Морозов А.В. Домашняя работа по алгебре за 7 класс
Подождите немного. Документ загружается.
71
Глава IV. Многочлены
§ 9. Сумма и разность многочленов
24. Многочлен и его стандартный вид
№616.
а) –6x
4
; y
3
; –5y; 11; б) 25ab; ab
2
; –a
2
b; 8a; –7b.
№617.
а) 10x – 8xy –3xy = 10x – 11xy; б) 2ab – 7ab + 7a
2
= –5ab + 7a
2
;
в) 3x
4
– 5x + 7x
2
–8x
4
+ 5x = –5x
4
+ 7x
2
;
г) 2a
3
+ a
2
– 17 – 3a
2
+ a
3
– a – 80 = 3a
3
– 2a
2
– a – 97;
д) 12ab
2
– b
3
– 6ab
2
+ 3a
2
b –5ab
2
+ 2b
3
= ab
2
+ b
3
+ 3a
2
b;
е) 2a
2
– ax
3
– a
4
– a
2
x
3
+ ax
3
+ 2a
4
= a
4
+ 2a
2
– a
2
x
3
.
№618.
а) –a
4
+ 2a
3
– 4a
4
+ 2a
2
– 3a
2
= –5a
4
+ 2a
3
– a
2
;
б) 1 + 2y
6
– 4y
3
– 6y
6
+ 4y
3
– y
5
– 9 = –4y
6
– y
5
– 8;
в) 10x
2
y – 5xy
2
– 2x
2
y + x
2
y – 3xy
2
= 9x
2
y – 8xy
2
;
г) 3ab
3
+ 6a
2
b
2
– ab
3
– 2a
2
b
2
– 4a
2
b
2
+ 7= 2ab
3
+ 7.
№619.
а) –8p
4
+ 12p
3
+ 4p
4
– 8p
2
+ 3p
2
= –4p
4
+ 12p
3
– 5p
2
;
б) 2aa
2
+ a
2
– 3a
2
+ a
3
– a = 3a
3
– 2a
2
– a;
в) 3xx
4
+ 3xx
3
–5x
2
x
3
– 5x
2
x = 3x
5
+ 3x
4
– 5x
5
– 5x
3
= –2x
5
+ 3x
4
– 5x
3
;
г) 3a ⋅ 4b
2
– 0,8b ⋅ 4b
2
– 2ab ⋅ 3b + b ⋅ 3b
2
– 1 =
= 12ab
2
– 3,2b
3
– 6ab
2
+ 3b
2
– 1 = 6ab
2
– 0,2b
3
– 1.
№620.
а) 2a
2
x
3
– ax
3
– a
4
– a
2
x
3
+ ax
3
+ 2a
4
= a
4
+ a
2
x
3
;
б) 5x ⋅ 2y
2
– 5x ⋅ 3xy – x
2
y + 6xy
2
=10xy
2
– 15x
2
y
– x
2
y + 6xy
2
= 16xy
2
– 16x
2
y.
№621.
а) 5x
6
– 3x
2
+ 7 – 2x
6
– 3x
6
+ 4x
2
= x
2
+ 7,
при x = –10, х
2
+ 7 = (–10)
2
+ 7 = 107;
б) 4a
2
b – ab
2
– 3a
2
b + ab
2
– ab + 6 = a
2
b – ab + 6,
при a = –3, b = 2, a
2
b – ab + 6 = (–3)
2
⋅ 2 – (–3) ⋅ 2 + 6 = 30.
№622.
а) 6a
3
– a
10
+ 4a
3
+ a
10
– 8a
3
+ a = 2a
3
+ a,
при a = –3, 2a
3
+ a = 2 ⋅ (–3)
3
+ (–3) = –57;
б) 4x
6
y
3
– 3x
6
y
3
+ 2x
2
y
2
– x
6
y
3
– x
2
y
2
+ y = x
2
y
2
+ y,
при x = –2, y = –1, x
2
+ y2 + y = (–2)
2
⋅ (–1)
2
+ (–1) = 3.
№623.
при x = 0, 2х
2
+ 1 = 2 ⋅ 0
2
+ 1 = 1, при x = –2, 2х
2
+ 1 = 2 ⋅ (–2)
2
+ 1 = 9,
при x = 3, 2х
2
+ 1 = 2 ⋅ 3
2
+ 1 = 19, при x=–4, 2х
2
+ 1 = 2 ⋅ (–4)
2
+ 1 = 33.
Для любого x 2x
2
+ 1 > 0, т.к. для любого x 2x
2
≥ 0 и 1 > 0.
Т.е. нет таких x, чтобы 2x
2
+ 1 ≤ 0.
№624.
x
2
+ y
2
+ 1 > 0 для любых x и y, т.к. x
2
+ y
2
≥ 0, 1 > 0, а сумма
неотрицательных чисел и положительного числа больше нуля.
72
№625.
а) a ⋅ 10 + b = 10a + b; б) a ⋅ 100 + b ⋅ 10 + c = 100a + 10b + c.
№626.
а) 17a
4
– 8a
5
+ 3a – a
3
– 1 = –8a
5
+ 17a
4
– a
3
+ 3a – 1;
б) 35 – c
6
+ 5c
2
– c
4
= –c
6
– c
4
+ 5c
2
+ 35.
№627.
а) x
4
– 5 – x
2
+ 12x = –5 + 12x – x
2
+ x
4
; б) 2y + y
3
– y
2
+ 1 =1 + 2y – y
2
+ y
3
.
№628.
а) 4a
6
– 2a
7
+ a – 1; 7 степень; б) 5p
3
– p – 2; 3 степень;
в) 1 – 3x; 1 степень; г) 4xy + xy
2
– 5x
2
+ y; 3 степень;
д) 8x
4
y + 5x
2
y
3
– 11; 5 степень; е) xy + yx + xz – 1; 2 степень.
№629.
а) x
4
+ 3x; б) x
2
y + y
3
x.
№630.
а) при x = 1,97, x
2
+ 4,23 = 1,97
2
+ 4,23 = 8,1109;
в) при a=2,3; b=138,9, a
4
+2b=2,3
4
+2⋅138,9 = 27,9841 + 277,8 = 305,7841.
№631.
а)0,3y = 70, y =
1
233
3
; б)
5
8
x = –1, x = –
8
5
; в)
13
97
a
=
−
, a = –
31 6
3
79 7
:
=
−
.
№632.
а)
32 34
88
525 55 1
555
⋅⋅
==
; б)
()
5538
4
48
2
28 22 2
1
42
2
⋅⋅
=
==; в)
58 108
999
43 2 3 2
6323
⋅⋅
=
=
⋅
.
№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) (4x
3
– 5x – 7) + (x
3
– 8x) = 4x
3
– 5x – 7 + x
3
– 8x = 5x
3
– 13x – 7;
б) (5y
2
– 9) – (7y
2
– y + 5) = 5y
2
– 9 – 7y
2
+ y – 5 = –2y
2
+ y – 14.
№636.
а) (2a
3
– 5a + 5) + (a
3
– 4a – 2) = 2a
3
– 5a + 5 + a
3
– 4a – 2 = 3a
3
– 9a + 3;
б) (2a
3
– 5a + 5) – (a
3
– 4a – 2) = 2a
3
– 5a + 5 – a
3
+ 4a + 2 = a
3
– a + 7;
в) (a
3
– 4a – 2) – (2a
3
– 5a + 5) = a
3
– 4a – 2 – 2a
3
+ 5a – 5 = –a
3
+ a – 7.
№637.
а) (1 + 3a) + (a
2
– 2a)= 1 + 3a + a
2
– 2a = a
2
+ a + 1;
б) (2x
2
+ 3x) + (–x + 4) = 2x
2
+ 3x – x + 4 = 2x
2
+ 2x + 4;
в) (y
2
– 5y) + (5y – 2y
2
) = y
2
– 5y + 5y – 2y
2
= –y
2
;
г) (b
2
– b + 7) – (b
2
+ b+ 8) = b
2
– b + 7 – b
2
– b – 8 = –2b – 1;
д) (8n
3
– 3n
2
) – (7 + 8n
3
– 2n
2
) = 8n
3
– 3n
2
– 7 – 8n
2
+ 2n
2
= –n
2
– 7;
е) (a
2
+ 5a + 4) – (a
2
+ 5a – 4) = a
2
+ 5a + 4 – a
2
– 5a + 4 = 8.
73
№638.
а) 5,2a – (4,5a + 4,8a
2
) = 0,7a – 4,8a
2
;
б) –0,8b
2
+ 7,4b + (5,6b – 0,2b
2
) = –b
2
+ 13b;
в) 8x
2
+ (4,5 – x
2
) – (5,4x
2
– 1) = 8x
2
+ 4,5 – x
2
– 5,4x
2
+ 1 =
= –6,4x
2
+ 8x
2
+ 5,5 = 1,6x
2
+ 5,5;
г) (7,3y – y
2
+ 4) + 0,5y
2
– (8,7y – 2,4y
2
) =
= 7,3y – y
2
+ 4 + 0,5y
2
– 8,7y + 2,4y
2
= 1,9y
2
– 1,4y + 4.
№639.
а) 18x
2
– (10x – 5 + 18x
2
) = 18x
2
– 10x + 5 – 18x
2
= –10x + 5;
б) –12c
2
+ 5c + (c + 11c
2
) = –c
2
+ 6c;
в) (b
2
+ b – 1) – (b
2
– b + 1) = b
2
+ b – 1 – b
2
+ b – 1 = 2b – 2;
г) (15 – 7y
2
) – (y
3
– y
2
– 15) = 15 – 7y
2
– y
3
+ y
2
+ 15 = –y
3
– 6y
2
+ 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) (a
2
– 5ab) – (7 –3ab) + (2ab – a
2
) =a
2
–5ab–7 +3ab + 2ab – a
2
= –7 верно.
№642.
а) M + (5x
2
– 2xy) = 6x
2
+ 9xy – y
2
, M = 6x
2
+ 9xy – y
2
– (5x
2
– 2xy),
M = 6x
2
+ 9xy – y
2
– 5x
2
+ 2xy, M = x
2
+ 11xy – y
2
;
б) M – (4ab – 3b
2
) = a
2
– 7ab + 8b, M = a
2
– 7ab + 8b
2
+ 4ab – 3b
2
,
M = a
2
– 3ab + 5b
2
;
в) (4c
4
– 7c
2
+ 6) – M = 0, M = 4c
4
– 7c
2
+ 6.
№643. а) 5x
2
– 3x – 9 + N = 0, N = –5x
2
+ 3x + 9;
б) 5x
2
– 3x – 9 + N = 18, N = –5x
2
+ 3x + 27;
в) 5x
2
– 3x – 9 + N = 2x – 3, N = 2x – 3 – 5x
2
+ 3x + 9 = –5x
2
+ 5x + 6;
г) 5x
2
– 3x – 9 + N = x
2
– 5x + 6, N = x
2
–5x+6–5x
2
+3x + 9 = –4x
2
– 2x + 15.
№644.
а) (a
2
– 0,45a + 1,2) + (0,8a
2
– 1,2a) – (1,6a
2
– 2a) =
= a
2
– 0,45a + 1,2 + 0,8a
2
– 1,2a – 1,6a
2
+ 2a = 0,2a
2
+ 0,35a + 1,2;
б) (y
2
– 1,75y – 3,2) – (0,3y
2
+ 4) – (2y – 7,2) =
= y
2
– 1,75y – 3,2 – 0,3y
2
– 4 – 2y + 7,2 = 0,7y
2
– 3,75y;
в) 6xy – 2x
2
– (3xy + 4x
2
+ 1) – (–xy – 2x
2
– 1) =
= 6xy – 2x
2
– 3xy – 4x
2
– 1 + xy + 2x
2
+ 1 = –4x
2
+ 4xy;
г) –(2ab
2
– ab + b) + 3ab
2
– 4b – (5ab – ab
2
) =
= –2ab
2
+ ab – b + 3ab
2
– 4b – 5ab + ab
2
= 2ab
2
– 4ab – 5b.
№645.
а) 8a
2
b+(–5a
2
b+4b
2
)+(a
2
b–5b
2
+2)=8a
2
b–5a
2
b + 4b
2
+ a
2
b – 5b
2
+2=4a
2
b–b
2
+ 2;
б) (xy + x
2
+ y
2
) – (x
2
+ y
2
– 2xy) – xy = xy + x
2
+ y
2
– x
2
– y
2
+ 2xy – xy = 2xy.
№646. (5,7a
2
b – 3,1ab + 8b
3
) – (6,9ab – 2,3a
2
b + 8b
3
) =
= 5,7a
2
b – 3,1ab + 8b
3
– 6,9ab + 2,3a
2
b – 8b
3
= 8a
2
b – 10ab;
а) при a = 2; b = 5, 8a
2
b – 10ab = 8 ⋅ 2
2
– 10 ⋅ 2 ⋅ 5 = 160 – 100 = 60;
б) при a = –2; b = 5, 8a
2
b–10ab=8 ⋅ (–2)
2
⋅ 3 – 10 ⋅ (–2) ⋅ 3 = 96 + 60 = 156.
74
№647.
5x
2
– (3xy – 7x
2
) + (5xy – 12x
2
) = 5x
2
– 3xy + 7x
2
+ 5xy – 12x
2
= 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.
(0,7x
4
+ 0,2x
2
–5) – (–0,3x
4
+
5
1
x – 8) =
= 0,7x
4
+ 0,2x
2
– 5 + 0,3x
4
– 0,2x
2
+ 8 = x
4
+ 3 > 0 при любом x.
№649.
(7a
3
– 6a
2
b + 5ab
2
) + (5a
3
+ 7a
2
b + 3ab
2
) – (10a
3
+ a
2
+ 8ab
2
) =
= 7a
3
– 6a
2
b + 5ab
2
+ 5a
3
+ 7a
2
b + 3ab
2
– 10a
3
– a
2
– 8ab
2
= 2a
3
.
При a = –0,25, 2 ⋅ (–0,25) = –2 ⋅
11
64 32
=−
.
Неправильно, т.к. значение не зависит от b.
№650.
а) x
2
+y
2
– 2xy + 1 + N = y
2
+ 1, N = y
2
+ 1 – (x
2
+ y
2
– 2xy + 1), N=–x
2
+2xy;
б) x
2
+y
2
–2xy + 1 + N = x
2
+ 1, N = x
2
+ 1 – (x
2
+ y
2
– 2xy + 1), N=–y
2
+ 2xy.
№651.
(
3
5
x
2
– 0,4xy – 1,5y + 1) – (y
2
–
2
5
xy + 0,6x
2
) =
3
5
x
2
– 0,4xy – 1,5y + 1 –
– y
2
+
5
2
xy – 0,6x
2
= –y
2
–1,5y + 1.
№652.
а) 1,7 – 10b
2
– (1 – 3b
2
) + (2,3 + 7b
2
) = 1,7 – 10b
2
– 1 + 3b
2
+ 2,3 + 7b
2
= 3;
б) 1 – b
2
– (3b – 2b
2
) + (1 + 3b – b
2
) = 1 – b
2
– 3b + 2b
2
+ 1 + 3b – b
2
= 2.
№653.
а) x + y + z = 5a
2
+ 6ab – b
2
– 4a
2
+ 2ab + 3b
2
+ 9a
2
+ 4ab = 10a
2
+ 12ab + 2b
2
;
б) x – y – z = 5a
2
+ 6ab – b
2
– (–4a
2
+ 2ab + 3b
2
) – (9a
2
+ 4ab) =
= 5a
2
+ 6ab – b
2
+ 4a
2
– 2ab – 3b
2
– 9a
2
– 4ab = –4b
2
.
№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,
30 – 3x = 25, 3x = 5, x =
3
5
, x =
3
2
1
;
в) (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,
4,2y + 0,8 = 5,1y + 0,4, 4,2y – 5,1y = 0,4 – 0,8, 0,9y = 0,4, y =
9
4
.
75
№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) 3x
3
– 2x
2
– x + 4 = (3x
2
– 2x
2
) + (4 –x);
б) –5y
4
+ 4y
3
+ 3y
2
– 2y = (–5y
4
+4y
3
) + (3y
2
– 2y).
№657.
а) x
3
+ 2x
2
– 3x – 5 = x
3
– (–2x
2
+ 3x + 5);
б) 3a
4
+ 2a
3
+ 5a
2
– 4 = 3a
4
– (–2a
3
– 5a
2
+ 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.
324 32
00125
324
,,
,
,
−
= = 1,25% — относительная погрешность.
№660.
а) при a =
2
3
, b =
5
6
, 6(2a – b) =6 ⋅
25
22
36
⎛⎞
⋅−
⎜⎟
⎝⎠
= 8 – 5 = 3;
б) при a =
3
1
, b = 0,2, 15 ⋅
53
ab
⎛⎞
+
⎜⎟
⎝⎠
= 3a + 5b = 3 ⋅
1
3
+ 5 ⋅ 0, 2 = 1 + 1 = 2.
№661.
а) (2x
2
)
3
⋅
1
4
x
2
= 8x
6
⋅
1
4
x
2
= 2x
8
;
б) (–3y
4
)
3
⋅
1
9
y
5
= –27y
12
⋅
1
9
y
5
= –3y
17
;
в) –0,2a
2
b
3
⋅ (–5a
3
b
2
)
2
= –0,2a
2
b
3
⋅ 25a
6
b
4
= –5a
8
b
7
;
г) (–0,5c
4
d)
3
⋅ (–4c
2
d
2
)
2
= –
1
8
c
12
d
3
. 16c
4
d
4
= –2c
16
d
7
.
№662.
при x = 1,4, y = 0,157; x
3
– y = 1,4
3
– 0,157 = 2,587.
76
§ 10. Произведение одночлена и многочлена
26. Умножение одночлена на многочлен
№663. а) 2x(x
2
– 7x – 3) = 2x
3
– 14x
2
– 6x;
б) –4b
2
(5b
2
– 3b – 2) = –20b
4
+ 12b
3
+ 8b
2
;
в) (3a
3
– a
2
+ a)(–5a
3
) = –15a
6
+ 5a
5
– 5a
4
;
г) (y
2
– 2,4y + 6) ⋅ 1,5y = 1,5y
3
– 3,6y
2
+ 9y;
д) –0,5x
2
⋅ (–2x
2
– 3x + 4) = x
4
+ 1,5x
3
– 2x
2
;
е) (–3y
2
+ 0,6y)(–1,5y
3
) = 4,5y
5
– 0,9y
4
.
№664. а) 3ab(a
2
– 2ab + b
2
) = 3a
3
b – 6a
2
b
2
+ 3ab
2
;
б) –x
2
y(x
2
y
2
– x
2
– y
2
) = –x
4
y
3
+ x
4
y + x
2
y
3
;
в) 2,5a
2
b(4a
2
– 2ab + 0,2b
2
) = 10a
4
b – 5a
3
b
2
+ 0,5a
2
b
3
;
г) (–2ax
2
+ 3ax – a
2
)(–a
2
x
2
) = 2a
2
x
4
– 3a
3
x
3
+ a
4
x
2
;
д) (6,3x
3
y – 3y
2
– 0,7x) ⋅ 10x
2
y
2
= 63x
5
y
3
– 30x
2
y
4
– 7x
3
y
2
;
е) –1,4p
2
q
6
(5p
3
q – 1,5pq
2
– 2q
2
) = –7p
5
q
7
+ 2,1p
3
q
8
+ 2,8p
2
q
9
.
№665.
а)
7
2
x ⋅ (1,4x
2
– 3,5y) = 0,4x
3
– xy;
б) –
3
1
c
2
⋅ (1,2d
2
– 6c) = –0,4c
2
d
2
+ 2c
3
;
в)
1
2
ab
223223
234 1 3 2
345 3 8 5
aabb ababab
⎛⎞
−+ = − +
⎜⎟
⎝⎠
;
г)
25 2 2 3 37 46 55
215 11
52
526 53
ay ay ay a ay ay ay
⎛⎞
−−−=−++
⎜⎟
⎝⎠
.
№666.
а) –3x
2
(–x
3
+ x – 5) = 3x
5
– 3x
3
+ 15x
2
;
б) (1 + 2a – a
2
) ⋅ 5a = 5a + 10a
2
– 5a
3
;
в)
3
2
x
2
y(15x – 0,9y + 6) = 10x
3
y – 0,6x
2
y
2
+ 4x
2
y;
г) –3a
4
x(a
2
– 2ax + x
3
– 1) = –3a
6
x + 6a
5
x
2
– 3a
4
x
4
+ 3a
4
x;
д) (x
2
y – xy + xy
2
+ y
3
) ⋅ 3xy
2
= 3x
3
y
3
– 3x
2
y
3
+ 3x
2
y
4
+ 3xy
5
;
е) –
7
3
a
4
(2,1b
2
– 0,7a + 35) = –0,9a
4
b
2
+ 0,3a
5
– 0,9a
4
.
№667. а) 3(2x – 1) + 5(3 – x) = 6x – 3 + 15 – 5x = x + 12,
при x = –1,5, x + 12 = –1,5 + 12 = 10,5;
б) 25a – 4(3a – 1) + 7(5 – 2a) = 25a – 12a + 4 + 35 – 14a = 39 – a,
при a = 11, 39 – a = 39 – 11 = 28.
в)
42(101)(82)4202828yy y yy y y−−+−=−++−=−
при
01 8 8 ( 01) 08y,, y ,,
=
−−=−⋅−=;
г)
12(23)35 9( 1)2436 35 9 9 10 15ppp ppp p
−
+−+=−+−−=−+
при
21015102155p, p
=
−+=−⋅+=−.
77
№668. а) 14b + 1 – 6(2 – 11b) = 14b + 1 – 12 + 66b = 80b – 11;
б) 25(2 – 3c) + 16(5c – 1) = 50 – 75c + 80c – 16 = 5c + 34;
в) 14(7x – 1) – 7(14x + 1) = 98x – 14 – 98x – 7 = –21;
г) 36(2 – y) – 6(5 – 2y) = 72 – 36y – 30 + 12y = 42 – 24y.
№669. а) 14y + 2y(6 – y) = 14y + 12y –2y
2
= 26y – 2y
2;
б) 3y
2
– 2y(5 + 2y) = 3y
2
– 10y – 4y
2
= –y
2
– 10y;
в) 4x(x – 1) – 2(2x
2
– 1) = 4x
2
– 4x – 4x
2
+ 2 = –4x + 2;
г) 5a(a
2
– 3a) – 3a(a
2
– 5a) = 5a
3
– 15a
2
– 3a
3
+ 15a
2
= 2a
3
;
д) 7b(4c – b) +4c(c – 7b) = 28bc – 7b
2
+ 4c
2
– 28bc = 4c
2
– 7b
2
;
е) –2y(x
3
– 2y) – (x
3
y + 4y
2
) = –2x
3
y + 4y
2
– x
3
y – 4y
2
= –3x
3
y;
ж) 3m
2
(m + 5n) – 2n(8m
2
– n) = 3m
2
+ 15m
2
n – 16m
2
n + 2n
2
= 3m
3
– m
2
n + 2n
2
;
з) 6m
2
n
3
– n
2
(6m
2
n + n – 1) = 6m
2
n
3
– 6m
2
n
3
– n
3
+ n
2
.
№670. а) 6x(x – 3) – x(2 – x) = 6x
2
– 18x – 2x + x
2
= 7x
2
– 20x;
б) –a
2
(3a – 5) + 4a(a
2
– a) = –3a
3
+ 5a
2
+ 4a
3
– 4a
2
= a
3
+ a
2
;
в) ax(2x – 3a) – x(ax + 5a
2
) = 2ax
2
– 3a
2
x – ax
2
– 5a
2
x = ax
2
– 8a
2
x;
г) –4m
2
(n
2
– m
2
) + 3n
2
(m
2
– n
2
) =
= –4m
2
n
2
+ 4m
4
+ 3m
2
n
2
– 3n
4
= 4m
4
– m
2
n
2
– 3n
4
.
№671.
а) –2x(x
2
– x + 3) + x(2x
2
+ x – 5) = –2x
3
+2x
2
– 6x + 2x
3
+ x
2
– 5x = 3x
2
– 11x,
при x = 3, 3x
2
– 11x = 3 ⋅ 3
2
– 11 ⋅ 3 = 27 – 33 = –6,
при x = –3, 3x
2
– 11x = 3 ⋅ (–3)
2
– 11 ⋅ (–3) = 27 + 33 = 60;
б) x(x – y) – y(y
2
– x) = x
2
– xy – y
2
+ xy = x
2
– y
2
,
при x = 4; y = 2, x
2
– y
2
= 4
2
– 2
2
= 16 – 8 = 8.
№672. а) 5x(2x – 6) – 2,5x(4x – 2) = 10x
2
– 30x – 10x
2
+ 5x = –25x,
при x = –8, –25x = –25 ⋅ (–8) = 200, при x = 10, –25x = –25 ⋅ 10 = –250;
б) 5a(a – 4b) – 4b(b – 5a) = 5a
2
– 20ab – 4b
2
+ 20ab = 5a
2
– 4b
2
,
при a = –0,6; b = –0,5, 5a
2
– 4b
2
= 5 ⋅ (–0,6)
2
– 4 ⋅ (–0,5)
2
= 1,8 – 1 = 0,8.
№673. а) (3a
2
)
2
– a
3
(1 – 5a) = 9a
4
– a
3
+ 5a
4
= 14a
4
– a
3
;
б)
3
23232
1111
12 2 2
2888
bb bb bbb b bb
⎛⎞⎛ ⎞
−−−−=−−++=−
⎜⎟⎜ ⎟
⎝⎠⎝ ⎠
;
в) x(16x – 2x
3
) – (2x
2
)
2
= 16x
2
– 2x
4
– 4x
4
= 16x
2
– 6x
4
;
г) (0,2c
3
)
2
– 0,01c
4
(4c
2
– 100) = 0,04c
6
– 0,04c
6
+ c
4
= c
4
.
№674.
A
B
C
D
E
F
a
b
c
a ⋅ (b + c) = ab + ac, a ⋅ (b + c) = S
ABCD
— площадь прямоугольника ABCD.
a ⋅ b = S
ABEF
a ⋅ c = S
EFDC
— площади прямоугольников ABEF и EFDC.
Площадь ABCD равна сумме площадей ABEF и EFDC: S
ABCD
= S
ABEF
+ S
EFDC
.
78
№675. x(2x + 1) – x
2
(x + 2) + (x
3
– x + 3) = 2x
2
+ x – x
3
– 2x
2
+ x
3
– x + 3 = 3.
№676. y(3y
2
– y + 5) – (2y
3
+ 3y – 16) – y(y
2
– y + 2) =
= 3y
3
– y
2
+ 5y – 2y
3
– 3y + 16 – y
3
+ y
2
– 2y = 16 не зависит от y.
№677.
а) a(b – c) + b(c – a) + c(a – b) = ab – ac + bc – ab + ac – bc = 0;
б) a(b+c–bc)–b(c+a–ac) + c(b – a) = ab + ac – abc – bc + abc + bc – ac = 0.
№678.
2x(x – 6) – 3(x
2
– 4x + 1) = 2x
2
– 12x – 3x
2
+ 12x – 3 = –x
2
– 3 т.к. –x
2
≤ 0;
–3 < 0, а сумма неотрицательного и положительного чисел меньше нуля.
№679.
а) 5x + 3(x – 1) = 6x + 11, 5x + 3x – 3 – 6x = 11,
2x – 3 = 11, 2x = 14, x = 7;
б) 3x – 5(2 – x) = 54, 3x – 10 + 5x = 54, 8x = 64, x = 8;
в) 8(y – 7) – 3(2y + 9) = 15, 8y – 56 – 6y – 27 = 15,
2y – 83 = 15, 2y = 98, y = 49;
г) 0,6 – 0,5(y – 1) = y + 0,5, 0,6 – 0,5y + 0,5 = y + 0,5,
–0,5y – y = 0,5 – 1,1, 1,5y = 0,6, y = 0,4;
д) 6 + (2 – 4x) + 5 = 3(1 – 3x), 6 + 2 – 4x + 5 = 3 – 9x,
–4x + 9x = 3 – 13, 5x = –10, x = –2;
е) 0,5(2y–1)–(0,5–0,2y) + 1 =0, y – 0,5 – 0,5 + 0,2y + 1 = 0, 1,2y
= 0, y = 0;
ж) 0,15(x – 4) = 9,9 – 0,3(x – 1), 0,15x – 0,6 = 9,9 – 0,3x + 0,3,
0,15x + 0,3x = 10,2 + 0,6, 0,45x = 10,8, x = 24;
з) 3(3x – 1) + 2 = 5(1 – 2x) – 1, 9x + 10x = 4 + 1, 19x = 5, x =
5
19
.
№680.
а) 3x(2x – 1) – 6x(7 + x) = 90, 6x
2
– 3x – 42x – 6x
2
= 90, 45x = –90, x = –2;
б) 1,5x(3 + 2x) = 3x(x + 1) – 30, 4,5x + 3x
2
= 3x
2
+ 3x – 30,
4,5x – 3x = –30, 1,5x = –30, x = –20;
в) 5x(12x– 7) – 4x(15x – 11) = 30 + 29x, 60x
2
– 35x – 60x
2
+ 44x = 30 + 29x,
9x – 29x = 30, 20x = –30, x = –1,5;
г) 24x – 6x(13x – 9) = –13 – 13x(6x – 1),
24x – 78x
2
+ 54x = –13 – 78x
2
+ 13x, 78x – 13x = –13, 65x = –13, x = –
5
1
.
№681. а) 3(–2x + 1) – 2(x + 13) = 7x – 4(1 – x),
–6x + 3 – 2x – 26 = 7x – 4 + 4x, –8x – 23 = 11x – 4, 19x = –19, x = –1;
б) –4(5 –2a) + 3(a – 4) = 6(2 – a) – 5a, –20 + 8a + 3a – 12 = 12 – 6a – 5a,
–32 + 11a + 11a = 12, 22a = 44, a = 2;
в) 3y(4y – 1) – 2y(6y – 5) = 9y – 8(3 + y),
12y
2
– 3y – 12y
2
+ 10y = 9y – 24 – 8y, 7y – y = –24, 6y = –24, y = –4;
г) 15x + 6x(2 – 3x) = 9x(5 – 2x) – 36, 15x + 12x – 18x
2
= 45x – 18x
2
– 36,
27x – 45x = –36, 18x = 36, x = 2.
№682. а) 4(1 – c) – 2(3 – 5c) = 1, 4 – 4c – 6 + 10c = 1, 6c = 3, c =0,5;
б) –3(2x + 1) –(8x + 5) = 20, –6x – 3 – 8x – 5 = 20,
–14x = 20 + 8, 14x = –28, x = –2;
в) 3(5x + 7) = 61 – 10x, 15x + 21 = 61 – 10x, 25x = 40, x = 1,6;
г) 2(7 + y) = 8 – y, 14 + 2y = 8 – y, 3y = –6, y = –2.
79
№683. а)
14
43
xx
+=
, 3x + 4x = 168, 7x = 168, x = 24;
б)
5
28
aa
−=, 4a – a = 40, 3a = 40, a =
40
3
, a =
1
13
3
;
в)
4
y
= y –1, y = 4y – 4, 3y = 4, y =
4
3
, y =
1
1
3
;
г) 2z + 3 =
2
5
z
, 10z + 15 = 2z, 8z = –15, z = –
15
8
, z = –
7
1
8
;
д)
24
7
35
cc
−=
, 10c – 12c = 105, 2c = –105, c = –52,5;
е)
5
40
93
xx
++=
, 5x + 3x + 36 = 0, 8x = –36, x = –4,5;
ж)
45
1
912
aa
+=
, 16a + 36 = 15a, a = –36;
з)
51
12 8 3
mm
−
=
, 10m – 3m = 8, 7m = 8, m =
8
7
, m =
1
1
7
;
и)
32
14 2 7
nn
+=, 3n + 7n = 4, 10n = 4, n = 0,4.
№684. а)
6521
2
73
xx−−
=+
, 3(6x – 5) = 7(2x – 1) + 42,
18x – 15 = 14x – 7 +42, 4x = 50, x = 12,5;
б)
531
4
25
xx−−
+
=
, 5(5x – x) +2(3x – 1) = 40,
25 – 5x + 6x – 2 = 40, x + 23 = 40, x = 17;
в)
57 5
5
12 8
xx−−
−=
, 2(5x – 7) – 3(x – 5) = 120,
10x – 14 – 3x + 15 = 120, 7x = 119, x = 17;
г)
411137
2
15 20
yy−−
+=
, 4(4y – 11) + 3(13 – 7y) = 120,
16y – 44 + 39 – 21y = 120, 5y = –125, y = –25;
д)
56
0
38
yy−
+=
, 8(5 – 6y) + 3y = 0, 40 – 48y + 3y = 0, 45y = 40, y =
8
9
;
е)
32
0
45
yy−
−=, 5y – 4(3 – 2y) = 0, 5y – 12 + 8y = 0, 13y = 12, y =
12
13
.
№685. а)
35 1
1
53
xx++
−=
, 3(3x + 5) – 5(x + 1) = 15, 9x + 15 – 5x – 5 = 15,
4x = 5, x = 1,25;
б)
21 1
63
pp
p
−+
+=
, 2p – 1 – 2(p + 1) = 6p, 2p – 1 – 2p – 2 – 6p = 0,
–3 – 6p = 0, 6p = –3, p = –0,5;
80
в)
61 2
15 5 3
yyy−
−=
, 6y – 1 – 3y = 10y, 7y = –1, y = –
1
7
;
г)
12 2
436
x
xx−−
−=
, 3(12–x)–4(2–x) = 2x, 36 – 3x – 8 + 4x = 2x, x = 28.
№686. а)
32
14
23
xx−−
−=+
, 6 – 3(x – 3) = 2(2 – x) + 24,
6 – 3x + 9 = 4 – 2x + 24, 15 – 3x = 28 – 2x, –3x +2x = 28 – 15, x = –13;
б)
13 2 3
10 5 15 2
aaaa+−
−= +, 3(a + 13) – 12a = 2(3 – a) + 15a,
3a + 39 – 12a = 6 – 2a + 15a, 39 – 9a = 6 – 39,
–9a – 13a = 6 – 39, 22a = 33, a = 1,5;
в)
21 6
3
4612
mmm
+
−
+= −
, 3(2m + 1) + 26 = 2m – 6 + m,
6m – 3m = – 6 – 39, 3m = –45, m = –15;
г)
11 3
2
96 2
xx x+− +
−=−
, 2(x + 1) – 3(x – 1) = 36 – 9x – 27,
5 – x = 9 – 9x, 8x = 4, x = 0,5.
№687. а)
6785
5
43
yy+−
+=
, 3(6y + 7) + 4(8 – 5y) = 60,
18y + 21 + 32 – 20y = 60, –2y + 53 = 60, 2y = –7, y = –3,5;
б)
512 3
1
35
aa−−
=−, 5(5a – 1) = 3(2a – 3) – 15,
25a – 5 = 6a – 9 – 15, 25a – 6a = 5 – 24, 19a = –19, a = –1;
в)
11 4 9
5
72
xx−−
−=
, 2(11x – 4) – 7(x – 9) = 70,
22x – 8 – 7x + 63 = 70, 15x = 70 – 63 + 8, 15x = 15, x = 1;
г)
21 3
946
ccc−+
+=
, 4(2c – 1) + 9c = 6(c + 3),
8c – 4 + 9c = 6c + 18, 17c – 6c = 18 + 4, 11c = 22, c = 2;
д)
312 6
10
24 36
pp−+
−−=
, 3(3p – 1) – 2(2p + 6) – 72 = 0,
9p – 3 – 4p – 12 – 72 = 0, 9p – 4p = 72 + 12 + 3, 5p = 87, p = 17,4;
е)
12 3 20
5
463
x
xx−+
−= +
, 60 – 3(1 – 2x) = 2(3x + 20) + 4x,
60 – 3 + 6x = 6x + 40 + 4x, 57 – 40 = 4x, 4x = 17, x = 4,25.
№688.
Цена Кол-во Стоимость Всего
открытки x – 1/2 р 15 15(x – 1/2)
конверты x р 10 10x
блокноты 4x р 1 4x
36 р
15(x –
2
1
) + 10x + 4x = 36, 29x = 43,5, x = 1,5 (р) стоит конверт.
Ответ: 1р; 1,5р; 6р.