Sabendo que [tex3](x+y)^5 = x^5+5x^4y+10x^3y^2+10x^2y+5xy^4+y^5[/tex3]
, desenvolveremos [tex3](a+b+c)^5[/tex3]
, adotando [tex3]x=a[/tex3]
e [tex3]y=b+c[/tex3]
:
[tex3](a+b+c)^5-a^5-b^5-c^5 = [a+(b+c)]^5-a^5-b^5-c^5[/tex3]
[tex3]= a^5+5a^4(b+c) + 10a^3(b+c)^2 + 10a^2(b+c)^3 + 5a(b+c)^4+ (b+c)^5- a^5-b^5-c^5[/tex3]
[tex3]= 5a^4(b+c)+10a^3(b+c)^2 + 10a^2(b+c)^3+5a(b+c)^4 + (b+c)^5 -b^5 - c^5[/tex3]
Como [tex3]b^5+c^5 = (b+c)(b^4-b^3c+b^2c^2 - bc^3 +c^4)[/tex3]
, então:
[tex3](a+b+c)^5 - a^5-b^5-c^5 = 5a^4(b+c)+ 10a^3(b+c)^+ 10a^2(b+c)^3 + 5a(b+c)^4 + (b+c)^5 - (b+c)(b^4-b^3c+b^2c^2-bc^3+c^4) [/tex3]
Colocando [tex3](b+c)[/tex3]
em evidência, temos:
[tex3](a+b+c)^5-a^5-b^5-c^5 = (b+c)[ 5a^4+10a^3(b+c)+10a^2(b+c)^2+5a(b+c)^3+(b+c)^4- b^4+b^3c-b^2c^2+bc^3-c^4][/tex3]
Como [tex3](b+c)^4 = b^4+4b^3c+6b^2c^2+4bc^3+c^4[/tex3]
, então:
[tex3](a+b+c)^5 - a^5-b^5-c^5 =(b+c)[ 5a^4+10a^3(b+c)+ 10a^2(b+c)^2 +5a(b+c)^3+b^4+4b^3c+6b^2c^2+4bc^3+c^4-b^4+b^3c-b^2c^2+bc^3-c^4][/tex3]
[tex3]= (b+c)[5a^4+10a^3(b+c)+ 10a^2(b+c)^2+5a(b+c)^3+5b^3c+5b^2c^2+5bc^3][/tex3]
[tex3]= 5(b+c) [a^4+2a^3(b+c)+2a^2(b+c)^2+ a(b+c)^3+b^3c+b^2c^2+bc^3][/tex3]
Como [tex3](b+c)^2 = b^2 +2bc+c^2[/tex3]
e [tex3](b+c)^3 = b^3 +3b^2c+3bc^2+c^3[/tex3]
, então:
[tex3](a+b+c)^5 - a^5-b^5-c^5 = 5(b+c)[a^4+2a^3(b+c)+2a^2(b^2+2bc+c^2) + a(b^3+3b^2c+3bc^2+c^3)+b^3c+b^2c^2+bc^3][/tex3]
Daí, concluímos que:
[tex3](a+b+c)^5-a^5-b^5-c^5 = 5(b+c)[a^4+2a^3b+2a^3c+2a^2b^2+4a^2bc+2a^2c^2+ab^3+3ab^2c+3abc^2+ac^3+b^3c+b^2c^2+bc^3][/tex3]
[tex3]=5(b+c)[(a^4+a^2b^2+a^2c^2)+ (a^3b+ab^3+abc^2)+(a^3c+ab^2c+ac^3)+a^2bc+b^3c+bc^3)+(a^3b+a^3c+a^2bc)+(a^2b^2+a^2bc+ab^2c)+ (a^2bc+a^2c^2+a^2bc)+(ab^2c+abc^2+b^2c^2)][/tex3]
[tex3]= 5(b+c)[a^2(a^2+b^2+c^2) + ab(a^2+b^2+c^2)+ ac(a^2+b^2+c~^2) +bc(a^2+b^2+c^2) + a^2(ab+ac+bc) +ab(ab+ac+bc)+ac(ab+ac+bc)+bc(ab+ac+bc)][/tex3]
[tex3]= 5(b+c) [(a^2+ab+ac+bc)(a^2+b^2+c^2)(a^2+ab+ac+bc)(ab+ac+bc)] \implies 5(b+c)[(a^2+ab+ac+bc)(a^2+b^2+c^2+ab+ac+bc)][/tex3]
Em [tex3]a^2+ab+ac+bc[/tex3]
, temos:
[tex3]a(a+b)+c(a+b) \implies (a+b)(a+c)[/tex3]
Logo
[tex3](a+b+c)^5-a^5-b^5-c^5 = 5(a+b)(a+c)(b+c)(a^2+b^2+c^2+ab+ac+bc)[/tex3]
Atenciosamente goncalves3718
Solução retirada de
https://www.youtube.com/watch?v=NtLLyLk ... R&index=22