Chöông I. PHEÙP NHAÂN VAØ PHEÙP CHIA CAÙC ÑA THÖÙC

§1. NHAÂN ÑÔN THÖÙC VÔÙI ÑA THÖÙC
NHAÂN ÑA THÖÙC VÔÙI ÑÔN THÖÙC

A/ KIEÁN THÖÙC VAØ KÓ NAÊNG CAÀN NHÔÙ

1. Muoán nhaân moät ñôn thöùc vôùi moät ña thöùc, ta nhaân ñôn thöùc vôùi töøng haïng töû cuûa ña thöùc roài coäng caùc tích vôùi nhau.

2. Muoán nhaân moät ña thöùc vôùi moät ña thöùc, ta nhaân moãi haïng töû cuûa ña thöùc naøy vôùi töøng haïng töû cuûa ña thöùc kia roài coäng caùc tích vôùi nhau.

B/ BAØI TAÄP

& BAØI TAÄP CÔ BAÛN

1. Laøm tính nhaân :

a) 5x²(4x³ – 7x + )                                                                                                            b) (8x³ – 6xy + 7)(–xy)

c) (4x² – 6x + 5)(2x + 3)                                                                                                       d) (4x² + 6xy + 9y²)(2x – 3y)

2. Tính giaù trò cuûa bieåu thöùc :

(2x + 3)(5x – 1) – 5x(2x – 7)

trong moãi tröôøng hôïp sau :

a) x = 0                                                                                                                        b) x = 2                                           c) x = –1                                                                       d) x = –50.

3. Ruùt goïn bieåu thöùc :

a) 2x(x – 3y) + 3y(2x – 5y)

b) (5x – 3y)(2x + y) – x(10x – y)

c) (x – y)(x² + xy + y²) – (x + y)(x² – xy + y²)

4. Chöùng minh raèng :

a) (3x + 2y)(5x – y) – y² = 15x² + 7xy – 3y²

b) 2x² + 5xy + 3y² = 4x² – (x – 3y)(2x + y)

c) (x + y)(x – y) – 9y² = (x – 2y)(x + 5y) – 3xy.

5. Tìm x :

a) 4x(3x – 7) – 6(2x² – 5x + 1) = 12

b) (5x + 3)(4x – 1) + (10x – 7)(–2x + 3) = 27

c) (8x – 5)(3x + 2) – (12x + 7)(2x – 1) = 17

d) (5x + 9)(6x – 1) – (2x – 3)(15x + 1) = –190.

6. Caùc bieåu thöùc sau bieåu thöùc naøo coù giaù trò cuûa bieåu thöùc khoâng phuï thuoäc vaøo giaù trò cuûa bieán ?

a) –5x(5x – 2) + (5x + 1)(5x – 1) – 10x

b) (x + 8)(x – 4) – x(x – 12) + 32

c) (2x + 3)(3x – 1) – 6x(x – 2) – 19(x – 5).

7. a) Tìm ba soá töï nhieân lieân tieáp, bieát tích cuûa hai soá sau lôùn hôn tích cuûa hai soá ñaàu laø 100.

b) Tìm ba soá töï nhieân chaün lieân tieáp, bieát tích cuûa hai soá sau lôùn hôn tích cuûa hai soá ñaàu laø 256.

c) Tìm ba soá töï nhieân leû lieân tieáp, bieát tích cuûa hai soá sau lôùn hôn tích cuûa hai soá ñaàu laø 68.

& BAØI TAÄP TRAÉC NGHIEÄM

8. Choïn caâu traû lôøi ñuùng :

Cho bieát 5x(2x – 7) – 10x² = –70

vaø (y + 3)(y – 2) – y(y – 5) = 12

Giaù trò cuûa x + y laø :

A. –1                                                                                                                 B. –5

C. 5                                                                                                                             D. Moät keát quaû khaùc.

& BAØI TAÄP NAÂNG CAO

9. Tính giaù trò cuûa bieåu thöùc :

a) x⁴ – 2224x³ + 2223x² – 2223x + 2223. taïi x = 2222

b) x¹⁴ – 2009x¹³ + 2009x¹² – 2009x¹¹ + ... + 2009x² – 2009x + 2009 taïi x = 2008.

10. Cho A = 123456.123457 – 123455.123458

B = 987654.987655 – 987653.987656

So saùnh A vaø B

11. a) Chöùng minh raèng (x – 3)² + 65 = x(x – 6) + 75

b) Tìm giaù trò nhoû nhaát cuûa bieåu thöùc M = x(x – 6) + 74

12. Xaùc ñònh a, b bieát :

(x + a)(x + 5) = x² + 3x + b vôùi moïi x.

13. a) Cho a, b laø hai soá töï nhieân. Bieát a chia cho 5 dö 3, b chia cho 5 dö 4. Chöùng minh raèng ab chia cho 5 dö 2.

b) Tìm caùc soá töï nhieân x, y sao cho

(5^x + 3)(5^y + 4) = 516

14. Cho p laø soá nguyeân toá, p ³ 5 thoaû maõn 2p + 1 laø soá nguyeân toá.

Chöùng minh raèng : p(p + 7) + 31 laø hôïp soá.

& BAØI THI CHOÏN HOÏC SINH GIOÛI TOAÙN

15. Ruùt goïn bieåu thöùc :

A = 75(4¹⁹⁹³ + 4¹⁹⁹² + ... + 4² + 5) + 25

(Ñeà thi choïn hoïc sinh gioûi Toaùn, lôùp 8, Quaän 3 Tp. Hoà Chí Minh, naêm hoïc 1995 – 1996)

§2. NHÖÕNG HAÈNG ÑAÚNG THÖÙC ÑAÙNG NHÔÙ

A/ KIEÁN THÖÙC VAØ KÓ NAÊNG CAÀN NHÔÙ

1. Bình phöông cuûa moät toång

(A + B)² = A² + 2AB + B²

2. Bình phöông cuûa moät hieäu

(A – B)² = A² – 2AB + B²

3. Hieäu hai bình phöông

A² – B² = (A + B)(A – B)

4. Laäp phöông cuûa moät toång

(A + B)³ = A³ + 3A²B + 3AB² + B³

5. Laäp phöông cuûa moät hieäu

(A – B)³ = A³ – 3A²B + 3AB² – B³

6. Toång hai laäp phöông

A³ + B³ = (A + B)(A² – AB + B²)

7. Hieäu hai laäp phöông

A³ – B³ = (A – B)(A² + AB + B²)

Vôùi A, B laø caùc bieåu thöùc tuøy yù.

B/ BAØI TAÄP

& BAØI TAÄP CÔ BAÛN

16. Tính :

a) (2x + 3)²                                                                                                                                                                                                                                                                           b) (5x – 4)²

c) (4x² + 3y)(4x² – 3y)                                                                                                    d) (2x + 1)³

e) (x – y²)³                                                                                                                                                                                                                                                               f) (2x + 3)(4x² – 6x + 9)

g) ( x – 2y)(x² + 2xy + 4y²)                                                                                    h) (x + y)(x² – xy + y²)(x³ – y³)

17. Vieát caùc bieåu thöùc sau döôùi daïng bình phöông cuûa moät toång hoaëc moät hieäu :

a) x² + 6x + 9                                                                                                                     b) 25x² – 10xy + y²

c) x² + x +                                                                                                                       d) x² + xy + 25y².

18. Tính :

a) (a + b + c)²                                                                                                                                                                                                    b) (a – b + c)²

19. Vieát caùc bieåu thöùc sau döôùi daïng laäp phöông cuûa moät toång hoaëc moät hieäu :

a) x³ + 6x² + 12x + 8                                                                                                     b) 8 – 12x + 6x² – x³

c) 8x³ – 12x²y² + 6xy⁴ – y⁶                                                                                   d) –x³ + 6x² – 12x + 8

20. Ruùt goïn caùc bieåu thöùc :

a) (3x – 2)² + (4x – 1)² + (2 + 5x)(2 – 5x)

b) (x + 2)(x² – 2x + 4) – (x – 3)(x² + 3x + 9)

c) (x + y – z)² + 2(x + y – z)(z – y) + (z – y)²

d) (a – b)³ + (b – c)³ + (c – a)³ + 3a²b + 3b²c + 3c²a.

21. Tìm x, bieát :

a) (x + 2)² + (x – 3)² = 2x(x + 7)

b) (x + 3)(x² – 3x + 9) = x(x² + 4) – 1

c) (x + 1)³ + (x – 1)³ = 2x³

d) x³ – 3x² + 3x – 126 = 0.

22. Tính nhanh :

a) 89² + 11² + 22.89

b) 73² + 23² – 46.73

c) 976² – 975² + 975² – 974².

& BAØI TAÄP TRAÉC NGHIEÄM

23. Choïn caâu traû lôøi ñuùng :

Ruùt goïn bieåu thöùc :

(2x + 7)² + (5 + 2x)(5 – 2x) – 25

A. 14x + 49                                                                                                                        B. –14x + 49

C. 14x – 49                                                                                                                        D. Moät keát quaû khaùc.

& BAØI TAÄP NAÂNG CAO

24. a) Vieát bieåu thöùc sau döôùi daïng toång cuøa hai bình phöông

4x² + y² – 4x + 10y + 26

b) Tìm x, y bieát 4x² + y² – 4x + 10y + 26 = 0.

25. Tính nhanh :

a) A = 100² – 99² + 98² – 97² + ... + 2² – 1²

b) B = 1² – 2² + 3² – 4² + ... – 2008² + 2009²

c) C = (2 + 1)(2² + 1)(2⁴ + 1)(2⁸ + 1)(2¹⁶ + 1) – 2³².

26. a) Chöùng minh raèng caùc bieåu thöùc sau döông vôùi moïi x

A = x² + 8x + 17

B = x² – 10x + 29

b) Chöùng minh raèng caùc bieåu thöùc sau aâm vôùi moïi x

C = –x² + 2x – 5

D = –x² + x – 1

27. a) Tìm giaù trò nhoû nhaát cuûa bieåu thöùc sau :

A = 4x² – 4x + 2009

b) Tìm giaø trò lôùn nhaát cuûa bieåu thöùc sau :

B = –x² + 5x – 127

28. Cho a + b + c = 0. Chöùng minh raèng a³ + b³ + c³ = 3abc

29. Cho a³ – 3ab² = 2 vaø b³ – 3a²b = –11 Tính a² + b².

& BAØI THI CHOÏN HOÏC SINH GIOÛI TOAÙN

30. a) Cho a, b, c, d Î Z thoaû maõn a + b = c + d. Chöùng minh raèng
a² + b² + c² + d² luoân laø toång cuûa ba soá chính phöông.

(Ñeà thi choïn hoïc sinh gioûi Toaùn, lôùp 8, Quaän 9 Tp. Hoà Chí Minh, naêm hoïc 2007 – 2008)

b) Chöùng minh raèng : Neáu p vaø 1 laø hai soá nguyeân toá thoaû maõn p² – q¹ = p – 3q + 2 thì
p² + q² cuõng laø soá nguyeân toá.

(Ñeà thi choïn hoïc sinh gioûi Toaùn, lôùp 8, Quaän 1 Tp. Hoà Chí Minh, naêm hoïc 2008 – 2009)

c) Chöùng minh raèng bieåu thöùc sau khoâng theå laø laäp phöông cuûa moät soá töï nhieân

1991³³³³ + 1990²²²² + 1989¹¹¹¹

(Ñeà thi choïn hoïc sinh gioûi Toaùn, lôùp 8, tröôøng chuyeân Vaên Toaùn huyeän Ñöùc Phoå, Tænh Quaûng Ngaõi, naêm hoïc 1990 – 1991)

d) Chöùng minh raèng hieäu caùc bình phöông cuûa hai soá leû baát kì thì chia heát cho 8.

(Ñeà thi choïn hoïc sinh gioûi Toaùn, lôùp 8,
Quaän 1 Tp. Hoà Chí Minh, naêm hoïc 2005 – 2006)