makarytchev-gdz-7-1-1289-2003 (542427), страница 20
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–26 ⋅ (–1) – 16 = 10, то A ∈ графику.B(0; 16) Т.к. –26 ⋅ 0 – 16 = –16 ≠ 16, то B ∉ графику.№1060.а) (3n – 1)(n + 1) + (2n – 1)(n – 1) – (3n + 5)(n – 2) == 3n2 + 3n – n – 1 + 2n2 – 2n – n + 1 – (3n2 – 6n + 5n – 10) == 5n2 – n – 3n2 + n + 10 = 2n2 + 10,при n = –3,5, 2n2 + 10 = 2(–3,5)2 + 10 = 24,5 + 10 = 34,5;б) (5y – 1)(2 – y) – (3y + )(1 – y) + (2y + 6)(y – 3) == 10y – 5y2 – 2 + y – (3y – 3y2 + 4 – 4y) = 2y2 – 6y + 6y – 18 == 3y2 + 11y – 20 – 3y2 + y – 4 = 12y – 24 = 12(y – 2),при y = 4, 12(y – 2) = 12 ⋅ (4 – 2) = 12 ⋅ 2 = 24.№1061.а) (a – 3)(a2 – 8a + 5) – (a – 8)(a2 – 3a + 5) == a3 – 8a2 + 5a – 3a2 + 24a – 15 – a3 + 3a2 – 5a + 8a2 – 24a + 40 = 25;б) (x2 – 3x + 2)(2x + 5) – (2x2 + 7x + 17)(x – 4) == 2x3 + 5x2 – 6x2 – 15x + 4x + 10 – 2x3 + 8x2 – 7x2 + 28x – 17x + 68 = 78.№1062.(a2 + b2)(ab + cd) – ab(a2 + b2 – c2 – d2) ==a3b+a2cd+ab3+b2cd–a3b–ab3 + abc2 + abd2 = a2cd + b2cd + abc2 + abd2;(ac+bd)(ad+bc) = a2cd + abc2 + abd2 + b2cd = a2cd + b2cd + abc2 + abd2.134№1063.(b + c – 2a)(c – b) + (c + a – 2b)(a – c) – (a + b – 2c)(a – b) == (bc – b2 + c2 – bc – 2ac + 2bc) + (ac – c2 + a2 – ac – 2ab + 2bc) –– (a2–ab+ab–b2 – 2ac + 2bc)=–b2 + c2 – 2ac + 2ab – c2 + a2 – ac – 2ab ++ 2bc – a2 + b2 + 2ac – 2bc = 0.№1064.а) (a+8)2–2(a+8)(a–2)+(a–2)2=((a+8)–(a–2))2=(a+8–a + 2)2 = 102 = 100;б) (y–7)2–2(y–7)(y–9)+(y–9)2=((y–7)–(y–9))2 =(y – 7 – y + 9)2 = 22 = 4.№1065.а) (ax–2(a+2))(a(x–1)+2)+2(4 – a2) + 3a2x =(ax – 2a – 4)(ax– a + 2)+8 –– 2a2 + 3a2x =a2x2–a2x + 2ax – 2a2 – 4a – 4ax + 4a – 8 + 8 – 2a2 + 3a2x == a2x2 – 2ax = ax(ax – 2);б) (3 – b(c – 1))(bc(bc + 3b + 1)=(3–bc+b)(bc+4b+4) + b2c2 + 3b2 + bc == 3bc + 12b + 12 – b2c2 – 4b2c + b2c2 +4b2 + 4b + b2c2 + 3b2c + bc == 4b2 + 16b + 12 = 4b(b + 4) + 12.№1066.а) 2(a2 – 1)2 – (a2 + 3)(a2 – 3) –1 2(a + a – 4)(2a2 + 3) =21(2a4 + 3a2 + 2a3 + 3a – 8a2 – 12) =233= 2a4 – 4a2 + 2 – a4 + 9 – a4 – a2 – a3 – a + 4a2 + 6 =223311= –a3 – a2 – a + 15 + 2 = –a3 – 1 a2 – 1 a + 17;2222б) 4(m3 – 3)2 – (m2 – 6)(m2 + 6) – 9(8 – m + m2)(1 –m) == 4(m6 –6m3 + 9) – m4 + 36 – 9(8 – 8m – m + m2 + m2 – m3) == 4m6 – 24m3 + 36 – m4 + 36 – 72 + 72m + 9m – 9m2 – 9m2 + 9m3 == 4m6 – m4 – 15m3 – 18m2 + 81m.№1067.(a(a + 2b) + b2)(a(a – 2b) + b2)((a2 – b2) + 4a2b2) == (a2 + 2ab + b2)(a2 – 2ab + b2)(a4 – 2a2b2 + b4 + 4a2b2) == (a + b)2(a – b)2(a2 + b2)2 = [(a2 – b2)(a2 + b2)]2 = (a4 – b4)2 = a8 – 2a4b4 + b4.№1068.а) (a+b)2(a–b)–2ab(b – a) – 6ab(a – b) = ((a + b)2 + 2ab – 6ab)(a – b) == (a2+2ab+b2–4ab)(a–b)=(a2–2ab + b2)(a – b) =(a – b)2(a – b) =(a – b)3;б) (a+b)(a–b)2+2ab(a + b)–2ab(–a – b) = (a + b)((a – b)2 + 2ab + 2ab) == (a+b)(a2–2ab+b2 + 4ab)=(a+b)(a2 + 2ab + b2)=(a+b)(a + b)2 =(a + b)2.№1069.(a2+b2)(a4 – a2b2 + b4) – (a3 – b3)(a3 + b3) = (a2)3 + (b2)3 – ((a3)2 – 9b3)2 == a6 + b6 – a6 + b6 = 2b6.№1070.а) (y + 5)(y2 – 5y + 25) – y(y2 + 3) = y3 + 125 – y3 – 3y = 125 – 3y,при y = –2, 125 – 3y = 125 – 3(–2) = 125 + 6 = 131;б) a2(a + 4) – (a + 2)(a2 – 2a + 4) = a3 + 4a2 – a3 – 8 = 4a2 – 8,при a = 3, 4a2 – 8 = 4 ⋅ 32 – 8 = 36 – 8 = 28;= 2(a4 – 2a2 + 1) – (a4 – 9) –135в) x(x + 3)2 – (x – 1)(x2 + x + 1) = x3 + 6x2 + 9x – x3 + 1 = 6x2 + 9x + 1,при x = –4, 6x2 + 9x + 1 = 6 ⋅ 16 + 9 ⋅ (–4) + 1 = 96 – 36 + 1 = 61;г) (2p–1)(4p2 + 2p + 1) – p(p – 1)(p + 1) = 8p3 – 1 – p3 + p = 7p3 + p – 1,328 3189 1 1931при p = , 7p3 + p – 1 = 7 ⋅ + − 1 =+ == 24 .28 28288№1071.
(a2 + b2)(c2 + d2) = a2c2 + a2d2 + b2c2 + b2d2;(ac + bd)2 + (ad – bc)2 = a2c2 + 2abcd + b2d2 + a2d2 – 2abcd + b2c2 == a2c2 + a2d2 + b2c2 + b2d2.№1072.(pr+cqs)2 + c(ps – qr)2 = p2r2 + 2prcqs + c2q2s2 + cp2s2 – 2pscqr + cq2r2 == r2(p2 + cq2) + cs2(cq2 + p2) = (p2 + cq2)(r2 + cs2).№1073.(x2 + x–1)(x – a) = x3 – ax2 + x2 – ax – x + a = x3 – (a – 1)x2 – (a +1)x + a;а) при a = 1, x3 – 0 ⋅ x2 – 2 ⋅ x + 1 = x3 – 2x + 1;б) при a = –1, x3 – (–2)x2 – 0 ⋅ x – 1 = x3 + 2x2 – 1.№1074.(x2 – 10x + 6)(2x + b) = 2x3 – 20x2 + 12x + bx2 – 10bx + 6b == 2x3 + (b – 20)x2 + (12 – 10b)x + 6b;а) при b = 20, 2x3 + 0 ⋅ x2 + (12 – 200)x + 120 = 2x3 – 188x + 120;б) при b = 1, 2x3 + (1 – 20)x2 + (12 – 10)x + 6 = 2x3 – 19x2 + 2x + 6.№1075.а) 2,1a2 – 2,1b2 = 2,1(a – b)(a + b);б) 1,7a2 + 1,7b2 = 1,7(a2 + b2);3322в) 1,1a – 1,1b = 1,1(a – b)(a + ab + b ); г) 7a3 + 7b3 = 7(a + b)(a2 – ab + b2);д) 2a4–2b4 = 2(a2–b2)(a2+b2) = 2(a – b)(a + b)(a2 + b2); е) 5a4+5b4 = 5(a4 + b4);ж) 2,5a6 – 2,5b6=2,5(a3–b3)(a3+b3)=2,5(a – b)(a2 + ab + b2)(a + b)(a2 – ab + b2);з) 1,2a6 + 1,2b6 = 1,2(a2 + b2)(a4 + a2b2 + b4);и) 3a8 – 3b8 = 3(a4 – b4)(a4 + b4) = 3(a2 – b2)(a2 + b2)(a4 + b4) == 3(a – b)(a + b)(a2 + b2)(a4 + b4).№1076.а) 9c15 – c13 = c13(9c2 – 1) = c13(3c – 1)(3c + 1);1 ⎞1 ⎞⎛1⎞1 20⎛⎛x = x20 ⎜ x 2 − ⎟ = x 20 ⎜ x − ⎟⎜ x + ⎟ ;б) x22 –49 ⎠7 ⎠⎝7⎠49⎝⎝в) a5 – 0,64a2 = a2(a3 – 0,64);716 ⎞4 ⎞⎛4⎞⎛⎛г) y7 – 1 y5 = y5 ⎜ y 2 − ⎟ = y5 ⎜ y − ⎟⎜ y + ⎟ .9933⎠⎝⎠⎝⎠⎝№1077.а) 2x8 – 12x4 + 18 = 2(x8 – 6x + 9) = 2(x4 – 3)2 = 2(х4 – 3)(х4 – 3);б) –2a6–8a3b–8b2=–2(a6+4a3b+4b2)=–2(a3+2b)2 = –2(а3 + 2b) (а3 + 2b);в)a4b+6a2b3+9b5=b(a4+6a2b2 + 9b4) = b(a2 + 3b2)2 = b(а3 + 3b) (а3 + 3b);г) 4x + 4xy6 + xy12 = x(4 + 4y6 + y12) = x(2 + y6)2 = x(2 + y6)(2 + y6).№1078.а) 70a–84b–20ab–24b2=10a(7 + 2b) – 12b(7 + 2b) = 2(7 + 2b)(5a – 6b);б) 21bc2 – 6c – 3c2 + 42b = 21b(c2 + 2) – 3c(c2 + 2) = 3(c2 + 2)(7b – c);в) 12y–9x2+36–3x2y=3y(4 – x2) + 9(4 – x2) = (3y + 9)(4 – x2)=3(y+3)(2–x)(2 + x);г) 30a2–18a2b–72b+120a=30a(a2 + 4) – 18b(a2 + 4) = 6(a2 + 4)(5a – 3b).136№1079.а) 3a3–3ab2+a2b – 3 = 3a(a2–b2)+b(a2–b2)=(3a + b)(a2 – b2)=(3a+b)(a–b)(a + b);б) 2x–a2y–2a2x+y = 2x(1 – a2) + y(1 – a2) = (2x+ y)(1– a2)=(2x + y)(1 – a)(1 + a);в) 3p–2c2–3c3p+2=3p(1–c3)+2(1 – c3) = (3p + 2)(1 – c3)=(3p+2)(1–c)(1 + c + c2);г) a4 –24 +8a –3a2 =a3(a –3) +8(a –3) =(a3 +8)(a –3)=(a + 2)(a2 – 2a + 4)(a – 3).№1080.а) x3 + 3x2 – 4x – 12 = 0, x2(x + 3) – 4(x + 3) = 0,(x + 3)(x – 2)(x + 2) = 0, x1 = –3; x2 = 2; x3 = –2;б) 2m3 – m2 – 18m + 9 = 0, m2(2m – 1) – 9(2m – 1) = 0,(m – 3)(m + 3)(2m – 1) = 0, m1 = 0,5; m2 = 3; m3 = –3;в) y3 – 6y2 = 6 – y,y2(y – 6) + (y – 6) = 0,(y2 + 1)(y – 6) = 0, y = 6, больше нет, т.к.
y2 + 1 > 0;г) 2a3 + 3a2 = 2a + 3, a2(2a + 3) – (2a + 3) = 0,(a – 1)(a + 1)(2a + 3) = 0, a1 = 1; a2 = –1; a3 = 1,5.№1081.а) x3 – 2x2 – x + 2 = 0,x2(x – 2) – (x – 2) = 0,(x – 1)(x + 1)(x – 2) = 0, x1 = 1; x2 = –1; x3 = 2;б) y3 – y2 = 16y – 16, y2(y – 1) – 16(y – 1) = 0,(y – 4)(y + 4)(y – 1) = 0, y1 = 4; y2 = –4; y3 = 1;в) 2y3 – y2 – 32y + 16 = 0, y2(2y – 1) – 16(2y – 1) = 0,(y – 4)(y + 4)(2y – 1) = 0, y1 = 4; y2 = –4; y3 = 0,5;г) 4x3 – 3x2 = 4x – 3, x2(4x – 3) – (4x – 3) = 0,(x – 1)(x + 1)(4x – 3) = 0, x1 = 1; x2 = –1; x3 = 0,75.№1082.а) x2 – y2 – 1,5(x – y) = (x – y)(x + y) – 1,5(x – y) = (x – y)(x + y – 1,5);б) x2 – a2 + 0,5(x + a) = (x – a)(x + a) + 0,5(x + a) = (x – a+ 0,5)(x + a);в) 4a2 – b2 – 2a + b = (2a – b)(2a + b) – (2a – b) = (2a – b)(2a + b – 1);г) p2 – 16c2 – p – 4c = (p – 4c)(p + 4c) – (p + 4c) = (p – 4c – 1)(p + 4c);д) a2 + 6a + 6b – b2 = (a – b)(a + b) + 6(a + b) =(a – b – 6)(a + b);е) x2 – 7x + 7y – y2 = (x – y)(x + y) – 7(x – y) = (x – y)(x + y – 7).№1083.а) x2(x + 2y) – x – 2y = (x2 – 1)(x + 2y) = (x – 1)(x + 1)(x + 2y);б) y2(2y–5)–8y+20=y2(2y–5)–4(2y–5)=(y2–4)(2y–5)=(y–2)(y + 2)(2y – 5);в) a3–5a2–4a+20=a2(a – 5) – 4(a – 5) = (a2 – 4)(a–5)=(a–2)(a+2)(a – 5);г) x3–4x2–9x+36=x2(x – 4) – 9(x – 4) = (x2 – 9)(x – 4)=(x–3)(x+3)(x – 4).№1084.а) a2–b2+2(a+ b)2 = (a – b)(a + b) + 2(a + b)2 = (a – b+ 2(a + b))(a + b) == (3a + b)(a + b);б) b2 – c2 – 10(b – c)2 = (b – c)(b + c) – 10(b – c)2 == (b – c)(b + c – 10(b – c)) = (b – c)(11c – 9b);в) 2(x – y)2 + 3x2 – 3y2 = 2(x – y)2 + 3(x – y)(x + y) == (x – y)(2(x – y) + 3(x + y) = (x – y)(5x + y);г) 5a2 – 5 – 4(a + 1)2 = 5(a – 1)(a + 1) – 4(a + 1)2 == (5(a – 1) – 4(a + 1))(a + 1) = (a – 9)(a + 1).137№1085.а) x2 + y2 + 2xy – 1 = (x + y)2 – 1 = (x + y – 1)(x + y + 1);б) a2 + b2 – 2ab – 25 = (a – b)2 – 25 = (a – b – 5)(a – b + 5);в) 36 – b2 – c2 + 2bc = 36 – (b – c)2 = (6 – b + c)(6 + b – c);г) 49 – 2ax – a2 – x2 = 49 – (x + a)2 = (7 – x – a)(7 + x + a);д) 1 – 25x2 + 10xy – y2 = 1 – (5x – y)2 = (1 – 5x + y)(1 + 5x – y);е) b2 – a2 – 12a – 36 = b2 – (a + 6)2 = (b – a – 6)(b + a + 6);ж) 81a2 + 6bc – 9b2 – c2 = 81a2 – (3b – c)2 = (9a – 3b + c)(9a + 3b – c);з) b2c2 – 4bc – b2 – c2 + 1 = (b2c2 – 2bc + 1) – (b2 + 2bc + c2) == (bc – 1)2 – (b + c)2 = (bc – 1 – b – c)(bc – 1 + b + c).№1086.а)x3 + y3 + 2xy(x + y) = (x + y)(x2 – xy+y2)+2xy(x+y)=(x+y)(x2 + xy + y2);б) x3–y3–5x(x2+xy+y2)=(x–y)(x2 + xy + y2) – 5x(x2 + xy + y2)=(–4x–y)(x2+xy + y2);в) a3 – b3 + 5a2b – 5ab2=(a–b)(a2 + ab + b2) + 5ab(a – b)=(a – b)(a2 + 6ab + b2);г) p3–2p2+2p–1=(p3–1)–2p(p–1)=(p – 1)(p2 + p + 1) – 2p(p – 1)=(p–1)(p2–p + 1);д) 8b3 + 6b2 + 3b + 1 = (8b3 + 1) + 3b(2b + 1) == (2b + 1)(4b2 – 2b + 1) + 3b(2b + 1) = (2b + 1)(4b2 + b + 1);е) a3 – 4a2 + 20a – 125 = (a3 – 125) – 4a(a – 5) == (a – 5)((a2 + 5a + 25) – 4a(a – 5) = (a – 5)(a2 + a + 25).№1087.а) x3 + y3 + 2x2 – 2xy + y2 = (x + y)(x2 – xy + y2) + 2(x2 – xy + y2) == (x2 – xy + y2)(x + y + 2);б) a3 – b3 + 3a2 + 3ab + 3b2 = (a – b)(a2 + ab + b2) + 3(a2 + ab + b2) == (a2 + ab +b2)(a – b + 3);в) a4 + ab3 – a3b – b4 = a(a3 + b3) – b(a3 + b3) = (a – b)(a3 + b3) == (a – b)(a + b)(a2 – ab + b2);г) x4 + x3y – xy3 – y4 = x3(x + y) – y3(x + y) = (x + y)(x3 – y3) == (x + y)(x – y)(x2 + xy + y2).№1088.а) x2 – 2xy + y2 + a2 = (x – y)2 + a2 ≥ 0;б) 4x2 + a2 – 4x + 1 = (2x – 1)2 + a2 ≥ 0;в) 9b2 – 6b + 4c2 + 1 = (3b – 1)2 + 4c2 ≥ 0;г) a2+2ab + 2b2 + 2b + 1 = (a2 + 2ab + b2) + (b2 + 2b + 1)=(a + b)2 + (b + 1)2 ≥ 0;д) x2 – 4xy + y2 + x2y2 + 1 = (x2 – 2xy + y2) = (x2y2 – 2xy + 1)=(x–y)2+(xy+1)2 ≥ 0;е) x2 + y2 + 2x + 6y + 19 = (x2 + 2x + 1) + (y2 + 6y + 9) = (x + 1)2 + (y + 3)3 ≥ 0.№1089.а) a2 + 16a + 64 = (a + 8)2 ≥ 0 не может;б) – b2 – 25 + 10b = –(b2 – 10 + 25) = –(b – 5)2 ≤ 0 не может;в) –x2 + 6x – 9 = –(x2 – 6x + 9) = –(x – 3)2 ≤ 0 только 0;г) (y + 10)2 – 0,1 < 0 ⇒ (y + 10)2 < 0,1 может;д) 0,001 – (a + 100)2 > 0 ⇒ (a + 100)2 < 0,001 может.№1090.а) (2n + 3)(3n – 7) – (n + 1)(n – 1) = 6n2 – 14n + 9n – 21 – n2 + 1 == 6n2 – 5n – 20 = 5(n2 – n – 4) кратно 5;138б) (7n + 8)(n–1)+(3n–2)(n + 2) = 7n2 – 7n + 8n – 8 + 3n2 + 6n – 2n – 4 ==10n2+5n–12=5n(2n+1)–12 не кратно 5, т.к.
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