105Глава V. Формулы сокращенного умножения§ 12. Квадрат суммы и квадрат разности31. Возведение в квадрат суммы и разности двух выражений№859.а) (x + y)2 = x2 + 2xy + y2;в) (b + 3)2 = b2 + 6b + 9;д) (y – 9)2 = y2 – 18y + 81;ж) (a + 12)2 = a2 + 24a + 144;и) (b – 0,5)2 = b2 – b + 0,25;№860.а) (m + n)2 = m2 + 2mn + n2;в) (x + 9)2 = x2 + 18x + 81;д) (a – 25)2 = a2 – 50a + 625;ж) (0,2 – x)2 = 0,04 – 0,4x + x2;№861.б) (p – q)2 = p2 – 2pq + q2;г) (10 – c)2 = 100 – 20c + c2;е) (9 – y)2 = 81 – 18y + y2;з) (15 – x)2 = 225 – 30x + x2;к) (0,3 – m)2 = 0,09 – 0,6m + m2.б) (c – d)2 = c2 – 2cd + d2;г) (8 – a)2 = 64 – 16a + a2;е) (40 + b2) = 1600 + 80b + b2;з) (k + 0,5)2 = k2 + k + 0,25.BCb23a14abDASABCD = (a + b)2, S1 = a2; S3 = b2; S2 = S4 = ab.Тогда SABCD = S1 + S3 + 2S2, т.е.

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