Curso Mentorwww.cursomentor.com

Professor: Leonardo Santos

Tema: Produtos Notáveis II

Data: 25 de janeiro de 2014

Q1. Demonstre as identidades a seguir:

1. (a+ b)2 ≡ a2 + 2ab+ b2

2. (a− b)2 ≡ a2 − 2ab+ b2

3. (a+ b)3 ≡ a3 + 3a2b+ 3ab2 + b3

4. (a− b)3 ≡ a3 − 3a2b+ 3ab2 − b3

5. (a+ b)2 − (a− b)2 ≡ 4ab

6. a3 + b3 ≡ (a+ b)3 − 3ab(a+ b)

7. a3 − b3 ≡ (a− b)(a2 + ab+ b2)

8. a3 + b3 ≡ (a+ b)(a2 − ab+ b2)

9. ab ≡(a+ b

2

)2

−(a− b

2

)2

10. (a+ b)2 + (a− b)2 ≡ 2(a2 + b2)

11. (a− b)(a+ b)(a2 + b2) ≡ a4 − b4

12. (a− b)(a+ b)(a2 + b2)(a4 + b4) ≡ a8 − b8

13. (a+ b)3 ≡ a(a− 3b)2 + b(b− 3a)2

14. ab(a− b) + bc(b− c) + ca(c− a) ≡ (a− b)(b− c)(a− c)

15. (a+ b+ c)(ab+ bc+ ca)− abc ≡ (a+ b)(b+ c)(a+ c)

16. (a− b)3 + (b− c)3 + (c− a)3 ≡ 3(a− b)(b− c)(c− a)

17. (a+ b− c)3 + (b+ c− a)3 + (c+ a− b)3 + 24abc ≡ (a+ b+ c)3

18. a3 + b3 + c3 − 3abc ≡ (a+ b+ c)(a2 + b2 + c2 − ab− bc− ca)

1

19. a3 + b3 + c3 − 3abc ≡ (a+ b+ c)

[(a− b)2

2+

(b− c)2

2+

(a− c)2

2

]20. a3(b− c) + b3(c− a) + c3(a− b) + (a+ b+ c)(a− b)(b− c)(c− a) ≡ 0

21. a3(b− c)3 + b3(c− a)3 + c3(a− b)3 ≡ 3abc(a− b)(b− c)(c− a)

22. (a2 + ab+ b2)(a2 − ab+ b2) ≡ a4 + a2b2 + b4

23. (a+ b+ c)2 ≡ a2 + b2 + c2 + 2ab+ 2ac+ 2bc

24. (a+b+c)3 ≡ a3+b3+c3+3a2b+3a2c+3b2a+3b2c+3c2a+3c2b+6abc

25. (a+ b)2 + (b+ c)2 + (c+ a)2 ≡ (a+ b+ c)2 + a2 + b2 + c2

26. (a− b)2(a+ b)2(a2 + b2)2 ≡ (a4 − b4)2

27. (a+ b+ c)2 + (a− b)2 + (b− c)2 + (c− a)2 ≡ 3(a2 + b2 + c2)

28. (a+ b+ c+ d)2 + (a+ b− c− d)2 + (a− b+ c− d)2 + (a− b− c+ d)2 ≡4(a2 + b2 + c2 + d2)

29.(x− b)(x− c)

(a− b)(a− c)+

(x− c)(x− a)

(b− c)(b− a)+

(x− a)(x− b)

(c− a)(c− b)≡ 1

30. (y + z)2 + (z + x)2 + (x+ y)2 − x2 − y2 − z2 ≡ (x+ y + z)2

31. (x2 + xy + z2)2 − 4xy(x2 + y2) ≡ (x2 − xy + y2)2

32.a2(x− b)(x− c)

(a− b)(a− c)+b2(x− c)(x− a)

(b− c)(b− a)+c2(x− a)(x− b)

(c− a)(c− b)≡ x2 a ̸= b ̸= c

33.a(x− b)(x− c)

(a− b)(a− c)+

b(x− c)(x− a)

(b− c)(b− a)+

c(x− a)(x− b)

(c− a)(c− b)≡ x a ̸= b ̸= c

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