GDZ_33 (1) / Алгебра 7кл_Алимов_Решебник_2002 1-801

349.1) x2 +3x +2 = x2 +2x + x +2 = (x2 +2x) +(x +2)=

=x (x +2)+(x + 2)= (x + 2) (x +1)

2)x2 −5x +6 = x2 −2x −3x +6 = x (x −2)−3 (x −2)=

= (x −2) (x −3)

3)x2 −7x −8 = x2 −8x + x −8 = x (x −8)+(x −8)= (x −8) (x +1)

4)x2 +9x −10 = x2 +10x − x −10 = (x2 +10x) −(x +10)=

=x (x +10)−(x +10)= (x +10) (x −1)

350.1) a3 + 2a2 −3 = a3 +3a2 −a2 −3 = (3a2 −3) +(a3 −a2 ) =

=3(a2 −1) + a2 (a −1) = 3(a −1)(a +1) +a2 (a −1) = (a −1)(3a +3 +a2 )

2) x3 −7x +6 = x3 − x − 6x + 6 = x (x −1) (x +1)−6 (x −1)=

=(x −1) (x 2 + x −6) = (x −1) (x 2 +3x −2x −6) =

=(x −1) [x (x +3)−2 (x + 3)]= (x −1) (x +3) (x − 2)

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3) a4 +2a3 +1 = a4 +a3 +a3 +1 = a3 (a +1)+(a3 +1) =

=a3 (a +1)+(a3 +a2 −a2 +1) = a3 (a +1)+a2 (a +1)−(a2 −1) =

=a3 (a +1)+a 2 (a +1)−(a −1) (a +1)= (a +1) (a 3 +a 2 −a +1) 4) 2a4 −a2 −1 = 2a4 −2a2 +a2 −1 = 2a2 (a2 −1) +(a2 +1) =

=(a 2 −1) (2a 2 +1) = (a −1) (2a 2 +1)(a +1)

§21. Формула разности квадратов

351.1) 4a2 = (2a)2 ; 9b2 = (3b)2 ; 16c2 = (4c)2 ; 0,04x2 = (0,2x)2

3)64 y2 −36x2 = (8y −6x) (8y +6x);

4)81a2 −16b2 = (9a −4b) (9a + 4b)

353. 1) 19 y 2 − 1625 x 2 −(13 y − 54 x) (13 y + 54 x)

2)94 a 2 −161 b 2 = ( 23 a − 14 b) ( 23 a + 14 b)

3)0,25a2 −0,49b2 = (0,5a −0,7b) (0,5a + 0,7b)

4)0,09x 2 −0,16 y 2 = (0,3x −0,4 y) (0,3x +0,4 y)

354.1) 36x2 y2 −1 = (6xy −1) (6xy +1)

2)x2 y4 −16 = (xy2 −4) (xy2 +4)

3)81a 6 −49b 4 = (9a3 −7b 2 ) (9a3 +7b 2 )

4)25a 2 −9b6 = (5a −3b3 ) (5a +3b3 )

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355.1) a4 −b4 = (a2 −b2 ) (a2 +b2 ) = (a −b) (a +b) (a2 +b2 )

2)a4 −b8 = (a2 −b4 ) (a2 +b4 ) = (a −b2 ) (a +b2 ) (a2 +b2 )

3)a 4 −16 = (a 2 −4) (a 2 +4) = (a −2) (a +2) (a 2 +4)

4)b4 −81 = (b2 −9) (b2 +9) = (b −3) (b +3) (b2 +4)

356. 1) (2b +a) (2b −a)= 4b2 −a2 ; 2) (c +3d ) (c −3d )= c2 −9d 2

3)(y +6x) (6x − y)= 36x2 − y2 ; 4) (3m −2n) (2n +3m)= 9m2 −4n2

357.1) (c2 +d 2 ) (c2 −d 2 )= c4 −d 4 ; 2) (a2 +b3) (a2 −b3)= a4 −b6

3)(x 4 − y 3 )(y 3 + x 4 )= x 2 − y 6 ; 4) (m3 −n3) (m3 +n3)= m6 −n6

358.1) (3a2 +4b3) (3a2 −4b3)= 9a4 −16b6

2)(2m 4 −5n 2 )(5n 2 +2m 4 )= 4m8 −25n 4

3)(0,2t3 +0,5p4 ) (0,5p4 −0,2t3)= 0,25p8 −0,04t6

4)(1,2a2 −0,3b2 ) (1,2a2 +0,3p2 )= 1,44a4 −0,09b4

359.1) 48 52 = (50 −2) (50 +2)= 2500 −4 = 2496

2)68 72 = (70 + 2) (70 −2)= 4900 −4 = 4896

3)43 37 = (40 + 3) (40 − 3)= 1600 −9 = 1591

4)47 53 = (50 −3) (50 +3)= 2500 −9 = 2491

360.1) 47 33 = (40 + 7) (40 −7)= 1600 −49 = 1551

2)44 36 = (40 + 4) (40 −4)= 1600 −16 = 1584

3)84 76 = (80 +4) (80 −4)= 6400 −16 = 6384

4)201 199 = (200 +1) (200 −1)= 40000 −1 = 39999

361.1) (a +b)2 −c2 = (a +b +c) (a +b −c)

2)(m −n)2 − k 2 = (m −n − k ) (m −n + k )

3)(a +2b)2 −9a2 = (a +2b +3a) (a +2b −3a)= 4 (2a +b) (b −a)

4)(3x − y)2 −4 y2 = (3x − y + 2 y) (3x − y −2 y) = 3 (3x + y) (x − y)

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362.1) (a −b)2 −(a −c)2 = (a −b −a +c) (a −b + a −c)=

=(c −b) (2a −b −c)

2) (a +b)2 −(b +c)2 = (a +b +b +c) (a +b −b −c)= = (a + 2b +c) (a −c)

3) (2a +b)2 −(2b +a)2 = (2a +b −2b −a) (2a +b + 2b + a)=

= 3 (a −b) (a +b)

4) (a +3b)2 −(3a +b)2 = (a +3b −3a −b) (a +3b +3a +b)=

=(2b −2a) (4a +4b)= 8 (b −a) (a +b)

363.1) 472 −372 = (47 +37) (47 −37)= 84 10 = 840 2) 542 −442 = (54 −44) (54 +44)= 10 98 = 980

3) 50,72 −50,62 = (50,7 −50,6) (50,7 +50,6)= 0,1 101,3 = 10,13 4) 29,42 −29,32 = (29,4 −29,3) (29,4 +29,3)= 0,1 58,7 = 5,87

6)7 95 2 − 4 49 2 = 7 95 −4 49 7 95 +4 49 = 3 91 12 = 37 13

364.1) (x −1) (x +1)= x2 −2 (x −3)

x2 −1 − x2 +2x −6 = 0 2x = 7 ; x = 3,5

2) 3 (x +5) − x2 = (2 − x) (2 + x) 3x + 15 – x2 = – x2 + 4

3x = – 11 x = −3 32

3)(2x +3) (2x +3) − 4 (x −1) (x +1) = 49

4x2 + 12x + 9 – 4x2 + 4 = 49 12x = 36; x = 3

4)(3x +1) (3x +1) − (3x − 2) (2 + 3x) =17

9x2 + 3x + 3x + 1 – 9x2 + 4 = 17 6x + 5 = 17

x = 2

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365. 1) (3 + x) (3 − x) (9 + x2 ) = (9 − x2 ) (9 + x2 ) = 81− x4

2) (4x2 + y2 ) (2x + y) (2x − y) = (4x2 + y2 ) (4x2 − y2 ) = =16x 4 − y 4 . (опечатка в ответе задачника).

3)(x2 +1) (x +1) (x −1) = (x2 +1) (x2 −1) = x4 −1

4)(3a − 2b) (3a + 2b) (9a2 + 4b2 ) = (9a2 − 4b2 ) (9a2 + 4b2 ) = = 81a4 −16b4

=4040 18762,,68 = 1878626 = 13

367.Пусть x – первое число, тогда следующее за ним x + 1.

|(x + 1)2 – x2| = |(x + 1 – x) (x + 1 + x)| = |2x + 1| – нечетное число.

368. (7n + 1) 2 – (2n – 4) 2 = (7n + 1 – 2n + 4) (7n + 1 + 2n – 4) =

=(5n + 5) (9n – 3) = 15 (n + 1) (3n – 1) M15, т. к. 15(n + 1)(3n – 1) : 15 = (n + 1)(3n – 1).

369.1) (a + b)3 – (a – b)3 – 8b3 =

=(a2 + 2ab + b2) (a + b) – (a2 – 2b + b2) (a – b) – 8b3 =

=a3 + 2a2b + ab2 + a2b + 2ab2 – a3 + 2a2b – ab2 + a2b – 2ab2 +

+ b3 – 8b3 = 6a2b −7b3 = 6b (a −b) (a +b)

2) (a2 + b2 )2 − (a2 −b2 )2 − a2 =

= (a2 +b2 − a2 +b2 ) (a2 +b2 + a2 −b2 ) − a2 = 2b2 2a2 − a2 = = a2 (4b2 −1) = a2 (2b −1) (2b +1) .

(опечатка в ответе задачника).

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3) (a4 +b4 )2 −(a4 −b4 )2 − a2b2 = 2b2 2a2 −a2 =

= (a4 +b4 − a4 +b4 ) (a4 +b4 + a4 −b4 ) − a2b2 = 2b4 2a4 − a2b2 = = a2b2 (2ab −1) (2ab +1)

4) 9a4 −13a2b2 + 4b4 = 9a4 −9a2b2 − 4a2b2 + 4b4 =

=9a2 (a2 −b2 ) − 4b2 (a2 −b2 ) = (a2 −b2 ) (9a2 − 4b2 ) =

=(a −b) (a +b) (3a −2b) (3a + 2b)

§ 22. Квадрат суммы. Квадрат разности

372.1) (0,2x + 0,3y)2 = 0,04x2 + 0,12xy + 0,09 y2

2)(0,4b −0,5c)2 = 0,16b2 −0,4bc +0,25c2

373.1) (−4ab −5a2 )2 =16a2b2 + 40a3b + 25a4

2)(−3b2 − 2ab)2 = 9b4 +12ab3 + 4a2b2

3)(0,2x2 +5xy)2 = 0,04x4 + 2x3 y +25x2 y2

4)(4xy +0,5y2 )2 =16x2 y2 +4xy3 +0,25y4

374.1) (90 −1)2 = 902 −2 90 +1 = 8100 −180 +1 = 7921

2)(40 +1)2 = 402 +2 40 +1 =1600 +80 +1 =1681

3)1012 = (100 +1)2 =10000 + 200 +1 =10201

4)982 = (100 − 2)2 =1000 − 400 + 4 = 9604

375.1) 722 = (70 + 2)2 = 4900 + 280 + 4 = 5184

2)572 = (60 −3)2 = 3600 −360 +9 = 3249

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3)9972 = (1000 −3)2 =1000000 − 6000 + 9 = 994009

4)10012 = (1000 +1)2 =1000000 + 2000 +1 =1002001

376.(a +1)2 ≈1 + 2a

1)1,0052 = (1+0,005)2 ≈1+2 0,005 =1,01

2)1,0042 = (1 + 0,004)2 ≈1 + 2 0,004 =1,008

3)1,0122 = (1 + 0,012)2 ≈1 + 2 0,012 =1,024

4)1,0112 = (1 + 0,011)2 ≈1 + 2 0,011 =1,022

5)0,9922 = (1 −0,008)2 ≈1 − 2 0,008 = 0,984

6)0,9942 = (1 −0,006)2 ≈1 − 2 0,006 = 0,988

7)0,9882 = (1 −0,012)2 ≈1 − 2 0,012 = 0,976

8)0,9892 = (1 − 0,011)2 ≈1 − 2 0,011 = 0,978

377.1) a2 + 4a + x = a2 + 4a + 4 = (a + 2)2

2)p2 −0,5 p + x = p2 −0,5 p + 161 = ( p − 14 )2

3)36a2 − x + 49b2 = 36a2 −84ab + 49b2 = (6a − 7b)2

4)a2 − 6ab + x = a2 − 6ab + 9b2 = (a −3b)2

378.1) m4 −3m2 + x = m4 −3m2 + 2,25 = (m2 −1,5)2

380.1) 9x2 + 24x +16 = (3x + 4)2 ;

2)100 −60a +9a2 = (10 −3a)2

3)36m2 +12nm + n2 = (6m + n)2 ;

4)a2 +10ab + 25b2 = (a + 5b)2

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381.1) x4 + 2x2 y + y2 = (x2 + y)2 ;

2)p4 − 2 p2q + q2 = ( p2 − q)2

3)4c4 +12c2d 3 + 9d 6 = (2c2 + 3d 3 )2

4)25a6 +30a3b +9b2 = (5a3 +3b)2

382.1) a4 −8a2 +16 = (a2 − 4)2 = (a − 2)2 (a + 2)2

2)b4 −18b2 +81 = (b2 −9)2 = (b −3)2 (b + 3)2

3)25a4 −10a2b +b2 = (5a2 −b)2

4)16 −8a2b2 + a4b4 = (4 − a2b2 ) = (2 − ab)2 (2 + ab)2

383.1) − a2 − 2a −1 = −(a +1)2 ;

2)−9 + 6b −b2 = −(3 −b)2

3)− 2a2 +8ab −8b2 = −2 (a − 2b)2

4)−12ab −3a2 −12b2 = −3 (a + 2b)2

3) −5x (x −3) +5 (x −1)2 = −20

−5x2 +15x + 5x2 −10x +5 = −20 5x = −25 4 x = −5

4) (2x −3)2 − (2x + 3)2 =12

4x2 −12x +9 − 4x2 −12x −9 =12 24x = −12 ; x = −12

385.1) (x − y)2 + (x + y)2 = x2 − 2xy + y2 + x2 + 2xy + y2 = 2x2 + 2 y2

2)(x + y)2 − (x − y)2 = x2 + 2xy + y2 − x2 + 2xy − y2 = 4xy

3)(2a + b)2 − (2a −b)2 = 4a2 + 4ab + b2 − 4a2 + 4ab −b2 = 8ab

4)(2a +b)2 +(2a −b)2 = 4a2 + 4ab +b2 + 4a2 −4ab +b2 =

= 8a2 + 2b2

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386.1) (a −b)2 = a2 − 2ab +b2 = b2 − 2ab + a2 = (b − a)2

2)(−a −b)2 = (−1)2 (a + b)2 = (a + b)2

3)(−1) (a + b) (a + b) = −(a + b)2

4)(−1)3 (−a +b)3 = −(b − a)3

5)(a + b + c)2 = a2 + b2 + c2 + 2ab + 2ac + 2bc

(a +b +c)2 = (a +b)2 + 2 (a +b) c +c2 = a2 + 2ab +b2 + 2ac + + 2bc +c2 = a2 +b2 +c2 + 2ab + 2ac + 2bc

a2 +b2 + c2 + 2ab + 2ac + 2bc = a2 +b2 + c2 + 2ab + 2ac + 2bc ч.т.д.

387.1) 5m2 −10mn + 5n2 = 5 (m2 − 2mn + n2 ) = 5 (m − n)2 m =142; n = 42

5 (142 − 42)2 = 5 10000 = 50000

2) 6m2 +12mn +6n2 = 6 (m2 +2mn +n2 ) = 6 (m +n)2

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389.1) (x + 2)3 = x3 + 6x2 +12x +8

2)(3 − y)3 = 27 − 27 y + 9 y2 − y3

3)(2a −b)3 = 8a3 −12a2b + 6ab2 −b3

4)(3b + 2a)3 = 27b3 +54b2a +36ba2 +8a3

390.1) 125 + 75a +15a2 + a3 = (5 + a)3

2)m3 −12m2 + 48m − 64 = (m − 4)3

3)x6 −3x4 y +3x2 y2 − y3 = (x2 − y)3

4)c6 + 3c4d 2 + 3c2d 4 + d 6 = (c2 + d 2 )3

391.Рассмотрим двузначные числа и их квадраты (после 20 все аналогично):

Цифра единиц двузначного числа, квадрат которого содержит нечетное число десятков, 4 или 6.

§ 23. Применение нескольких способов разложения многочлена на множители

392.1) 2a2 − 2 = 2 (a2 −1) = 2 (a −1) (a +1)

2)3x2 −12 = 3 (x2 − 4) = 3 (x − 2) (x + 2)

3)9x3 −81x = 9x (x2 −9) = 9x (x −3) (x +3)

4)16x − 4x3 = 4x (4 − x2 ) = 4x (2 − x) (2 + x)

5)8 −72x6 y2 = 8 (1 −9x6 y2 ) = 8 (1 −3x3 y) (1 +3x3 y)

6)32a4b − 2a2b = 2a2b (16a2 −1) = 2a2b (4a −1) (4a +1)

393.1) 2a2 +4ab +2b2 = 2 (a2 +2ab +b2 ) = 2 (a +b)2

2)2m2 + 2n2 − 4mn = 2 (m2 + n2 − 2mn) = 2 (m − n)2

3)5x2 +10xy + 5y2 = 5 (x2 + 2xy + y2 ) = 5 (x + y)2

4)8 p2 −16 p +8 = 8 ( p2 − 2 p +1) = 8 ( p −1)2

5)27a2b2 −18ab +3 = 3 (9a2b2 −6ab +1) = 3 (3ab −1)2

6)12m5n + 24m4n +12m3n =12m3n (m2 + 2m +1) =12m3n (m +1)2

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