makarytchev-gdz-7-1-1289-2003 (542427), страница 19
Текст из файла (страница 19)
а) x2 – y2 – x – y = (x – y)(x + y) – (x + y) = (x +y)(x – y – 1);б) a2 – b2 – a + b = (a – b)(a + b) – (a – b) = (a – )(a + b – 1);в) m + n + m2 – n2 = (m + n) + (m – n)(m + n) = (1 + m – n)(m + n);г) k2 – k – p2 – p = (k – p)(k + p) – (k + p) = (k + p)(k – p – 1).№1004. а) a – b + a2 – b2 = (a – b) + (a – b)(a + b) = (a – b)(1 + a + b);б) c2 + d – d2 + c = (c – d)(c + d) + (c + d) = (c + d)(c – d + 1).№1005.а) ab2–a–b3+b=b2(a–b) – (a – b) = (b2 – 1)(a – b) = (b – 1)(b + 1)(a – b);б) bx2+2b2–b3–2x2=x2(b–2)+b2(2–b)=(x2 – b2)(b – 2)=(x–b)(x + b)(b – 2);в) x3+x2y–4y–4x=x2(x+y)–4(x+y)=(x2 – 4)(x + y) = (x – 2)(x + 2)(x + y);г) x3–3y2+3x2–xy2=x2(x+3)–y2(x + 3) = (x2 – y2)(x + 3)=(x–y)(x+y)(x + 3).№1006.а) x3 – x = 0, x(x2 – 1) = 0, x(x – 1)(x + 1) = 0, x1 = 0; x2 = 1; x3 = –1;б) 9x – x3 = 0, x(9 – x2) = 0, x(3 – x)(3 + x) = 0, x1 = 0; x2 = 3; x3 = –3;в) x3 + x2 = 0, x2(x + 1) = 0, x1 = 0; x2 = –1;г) 5x4–20x2=0, 5x2(x2–4)=0, 5x2(x – 2)(x + 2) = 0, x1=0; x2=2; x3=–2.№1007.а) x3 + x = 0, x(x2 + 1) = 0, x1 = 0; больше нет, т.к.
x2 + 1 ≠ 0 для любого x;б) x3 – 2x2 = 0, x2(x – 2) = 0, x1 = 0; x2 = 2.№1008.x3 – x = x(x2 – 1) = x(x– 1)(x + 1).Выражение делится на 6, т.к. хотя бы одно из x, x + 1, x + 2 четно и одно делится на 3.№1009.Если 2a – 1, 2a + 1 — два последовательных нечетных числа, то(2a + 1)2 – (2a – 1)2 = (2a + 1 – 2a + 1)(2a + 1 + 2a – 1) = 2(4a) = 8aкратно 8.№1010.а) (6x–1)(6x+1) – (12x – 5)(3x + 1) = 36x2 – 1 – (36x2 + 12x – 15x – 5) == 36x2 – 1– 36x2 + 3x + 5 = 3x + 4, при x=0,2, 3x+4=3 ⋅ 0,2 + 4 = 4,6;б) (5+2x)2–2,5x(8x+7)=25 + 20x + 4x2 – 20x2 – 17,5x=–16x2 + 2,5x + 25,при x = –0,5, –16x2 + 2,5x + 25 = –16 ⋅ 0,25 + 2,5 ⋅ (–0,5) + 25 = 19,75.№1011. y = 0,02x2;а) A(15; 4,5) 0,02 ⋅ 152 = 4,5 ⇒ A ∈ графику;б) B(–2,05; –0,12) 0,02 ⋅ (–2,02)2 = 0,08405 ≠ –0,12 ⇒ B ∉ графику;в) C(50; 50) 0,02 ⋅ 502 = 50 ⇒ C ∈ графику.№1012.
y = 0,24x + 6;Т.к. график функции пересекается с Ox, то y = 0,0,24x + 6 = 0, 0,24x = –6, x = 25, A(25; 0);Т.к. график функции пересекается с Oy, то x = 0,y = 0,24 ⋅ 0 + 6, y = 6, B(0; 6).126№1013.а) y = –0,9x + 4.б) y = 2,3x.yy40в) y =x0xx.10г) y = –9.yy00xx–938. Применение преобразований целых выражений№1014.а) x2+2x+2 = (x2 + 2x + 1) + 1 = (x + 1)2 + 1 > 0, т.к. (x + 1)2 ≥ 0, 1 > 0;б) 4x2–4x+6=(4x2–4x+1) + 5 = (2x – 1)2 + 5 > 0, т.к.
(2x – 1)2 ≥ 0, 5>0;в) a2 + b2 – 2ab + 1 = (a – b)2 + 1 > 0, т.к. (a – b)2 ≥ 0, 1 > 0;г) x2 + y2 + z2 + 2xy + 5 = (x2 + 2xy + y2) + z2 + 5 = (x + y)2 + z2 + 5 > 0,т.к. (x + y)2 ≥ 0, z2 ≥ 0, 5 > 0.№1015. 2b–b2–2 = –(b2–2b+1) – 1 = –(b – 1)2 – 1 < 0, т.к. (b – 1)2 ≥ 0, –1 < 0.№1016. а) y2–10y+30 = y2–10y + 25 + 5=(y – 5)2 + 5>0, т.к. (y – 5)2 ≥ 0, 5 > 0;б) c2 + 4cd + 4d2 + 4 = (c + 2d)2 + 4 > 0, т.к.
(c + 2d)2 ≥ 0, 4 > 0.№1017.Пусть 2k + 1 — нечетное число.(2k + 1)2 = 4k2 + 4k + 1, 4k 2 + 4k четное, 4k2 + 4k + 1 нечетное.№1018.(2n + 10)(n + 5) – 2(n + 3)(n – 3) – (5n + 13) == 2n2 + 10n + n + 5 – 2n2 + 18 – 5n – 13 = 6n – 10.Т.к. 6n кратно 6, а 10 на 6 без остатка не делится, то и исходное число некратно 6.№1019. (n+8)(n–4)–(n+3)(n – 2) + 27 = n2 + 4n – 32 – n2 – n + 6 + 27 = 3n + 1.Т.к. 3n кратно 3, а 1 не кратна 3, то и исходное число не кратно 3.127№1020. а) 3a2b + 2ab2 = ab(3a + 2b),212 1 ⎛ ⎛ 2⎞1⎞11при a=– , b= , ab(3a+2b)=– ⋅ ⋅ ⎜ 3 ⋅ ⎜ − ⎟ + 2 ⋅ ⎟ = − ⋅ ( − 2 + 1) = ;2⎠333 2 ⎝ ⎝ 3⎠3212, n = , mn(2n – 3m) + 1 =231 2 ⎛ 24 14 1 11 ⎛ 1 1⎞⎛ 1 ⎞⎞= – ⋅ ⋅ ⎜ 2 ⋅ − 3 ⋅ ⎜ − ⎟ ⎟ + 1 = − ⋅ ⎜1 + 1 ⎟ + 1 = − − + 1 = − + = .3 ⎝ 3 2⎠9 29 2 182 3 ⎝ 3⎝ 2 ⎠⎠№1021.
(a2 + b2)(a + b)(a – b) = (a2 + b2)(a2 – b2) = a4 – b4;б) 2mn2 – 3m2n + 1 = mn(2n – 3m) + 1, при m = –а) при a = 2, b = 0,1, удобнее использовать a4–b4=24–(0,1)4 = 15,9999;43б) при a = , b = , удобнее использовать (a2 + b2)(a + b)(a – b) =411⎛ 9 1 ⎞⎛ 3 1 ⎞⎛ 3 1 ⎞ 10 4 2 5 1 5= ⎜ + ⎟⎜ + ⎟⎜ − ⎟ = ⋅ ⋅ = ⋅ =.⎝ 16 16 ⎠⎝ 4 4 ⎠⎝ 4 4 ⎠ 4 4 4 8 2 161022. Для вычислений на калькуляторе удобнее использовать формулуx(a + b – c).
Вычислим: при a = 3,17; b = 1,12; c = 0,97; x = 4,11,x(a + b – c) = 4,11(3,17 + 1,12 – 0,97) = 13,6452.№1023.при x = 3,7, 3,5x3–2,1x2+1,9x–16,7=3,5⋅3,73–2,1⋅3,72+1,9⋅3,7–16,7 = 138,8665.№1024. x4 – 20x3 – 19x2 – 32x + 40 = x4 – 20x3 – 20x2 + x – 40x + 8x + 40 == x4–20x2(x+1)–40(x–1)+x(x+8)=x2(x2–20(x+1))–40(x–1)+x(x + 8), при x = 21;x2(x2 – 20(x + 1)) – 40(x – 1) + x(x + 8), 212(212 – (21 – 10)(21 + 1)) –– 40 ⋅ 20 + 21 ⋅ 29 = 212 – 212 + 1 – 800 + 21 ⋅ 29 = 21(21 + 29) – 800 == 21 ⋅ 500 – 800 = 1050 – 800 = 250.№1025. y = 4(3 – 2x) – 5; y = 12 – 8x – 5; y = –8x + 7 — линейная;y = x – 8(x – 8); y = x – 8x + 64; y = –7x + 64 — линейная;–8x + 7 = –7x + 64, –x = 57, x = –57, y = –8(–57) + 7 = 463.Точка пересечения графиков (–57; 463).№1026.
y = x2; y = –x + 6.y964L–312802xОтвет: (2; 4) и (–3; 9).№1027.а) a2+ b2 – c2 + 2ab = (a +b )2 – c2 = (a + b + c)(a + b – c);б) m2 – x2 – y2 + 2xy = m2 – (x – y)2 = (m –x + y)(m + x – y);в) a3 + a2 – ab2 – b2=a(a2 – b2) + (a2 – b2)=(a + 1)(a2 – b2)=(a + 1)(a – b)(a + b);г) 9n+m3–m2n – 9m = 9(n – m) – m2(n – m) = (n – m)(9 – m2)=(n–m)(3–m)(3+m).Дополнительные упражнения к главе VК параграфу 12№1028.21⎛1⎞а) ⎜ x + 9 ⎟ = x2 + 6x + 81;9⎝3⎠225 2⎛5⎞б) ⎜ y − 3 ⎟ =y – 5y + 9;36⎝6⎠22111 ⎞1 ⎞⎛⎛в) ⎜ −2a + b ⎟ = 4a2 – 2ab + b2; г) ⎜ −3 x − y ⎟ = 9x2 + 2xy + y2;2 ⎠3 ⎠49⎝⎝д) (5xy – 0,8y2)2 = 25x2y2 – 8xy3 + 0,64y4;е) (0,4a + 10ab)2 = 0,16a2 + 8a2b + 100a2b2;ж) (3a2–5ab)2=9a4–30a3b+25a2b2; з) (8xy+3y2)2 = 64x2y2 + 48xy3 + 9y4;и) (a3b3 – 1)2 = a6b6 – 2a3b3 + 1; к) (2 + x4y2)2 = 4 + 4x4y2 + x8y4;л) (x6 – 3xy2)2 = x12 – 6x7y2 + 9x2y4; м) (y8 – 2x4y)2 = y16 – 4x4y + 4x8y2.№1029.а) (0,7x3y – 2xy3)2 = 0,49x6y2 – 2,8x4y4 + 4x2y6;2429 6 2⎛3⎞б) ⎜ a3b − ab3 ⎟ =a b – a4b4 + a2b6;39⎝4⎠ 16в) (0,2p3q + 0,3pq3)2 = 0,04p6q2 + 0,12p4q4 + 0,09p2q6;281264⎛1⎞г) ⎜ bc 4 + b2c3 ⎟ = b2c8 + b3c7 + b4c6 .964981⎝8⎠№1030.а) (2m3n + 0,3mn4)2 = 4m6n2 + 1,2m4n5 + 0,09m2n8;23 ⎞129⎛1б) ⎜ a 4b2 − ab ⎟ = a8b4 − a5b3 + a 2b 2 ;5 ⎠9525⎝3в) (0,1a6b + 0,2ab6)2 = 0,01a12b2 + 0,04a7b7 + 0,04a2b12;231 10 4 1 6 8 9 2 12⎛1⎞г) ⎜ x5 y 2 − xy 6 ⎟ =x y − x y + x y .436416⎝6⎠№1031.(a + b + c)2 = ((a + b) + c)2 = (a + b)2 + 2(a + b)c + c2 == a2 + 2ab + b2 + 2ac + 2bc + c2 = a2 + b2 + c2 + 2ab + 2ac + 2bc.№1032.а) (a + b)4 = ((a + b)2)2 = (a2 + 2ab + b2)2 == a4 + 4a2b2 + b4 + 4a3b + 4ab3 + 2a2b2 = a4 + 4a3b + 6a2b2 + 4ab3 + b4;129б) (a – b)4 = ((a – b)2)2 = (a2 – 2ab + b2)2 == a4 + 4a2b2 + b4 – 4a3b – 4ab3 + 2a2b2 = a4 – 4a3b + 6a2b2 – 4ab3 + b4.№1033.а) (x + 7)2 – (x – 5)(x + 19) = x2 + 14x + 49 – x2 – 14x + 95 = 144;б) (x + 9)2 + (8 – x)(x + 26) = x2 + 18x + 81 + 8x + 208 – x2 – 26x = 289.№1034.а) b2 + 10b + 25 = (b + 5)2; б) c2 – 8c + 16 = (c – 4)2;в) 16x2 – 8x + 1 = (4x – 1)2; г) 4c2 + 12c + 9 = (2c + 3)2;д) x4 + 2x2y + y2 = (x2 + y)2; е) a6 – 6a3b2 + 9b4 = (a3 – 3b2)2.№1035.а) a4 – 8a2 + 16 = (a2 – 4)2; б) –4 – 4b – b2 = –(b + 2)2;г) c4d2 + 1 – 2c2d = (c2d – 1)2;в) 10x – x2 – 25 = –(x – 5)2;2д) a6b2 + 12a3b + 36 = (a3b + 6)2;е) x + 1 +1 2 ⎛1⎞x = ⎜ x + 1⎟ ;4⎝2⎠2ж) y – y2 – 0,25 = –(y – 0,5)2;з) 9 – m +и) –25 – 2n – 0,04n2 = –(5 + 0,2n)2.1 2 ⎛ 1 ⎞m = ⎜3 − m⎟ ;6 ⎠36⎝К параграфу 13№1036.а) (x2 – 11)(11 + x2) = x4 – 121; б) (y2 + 10)(–10 + y2) = y4 – 100;г) (b7 + 3)(–b7 + 3) = 9 – b14;в) (a5 – 1)(a5 + 1) = a10 – 1;6612д) (–c – 8)(c – 8) = 64 – c ; е) (d9 – 5)(–5 – d9) = 25 – d18.№1037.а) 1005 ⋅ 995 = (1000 + 5)(1000 – 5) = 1000000 – 25 = 999975;б) 108 ⋅ 92 = (100 + 8)(100 – 8) = 10000 – 64 = 9936;в) 0,94 ⋅ 1,06 = (1 – 0,06)(1 + 0,06) = 1 – 0,0036 = 0,9964;г) 1,09 ⋅ 0,91 = (1 + 0,09)(1 – 0,09) = 1 – 0,0081 = 0,9919;1 6 ⎛1 ⎞⎛1⎞148д) 10 ⋅ 9 = ⎜ 10 + ⎟⎜10 − ⎟ = 100 −= 99 ;7 7 ⎝7 ⎠⎝7⎠494972 ⎛2 ⎞⎛2⎞477е) 99 ⋅ 100 = ⎜100 − ⎟⎜100 + ⎟ = 10000 − = 9999 .99 ⎝9 ⎠⎝9⎠8181№1038.а) 5y(y2 – 3)(y2 + 3) = 5y(y4 – 9) = 5y5 – 45y;б) –8x(4x – x3)(4x + x3) = –8x(16x2 – x6) = – 128x3 + 8x7;в) (a4 – 3)(a4 + 3)(a8 + 9) = (a8 – 9)(a8 + 9) = a16 – 81;г) (1 – b3)(1 + b3)(1 + b6) = (1 – b6)(1 + b6) = 1 – b12.№1039.а) (a + 2)(a – 2) – a(a – 5) = a2 – 4 – a2 + 5a = 5a – 4;б) (a – 3)(3 + a) + a(7 – a) =a2 – 9 + 7a – a2 = 7a – 9;в) (b – 4)(b 4) – (b – 3)(b + 5) = b2 – 16 – b2 – 2b + 15 = –1 – 2b;г) (b + 8)(b – 6) – (b – 7)(b + 7) = b2 + 2b – 48 – b2 + 49 = 2b + 1;130д) (c – 1)(c + 1) + (c – 9)(c + 9) = c2 – 1 + c2 – 81 = 2c2 – 82;е) (5 + c)(c – 5) – (c – 10)(c + 10) = c2 – 25 – c2 + 100 = 75.№1040.а) (x – 8)(x + 8) – (x – 12)(x + 12) = x2 – 64 – x2 + 144 = 80;5 ⎞⎛5⎞ ⎛ 225 411⎛⎞⎛ 2⎞б) ⎜ y − ⎟⎜ y + ⎟ + ⎜ − y ⎟⎜ + y ⎟ = y 2 −+ − y2 =.9 ⎠⎝9⎠ ⎝ 381 981⎝⎠⎝ 3⎠№1041.а) (x – 5)2 + 2x(x – 3) = x2 – 10x + 25 + 2x2 – 6x = 3x2 – 16x + 25;б) (y + 8)2 – 4y(y – 2) = y2 + 16y + 64 – 4y2 + 8y = –3y2 + 24y + 64;в) (a – 4)(a + 4) + (2a – 1)2 = a2 – 16 + 4a2 – 4a + 1 = 5a2 – 4a – 15;г) (b – 3)(b + 3) – (b + 2)2 = b2 – 9 – b2 – 4b – 4 = –4b – 13;д) (2a – 5)2 – (5a – 2)2 = 4a2 – 20a + 25 – 25a2 + 20a – 4 = –21a2 + 21;е) (3b – 1)2 + (1 – 3b)2 = 9b2 – 6b + 1 + 1 – 6b + 9b2 = 18b2 – 12b + 2;ж) (2x + 1)2 – (x + 7)(x – 3) = 4x2 + 4x + 1 – x2 – 4x + 21 = 3x2 + 22;з) (3y – 2)2 – (y – 9)(9 – y) = (3y – 2)2 + (y – 9)2 == 9y2 – 12y + 4 + y2 – 18y + 81 = 10y2 – 30y + 85.№1042.2(x+2)(x – 2) + 16 = (x + 2)2 + (x – 2)2, 2(x2–4)+16=x2+4x+4+x2–4x+ 4,2x2 – 8 + 16 = 2x2 + 8, 2x2 + 8 = 2x2 + 8, 0=0 для любого x это верно.№1043.а) (x+y+1)(x+y–1)=((x+y)+1)((x + y) – 1) =(x + y)2 – 1=x2 + 2xy + y2 – 1;б) (m + n – 3)(m + n + 3) = (m +n )2 – 32 = m2 + 2mn + n2 – 9;в) (a – b – 5)(a – b + 5) = (a – b)2 – 52 = a2 – 2ab + b2 – 25;г) (c – d + 8)(c – d – 8) = (c – d)2 – 82 = c2 – 2cd + d2 – 64;д) (p + 2q – 3)(p – 2q – 3) = ((p – 3) + 2q)((p – 3) – 2q) == (p – 3)2 – (2q)2 = p2 – 6p + 9 – 4q2;е) (a – 3x + 6)(a + 3x + 6) = ((a + 6) – 3x)((a + 6) + 3x) == (a + 6)2 – (3x)2 = a2 + 12a + 36 – 9x2.№1044.а) (x – 7)2 +3 = (x – 2)(x + 2), x2 – 14x + 49 + 3 = x2 – 4,–14x = –4 – 52, 14x = 56, x = 4;б) (x + 6)2 – (x – 5)(x + 5) = 79, x2 + 12x + 36 – x2 + 25 = 79,12x + 61 = 79, 12x = 18, x = 2,5;в) (2x – 3)2 – (7 – 2x)2 = 2, 4x2 – 12x + 9 – 49 + 28 – 4x2 = 2,16x = 2 + 40, 16x = 42, x = 2,625;г) (5x – 1)2 – (1 – 3x)2 = 16x(x – 3), 25x2 – 10x + 1 – 1 + 6x – 9x2 = 16x2 – 48x,16x2 – 4x = 16x2 – 48x, 44x = 0, x = 0.№1045.а) 1 – a2b2 = (1 – ab)(1 + ab);б) 4x2y4 – 9 = (2xy2 – 3)(2xy2 + 3);в) –0,64 + x4 = (x2 – 0,8)(x2 + 0,8);г) 0,09x6 – 0,49y2 = (0,3x3 – 0,7y)(0,3x3 + 0,7y);д) 1,21a2 – 0,36b6 = (1,1a – 0,6b3)(1,1a + 0,6b3);142 ⎞⎛ 12 ⎞⎛ 1е) 2 b2 – c 2 = ⎜1 b − c ⎟⎜ 1 b + c ⎟ ;92323 ⎠4⎝⎠⎝131793 ⎞⎛ 13 ⎞⎛ 1ж) 1 x 2 − y 2 = ⎜1 x − y ⎟⎜1 x + y ⎟ ;9164 ⎠⎝ 34 ⎠⎝ 3з) 0,01a2b4 – 1 = (0,1ab2 – 1)(0,1ab2 + 1);и) –9m2 + 1,44n6 = (1,2n3 – 3m)(1,2n3 + 3m).№1046.382 − 172 (38 − 17)(38 + 17) 21 ⋅ 55 15===;а)722 − 162 (72 − 16)(72 + 16) 56 ⋅ 88 64б)39 ,52 − 3,52(39 ,5 − 3,5)(39 ,5 + 3,5)36 ⋅ 43 1=== ;57 ,52 − 14 ,52 (57 ,5 − 14 ,5)(57 ,5 + 14 ,5) 43 ⋅ 72 217 ,52 − 9 ,52(17 ,5 − 9,5)(17 ,5 + 9 ,5)27 ⋅ 81===.131,52 − 3,52 (131,5 − 3,5)(131,5 + 3,5) 128 ⋅ 135 80№1047.а) x10 – 1 = (x5 – 1)(x5 + 1);б) y12 – 16 = (y6 – 4)(y6 + 4) = (y3 – 2)(y3 + 2)(y6 + 4);в) a2x8 – 81 = (ax4 – 9)(ax4 + 9); г) 36 – b4y6 = (6 – b2y3)(6 + b2y3);д) 25p4q4 – 1 =(5p2q2 – 1)(5p2q2 + 1);е) –9 + 121m8n8 = (11m4n4 – 3)(11m4n4 + 3);ж) 0,01x16–0,16=0,01(x16–16)=0,01(x8–4)(x8+4)=0,01(x4–2)(x4+2)(x8+4);з) 1,69y14 – 1,21 = (1,3y7 – 1,1)(1,3y7 + 1,1);425 ⎛ 2 3 5 ⎞⎛ 2 3 5 ⎞и) m6 −= ⎜ m − ⎟⎜ m + ⎟ .936 ⎝ 36 ⎠⎝ 36⎠№1048.а) (x – 5)2 – 16 = (x – 5 – 4)(x – 5 + 4) = (x – 9)(x – 1);б) (b + 7) – 9 = (b – 7 – 3)(b + 7 + 3) = (b + 4)(b + 10);в) 25 – (3 – x)2 = (5 – 3 + x)(5 + 3 – x) = (x + 2)(8 – x);г) 81 – (a + 7)2 = (9 – a – 7)(9 + a + 7) = (2 – a)(a + 16);д) (5x–12)2–x2=(5x – 12 – x)(5x–12+x)=(4x–12)(6x–12)=24(x – 3)(x – 2);е) 36p2 – (5p – 3)2 = (6p – 5p + 3)(6p + 5p – 3) = (p + 3)(11p – 3)ж) (7x – 4)2 – (2x + 1)2 = (7x – 4 – 2x – 1)(7x – 4 + 2x + 1) == (5x – 5)(9x – 3) = 15(x – 1)(3x – 1);з) (n–2)2–(3n + 1)2 = (n – 2 – 3n – 1)(n – 2 + 3n + 1) = (–2n – 3)(4n – 1);и) 9(a + 1)2 – 1 = (3a + 3 – 1)(3a + 3 + 1) = (3a + 2)(3a + 4);к) 4 – 25(x – 3)2 = (2 – 5x + 15)(2x + 5x – 15) = (17 – 5x)(5x – 13);л) 9(x + 5) – (x – 7)2 = (3x + 15 – x + 7)(3x + 15 + x – 7) == (2x + 22)(4x + 8) = 8(x + 11)(x + 2);м) 49(y – 4)2 – 9(y + 2) = (7y – 28 – 3y – 6)(7y – 28 + 3y + 6) ==(4y – 34)(10y – 22) = 2(2y – 17)(5y – 11).№1049.а) (n + 1)2 – (n – 1)2 = n2 + 2n + 1 – n2 + 2n – 1 = 4n кратно 4;б) (2n+3)2–(2n–1)2=4n2+12n+9–4n2+4n–1=16n + 8=8(2n + 1) кратно 8;в) (3n + 1)2 – (3n – 1)2 = 9n2 + 6n + 1 – 9n2 + 6n – 1 = 12n кратно 12;г) (5n + 1)2 – (2n – 1)2 = 25n2 + 10n + 1 – 4n2 + 4n – 1 == 21n2 + 14n = 7n(3n + 2) кратно 7.в)132№1050.а) (3a – 2b)2 – (2a – b)2 = 9a2 – 12ab + 4b2 – 4a2 + 4ab – b2 ==5a2 – 8ab + 3b2 = 5a(a – b) – 3b(a – b) = (a – b)(5a – 3b),при a = 1,35, b = –0,65,(a–b)(5a–3b)=(1,35+0,65)(5⋅1,35+1,95)=2⋅(6,75+1,95) = 2 ⋅ 8,7 = 17,4;б) (2y–c)2+(y+2c)2=4y2 – 4yc + c2 + y2 + 4yc + 5c2=5y2 + 5c2 = 5(y2 + c2),при y= –1,4, c = 1,2, 5(y2 + c2) = 5(1,96 + 1,44) = 5 ⋅ 3,4 =17.№1051.а) 0,027x3 + 1 = (0,3x + 1)(0,09x2 – 0,3x + 1);б) y6 – 0,001x3 = (y2 – 0,1x)(y4 + 0,1xy2 + 0,01x2);в) d3 + 0,008c3 = (d + 0,2c)(d2 – 0,dc + 0,04c2);г) 125 – 0 064 p3 = (5 – 0,4p)(25 + 2p + 0,16p2).№1052.27⎛3⎞⎛ 9 3⎞а)− y12 = ⎜ − y 4 ⎟⎜ + y 4 + y8 ⎟ ;64⎝4⎠⎝ 16 4⎠1111⎛⎞⎛⎞= ⎜ − x5 ⎟⎜ + x5 + x10 ⎟ ;б) − x15 +27 ⎝ 393⎠⎝⎠33⎛3⎞⎛ 9⎞в) 3 a15 + b12 = ⎜ a5 + b4 ⎟⎜ a10 − a5b4 + b8 ⎟ ;82⎝2⎠⎝ 4⎠615⎛5⎞⎛ 25⎞г) 1 x18 + y3 = ⎜ x6 + y ⎟⎜ x12 − x6 y + y 2 ⎟ .644⎝6⎠⎝ 16⎠№1053.а) 413+193=(41+19)(412–41⋅19+192)=60⋅(412 – 41 ⋅ 19+192) кратно 60;б) 793–293=(79–29)(792+79⋅29+292)=50⋅(792 + 79 ⋅ 29 + 292) кратно 50;в) 663 + 343 = (66 + 34)(662 – 66 ⋅ 34+342)= 100 ⋅ (662 – 66 ⋅ 34 + 342) == 100 ⋅ 4 (332 – 33 ⋅ 17 + 172) = 400 (332 – 33 ⋅ 17 + 172) кратно 400;г) 543 – 243 = (54 – 24)(542 + 54 ⋅ 24 + 242) = 30 ⋅ (542 + 54 ⋅ 24 + 242) == 1080 ⋅ (92 + 4 ⋅ 4 + 42) кратно 1080.№1054.a3–b3 = a3+(–b)3 = (a + (–b))(a2 – a(–b) + (–b)2) = (a – b)(a2 + ab + b2).№1055.а) (x + 1)3 + x3 = (x + 1 + x)((x + 1)2 – (x + 1)x + x2) == (2x + 1)(x2 + 2x + 1 – x2 – x + x2) = (2x + 1)(x2 + x + 1);б) (y – 2)3 – 27 = (y – 2 – 3)((y – 2)2 + 3(y – 2) + 9) == (y – 5)(y2 – 4y + 4 + 3y – 6 + 9) = (y – 5)(y2 – y + 7);в) (a – b)3 + b3 = (a – b + b)((a – b)2 + (a – b)b + b2) == a(a2 – 2ab + b2 – ab + b2 + b2) = a(a2 – 3ab + 3b2);г) 8x3 + (x – y)2 = (2x + x – y) (4x2 – 2x(x – y) + (x – y)2) == (3x – y)(4x2 – 2x2 + 2xy + x2 – 2xy + y2) = (3x – y)(3x2 + y2);д) 27a3 – (a – b)3 = (3a – a + b)(9a2 + 3a(a – b) + (a – b)2) == (2a + b)(9a2 + 3a2 – 3ab+ a2 – 2ab + b2) = (2a + b)(13a2 – 5ab + b2);е) 1000 + (b – 8)2 = (10 + b – 8)(100 – 10(b – 8) + (b – 8)2) == (2 + b)(100 – 10b + b2 – 16b + 64) = (2 + b)(b2 – 26b + 244).133К параграфу 14№1056.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.№1057.а) (x + 4)(x2 – 4x + 16) =x3 – 4x2 + 16x + 4x2 – 16x + 64 = x3 + 64;б) (3a + 5)(9a2 – 15a + 25) = 27a3 + 125.№1058.а) (x + 1)(x + 2) – (х – 3)(x + 4) = 6, x2 + 3x + 2 – x2 – x + 12 = 6,2x = 6 – 14, 2x = –8, x = –4;б) (3x – 1)(2x + 7) – (x + 1)(6x – 5) = 7,6x2 + 21x – 2x – 7 – 6x2 + 5x – 6x + 5 = 7, 18x = 9, x = 0,5;в) 24 – (3y + 1)(4y – 5) = (11 – 6y)(2y – 7),24 – 12y2 + 12y – 4y + 5 = 22y – 77 – 12y2 + 42y,–12y2 + 11y – 22y + 12y2 – 42y = –77 – 24 – 5, 53y = 106, y = 2;г) (6y + 2)(5 – y) = 47 – (2y – 3)(3y – 1),30y – 6y2 + 10 – 2y = 47 – 6y2 + 2y + 9y – 3,28 – 6y2 + 6y2 – 11y = 47 – 3 – 10, 17y = 34, y = 2.№1059.y = (2x – 5)(3 – 8x) – (1 – 4x)2 = 6x + 16x2 – 15 – 40x – 1 + 8x – 16x2 == – 26x – 26 — линейная функция.A(–1; 10) Т.к.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
