makarytchev-gdz-7-1-1289-2003 (542427), страница 10
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P(–4; b) ∈ графику, значит, y = (–4)2; y = 16, т.е. P(–4; 16);точка Q(4; b) ∈ графику, т.к. 42 = 16.№603.а) 0,23 > 0,232; 0,23 > 0,233; 0,232 > 0,233;б) 1,47 < 1,472; 1,47 < 1,473; 1,472 < 1,473.№604.а) A(a; b) ∈ графику; B(–a; b) — принадлежит графику;C(a; –b); D(–a; –b) — не принадлежит графику;б) A(a; b) ∈ графику; D(–a; –b) — принадлежит, т.к. –b = –a3 = (–a)3;B(–a; b); C(a; –b) — не принадлежит.№605.а) пусть 0 < a < 1, тогда a3 < a2 < a; б) пусть a > 1, тогда a < a2 < a3;в) пусть –1 < a < 0, тогда a < a3 < a2; г) пусть a < –1, тогда a3 < a < a2.К параграфу 8№606.1а) y = = 0,(1);911 11− 0,1 = −=, если y ≈ 0,1;99 10 9011 111− 0,11 = −=, если y ≈ 0,11;99 100 90011 1111− 0,111 = −=, если y ≈ 0,11;99 1000 9000б) y =4= 0,(36);1144 442− 0,4 =−==, если y ≈ 0,4;1111 10 110 5544 91− 0 ,36 =−=, если y ≈ 0,36;1111 25 27544911− 0 ,364 =−=, если y ≈ 0,364.1111 250 2750№607.22 91− 0 ,18 =−=;1111 50 55022 199129− 0 ,19 =−==<, т.е.1111 100 1100 550 1100 11002≈ 0,18 — точнее, чем 0,19.1169№608.π = 3,14159;3,14159 − 3,141 = 0 ,00059 ;3,14159 − 3,142 = 0 ,00041 ;3,14159 − 3114159 199113 − 100000887=− ==≈ 0,00127 ;7 100000 77000007000003,14159 − 31014159 10 1005289 − 10000005289=−==≈ 0 ,00007 .71 100000 717100000071000000Значит, самое точное приближение числа π : 3№609.№610.10.71|1,361 –1,4| = 0,039 < 0,1, ч.т.д.77= 0,4375; а)≈ 0,41616б)7≈ 0,4416в)7≈ 0,4381635 − 327 23− === 0 ,0375 < 0 ,1 ;16 580807 11 175 − 1761−=== 0 ,0025 < 0 ,01 ;16 254004003500 − 35047 2194−=== 0 ,0005 < 0,001 .16 50080008000№611.2 59252 59+=≈ 0 ,3 ; ≈ 0,3;≈ 0 , 4; + = ; 0 ,3 + 0 , 4 = 0 ,7;7 14 147147 14 145≈ 0,4 0,3 + 0,4 = 0,7;1422 3155 239 74− 0 ,3 = −=;− 0,4 =− =;−=.77 10 701414 5 7014 10 70№612.a+b;c=22a − a − ba−b2b − a − bb−aa+ba+b==;a−b−==.222222№613.38,9 − 40а) 38,9 ≈ 40,404219 − 4220=1,1= 0,0275 = 2 ,75% ;401= 0,00024 = 0 ,02% .42200,010,5№614.≈ 0,0667 = 6,67% ;≈ 0,0079 = 0 ,79% ;0,1563т.к.
0,79% < 6,67%, то вагон измерили точнее.xx№615.= 0,1% ,= 0,001 , x = 0,492 < 0,5.492492Значит, точность измерения 0,5.б) 4219 ≈ 4220,704220=Глава IV. Многочлены§ 9. Сумма и разность многочленов24. Многочлен и его стандартный вид№616.а) –6x4; y3; –5y; 11; б) 25ab; ab2; –a2b; 8a; –7b.№617.а) 10x – 8xy –3xy = 10x – 11xy; б) 2ab – 7ab + 7a2 = –5ab + 7a2;в) 3x4 – 5x + 7x2 –8x4 + 5x = –5x4 + 7x2;г) 2a3 + a2 – 17 – 3a2 + a3 – a – 80 = 3a3 – 2a2 – a – 97;д) 12ab2 – b3 – 6ab2 + 3a2b –5ab2 + 2b3 = ab2 + b3 + 3a2b;е) 2a2 – ax3 – a4 – a2x3 + ax3 + 2a4 = a4 + 2a2 – a2x3.№618.а) –a4 + 2a3 – 4a4 + 2a2 – 3a2 = –5a4 + 2a3 – a2;б) 1 + 2y6 – 4y3 – 6y6 + 4y3 – y5 – 9 = –4y6 – y5 – 8;в) 10x2y – 5xy2 – 2x2y + x2y – 3xy2 = 9x2y – 8xy2;г) 3ab3 + 6a2b2 – ab3 – 2a2b2 – 4a2b2 + 7= 2ab3 + 7.№619.а) –8p4 + 12p3 + 4p4 – 8p2 + 3p2 = –4p4 + 12p3 – 5p2;б) 2aa2 + a2 – 3a2 + a3 – a = 3a3 – 2a2 – a;в) 3xx4 + 3xx3 –5x2x3 – 5x2x = 3x5 + 3x4 – 5x5 – 5x3 = –2x5 + 3x4 – 5x3;г) 3a ⋅ 4b2 – 0,8b ⋅ 4b2 – 2ab ⋅ 3b + b ⋅ 3b2 – 1 == 12ab2 – 3,2b3 – 6ab2 + 3b2 – 1 = 6ab2 – 0,2b3 – 1.№620.а) 2a2x3 – ax3 – a4 – a2x3 + ax3 + 2a4 = a4 + a2x3;б) 5x ⋅ 2y2 – 5x ⋅ 3xy – x2y + 6xy2=10xy2 – 15x2y – x2y + 6xy2 = 16xy2 – 16x2y.№621.а) 5x6 – 3x2 + 7 – 2x6 – 3x6 + 4x2 = x2 + 7,при x = –10, х2 + 7 = (–10)2 + 7 = 107;б) 4a2b – ab2 – 3a2b + ab2 – ab + 6 = a2b – ab + 6,при a = –3, b = 2, a2b – ab + 6 = (–3)2 ⋅ 2 – (–3) ⋅ 2 + 6 = 30.№622.а) 6a3 – a10 + 4a3 + a10 – 8a3 + a = 2a3 + a,при a = –3, 2a3 + a = 2 ⋅ (–3)3 + (–3) = –57;б) 4x6y3 – 3x6y3 + 2x2y2 – x6y3 – x2y2 + y = x2y2 + y,при x = –2, y = –1, x2 + y2 + y = (–2)2 ⋅ (–1)2 + (–1) = 3.№623.при x = 0, 2х2 + 1 = 2 ⋅ 02 + 1 = 1, при x = –2, 2х2 + 1 = 2 ⋅ (–2)2 + 1 = 9,при x = 3, 2х2 + 1 = 2 ⋅ 32 + 1 = 19, при x=–4, 2х2 + 1 = 2 ⋅ (–4)2 + 1 = 33.Для любого x 2x2 + 1 > 0, т.к.
для любого x 2x2 ≥ 0 и 1 > 0.Т.е. нет таких x, чтобы 2x2 + 1 ≤ 0.№624.x2 + y2 + 1 > 0 для любых x и y, т.к. x2 + y2 ≥ 0, 1 > 0, а сумманеотрицательных чисел и положительного числа больше нуля.71№625.а) a ⋅ 10 + b = 10a + b; б) a ⋅ 100 + b ⋅ 10 + c = 100a + 10b + c.№626.а) 17a4 – 8a5 + 3a – a3 – 1 = –8a5 + 17a4 – a3 + 3a – 1;б) 35 – c6 + 5c2 – c4 = –c6 – c4 + 5c2 + 35.№627.а) x4 – 5 – x2 + 12x = –5 + 12x – x2 + x4; б) 2y + y3 – y2 + 1 =1 + 2y – y2 + y3.№628.а) 4a6 – 2a7 + a – 1; 7 степень; б) 5p3 – p – 2; 3 степень;в) 1 – 3x; 1 степень; г) 4xy + xy2 – 5x2 + y; 3 степень;д) 8x4y + 5x2y3 – 11; 5 степень; е) xy + yx + xz – 1; 2 степень.№629.а) x4 + 3x; б) x2y + y3x.№630.а) при x = 1,97, x2 + 4,23 = 1,972 + 4,23 = 8,1109;в) при a=2,3; b=138,9, a4+2b=2,34+2⋅138,9 = 27,9841 + 277,8 = 305,7841.№631.158133 16а)0,3y = 70, y = 233 ; б) x = –1, x = – ; в) a = − , a = – : = −3 .385977 97№632.25 ⋅ 8 25 ⋅ 23 2853 ⋅ 252 53 ⋅ 54 145 ⋅ 38 210 ⋅ 38 2= 8 = ; б)а)== 8 = 1 ; в)= 9 9 = .44842555693 ⋅2322( )№633.
y = 0,01x;а) при y = 240, 240 = 0,01 ⋅ x; x = 24000;б) при y = –100, –100 = 0,01 ⋅ x; x = –10000.№634.а) (2n – 2) + 2n = 4n – 2; б) (2n – 1) + (2n + 1) + (2n + 3) = 6n + 3.25. Сложение и вычитание многочленов№635.а) (4x3 – 5x – 7) + (x3 – 8x) = 4x3 – 5x – 7 + x3 – 8x = 5x3 – 13x – 7;б) (5y2 – 9) – (7y2 – y + 5) = 5y2 – 9 – 7y2 + y – 5 = –2y2 + y – 14.№636.а) (2a3 – 5a + 5) + (a3 – 4a – 2) = 2a3 – 5a + 5 + a3 – 4a – 2 = 3a3 – 9a + 3;б) (2a3 – 5a + 5) – (a3 – 4a – 2) = 2a3 – 5a + 5 – a3 + 4a + 2 = a3 – a + 7;в) (a3 – 4a – 2) – (2a3 – 5a + 5) = a3 – 4a – 2 – 2a3 + 5a – 5 = –a3 + a – 7.№637.а) (1 + 3a) + (a2 – 2a)= 1 + 3a + a2 – 2a = a2 + a + 1;б) (2x2 + 3x) + (–x + 4) = 2x2 + 3x – x + 4 = 2x2 + 2x + 4;в) (y2 – 5y) + (5y – 2y2) = y2 – 5y + 5y – 2y2 = –y2;г) (b2 – b + 7) – (b2 + b+ 8) = b2 – b + 7 – b2 – b – 8 = –2b – 1;д) (8n3 – 3n2) – (7 + 8n3 – 2n2) = 8n3 – 3n2 – 7 – 8n2 + 2n2 = –n2 – 7;е) (a2 + 5a + 4) – (a2 + 5a – 4) = a2 + 5a + 4 – a2 – 5a + 4 = 8.72№638.а) 5,2a – (4,5a + 4,8a2) = 0,7a – 4,8a2;б) –0,8b2 + 7,4b + (5,6b – 0,2b2) = –b2 + 13b;в) 8x2 + (4,5 – x2) – (5,4x2 – 1) = 8x2 + 4,5 – x2 – 5,4x2 + 1 == –6,4x2 + 8x2 + 5,5 = 1,6x2 + 5,5;г) (7,3y – y2 + 4) + 0,5y2 – (8,7y – 2,4y2) == 7,3y – y2 + 4 + 0,5y2 – 8,7y + 2,4y2 = 1,9y2 – 1,4y + 4.№639.а) 18x2 – (10x – 5 + 18x2) = 18x2 – 10x + 5 – 18x2 = –10x + 5;б) –12c2 + 5c + (c + 11c2) = –c2 + 6c;в) (b2 + b – 1) – (b2 – b + 1) = b2 + b – 1 – b2 + b – 1 = 2b – 2;г) (15 – 7y2) – (y3 – y2 – 15) = 15 – 7y2 – y3 + y2 + 15 = –y3 – 6y2 + 30.№640.а) (a + b) + (a – b) = 2a, (a + b) – (a – b) = 2b;б) (a – b) + (a + b) = 2a, (a – b) – (a + b) = –2b;в) (–a – b) + (a – b) = –2b, (–a – b) – (a – b) = –2a;г) (a – b) + (b – a) = 0, (a – b) – (b – a) = 2a – 2b.№641.а) (x –y) + (y – z) + (z – x) = x –y + y – z + z – x = 0 верно;б) (a2 – 5ab) – (7 –3ab) + (2ab – a2) =a2–5ab–7 +3ab + 2ab – a2 = –7 верно.№642.а) M + (5x2 – 2xy) = 6x2 + 9xy – y2, M = 6x2 + 9xy – y2 – (5x2 – 2xy),M = 6x2 + 9xy – y2 – 5x2 + 2xy, M = x2 + 11xy – y2;б) M – (4ab – 3b2) = a2 – 7ab + 8b, M = a2 – 7ab + 8b2 + 4ab – 3b2,M = a2 – 3ab + 5b2;в) (4c4 – 7c2 + 6) – M = 0, M = 4c4 – 7c2 + 6.№643.
а) 5x2 – 3x – 9 + N = 0, N = –5x2 + 3x + 9;б) 5x2 – 3x – 9 + N = 18, N = –5x2 + 3x + 27;в) 5x2 – 3x – 9 + N = 2x – 3, N = 2x – 3 – 5x2 + 3x + 9 = –5x2 + 5x + 6;г) 5x2 – 3x – 9 + N = x2 – 5x + 6, N = x2–5x+6–5x2+3x + 9 = –4x2 – 2x + 15.№644.а) (a2 – 0,45a + 1,2) + (0,8a2 – 1,2a) – (1,6a2 – 2a) == a2 – 0,45a + 1,2 + 0,8a2 – 1,2a – 1,6a2 + 2a = 0,2a2 + 0,35a + 1,2;б) (y2 – 1,75y – 3,2) – (0,3y2 + 4) – (2y – 7,2) == y2 – 1,75y – 3,2 – 0,3y2 – 4 – 2y + 7,2 = 0,7y2 – 3,75y;в) 6xy – 2x2 – (3xy + 4x2 + 1) – (–xy – 2x2 – 1) == 6xy – 2x2 – 3xy – 4x2 – 1 + xy + 2x2 + 1 = –4x2 + 4xy;г) –(2ab2 – ab + b) + 3ab2 – 4b – (5ab – ab2) == –2ab2 + ab – b + 3ab2 – 4b – 5ab + ab2 = 2ab2 – 4ab – 5b.№645.а) 8a2b+(–5a2b+4b2)+(a2b–5b2+2)=8a2b–5a2b + 4b2 + a2b – 5b2+2=4a2b–b2 + 2;б) (xy + x2 + y2) – (x2 + y2 – 2xy) – xy = xy + x2 + y2 – x2 – y2 + 2xy – xy = 2xy.№646.
(5,7a2b – 3,1ab + 8b3) – (6,9ab – 2,3a2b + 8b3) == 5,7a2b – 3,1ab + 8b3 – 6,9ab + 2,3a2b – 8b3 = 8a2b – 10ab;а) при a = 2; b = 5, 8a2b – 10ab = 8 ⋅ 22 – 10 ⋅ 2 ⋅ 5 = 160 – 100 = 60;б) при a = –2; b = 5, 8a2b–10ab=8 ⋅ (–2)2 ⋅ 3 – 10 ⋅ (–2) ⋅ 3 = 96 + 60 = 156.73№647.5x2 – (3xy – 7x2) + (5xy – 12x2) = 5x2 – 3xy + 7x2 + 5xy – 12x2 = 2xy;а) при x = –0,25; y = 4, 2xy = 2 ⋅ (–0,25) ⋅ 4 = –2;б) при x = –5; y = 0,1, 2xy = 2 ⋅ (–5) ⋅ 0,1 = 1.№648.(0,7x4 + 0,2x2 –5) – (–0,3x4 +1x – 8) =5= 0,7x4 + 0,2x2 – 5 + 0,3x4 – 0,2x2 + 8 = x4 + 3 > 0 при любом x.№649.(7a3 – 6a2b + 5ab2) + (5a3 + 7a2b + 3ab2) – (10a3 + a2 + 8ab2) == 7a3 – 6a2b + 5ab2 + 5a3 + 7a2b + 3ab2 – 10a3 – a2 – 8ab2 = 2a3.11.При a = –0,25, 2 ⋅ (–0,25) = –2 ⋅=−6432Неправильно, т.к.
значение не зависит от b.№650.а) x2+y2 – 2xy + 1 + N = y2 + 1, N = y2 + 1 – (x2 + y2 – 2xy + 1), N=–x2+2xy;б) x2+y2–2xy + 1 + N = x2 + 1, N = x2 + 1 – (x2 + y2 – 2xy + 1), N=–y2 + 2xy.№651.323( x2 – 0,4xy – 1,5y + 1) – (y2 – xy + 0,6x2) = x2 – 0,4xy – 1,5y + 1 –555– y2 +2xy – 0,6x2 = –y2 –1,5y + 1.5№652.а) 1,7 – 10b2 – (1 – 3b2) + (2,3 + 7b2) = 1,7 – 10b2 – 1 + 3b2 + 2,3 + 7b2 = 3;б) 1 – b2 – (3b – 2b2) + (1 + 3b – b2) = 1 – b2 – 3b + 2b2 + 1 + 3b – b2 = 2.№653.а) x + y + z = 5a2 + 6ab – b2 – 4a2 + 2ab + 3b2 + 9a2 + 4ab = 10a2 + 12ab + 2b2;б) x – y – z = 5a2 + 6ab – b2 – (–4a2 + 2ab + 3b2) – (9a2 + 4ab) == 5a2 + 6ab – b2 + 4a2 – 2ab – 3b2 – 9a2 – 4ab = –4b2.№654.а) (23 + 3x) + (8x – 41) = 15, 23 + 3x + 8x – 41 = 15, 11x – 18 = 15,11x = 33, x = 3;б) (19 + 2x) – (5x – 11) = 25, 19 + 2x – 5x + 11 = 25,52, x =1 ;33в) (3,2y – 1,8) – (5,2у + 3,4) = –5,8, 3,2y – 1,8 – 5,2у – 3,4 = –5,8,–2y – 5,2 = –5,8, –2y = –0,6, y = 0,3;г) 1 – (0,5x – 15,8) = 12,8 – 0,7x, 1 – 0,5x + 15,8 = 12,8 – 0,7x,–0,5x + 0,7x = 12,8 – 1 – 15,8, 0,2x = –4, x = –20;д) 3,8 – 1,5y + (4,5y – 0,8) = 2,4y + 3, 3,8 – 1,5y + 4,5y – 0,8 — 2,4y = 3,3 + 0,6y = 3, 0,6y = 0, y = 0;е) 4,2y + 0,8 = 6,2y – (1,1y + 0,8) + 1,2, 4,2y + 0,8 = 6,2y – 1,1y – 0,8 + 1,2,44,2y + 0,8 = 5,1y + 0,4, 4,2y – 5,1y = 0,4 – 0,8, 0,9y = 0,4, y = .930 – 3x = 25, 3x = 5, x =74№655.а) 8y –3 – (5 – 2y) = 4,3, 8y –3 – 5 + 2y = 4,3, 10y – 8 = 4,3,10y = 12,3, y = 1,23.б) 0,5y – 1 – (2y + 4) = y, 0,5y – 1 – 2y – 4 – y = 0, 2,5y = –5, y = –2;в) –8x + (4 + 3x) = 10 – x, –8x + 4 + 3x + x = 10, 4x = –6, x = –1,5;г) 1,3x – 2 – (3,3x + 5) = 2x + 1, 1,3x – 2 – 3,3x – 5 – 2x = 1,4x – 7 = 1, 4x = –8, x = –2.№656.а) 3x3 – 2x2 – x + 4 = (3x2 – 2x2) + (4 –x);б) –5y4 + 4y3 + 3y2 – 2y = (–5y4 +4y3) + (3y2 – 2y).№657.а) x3 + 2x2 – 3x – 5 = x3 – (–2x2 + 3x + 5);б) 3a4 + 2a3 + 5a2 – 4 = 3a4 – (–2a3 – 5a2 + 4).№658.а) Если I число — x, то II число — (x + 1), а III число — (x + 2).Запишем их сумму:x + (x + 1) + (x + 2) = 3x + 3 — кратно 3.б) Если I число — x, то II число — (x + 1), а III число — (x + 2), IV число —(x + 3).
Запишем сумму:x + (x + 1) + (x + 2) + (x + 3) = 4x + 6;4x кратно 4, 6 не кратно 4, значит 4x + 6 не кратно 4.№659.если x = 1,8, то y ≈ 3,2; y = 3,24,|3,24 – 3,2| = 0,04 — абсолютная погрешность.3,24 − 3,2= 0,0125 = 1,25% — относительная погрешность.3,24№660.2 5⎞25⎛а) при a = , b = , 6(2a – b) =6 ⋅ ⎜ 2 ⋅ 2 − ⎟ = 8 – 5 = 3;3 6⎠36⎝11⎛a b⎞б) при a = , b = 0,2, 15 ⋅ ⎜ + ⎟ = 3a + 5b = 3 ⋅ + 5 ⋅ 0, 2 = 1 + 1 = 2.33⎝ 5 3⎠№661.11а) (2x2)3 ⋅ x2 = 8x6 ⋅ x2 = 2x8;4411б) (–3y4)3 ⋅ y5 = –27y12 ⋅ y5 = –3y17;99в) –0,2a2b3 ⋅ (–5a3b2)2 = –0,2a2b3 ⋅ 25a6b4 = –5a8b7;1г) (–0,5c4d)3 ⋅ (–4c2d2)2 = – c12d3. 16c4d4 = –2c16d7.8№662.при x = 1,4, y = 0,157; x3 – y = 1,43 – 0,157 = 2,587.75§ 10.
Произведение одночлена и многочлена26. Умножение одночлена на многочлен№663. а) 2x(x2 – 7x – 3) = 2x3 – 14x2 – 6x;б) –4b2(5b2 – 3b – 2) = –20b4 + 12b3 + 8b2;в) (3a3 – a2 + a)(–5a3) = –15a6 + 5a5 – 5a4;г) (y2 – 2,4y + 6) ⋅ 1,5y = 1,5y3 – 3,6y2 + 9y;д) –0,5x2 ⋅ (–2x2 – 3x + 4) = x4 + 1,5x3 – 2x2;е) (–3y2 + 0,6y)(–1,5y3) = 4,5y5 – 0,9y4.№664.
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