Question

descompunere:
a)8a³+12a²+6a+1=?
b)a³-3a²b+3ab²-b³=?
c)x³-(2x-y)³=?

Answer 1

a)
8a³+12a²+6a+1=8a^3 +(4a^2 +8a^2) +(4a +2a) +1=

(8a^3 +4a^2) +(8a^2 +4a) +(2a +1)=

4a^2 (2a+1) +4a(2a+1) +(2a+1)=

(2a+1)(4a^2 +4a +1)=

(2a+1)(2a+1)^2=

(2a+1)^3

b)
b)a³-3a²b+3ab²-b³=

a^3 -a^2 *b -2a^2 *b +2a*b^2 +a*b^2 -b^3=

(a^3 -a^2 *b) -(2a^2 *b -2a*b^2) +(a*b^2 -b^3)=

a^2 *(a-b) -2ab*(a-b) +b^2 *(a-b)=

(a-b)(a^2 -2ab +b^2)=

(a-b)^3

c)
x³-(2x-y)³=

Aplicam: a^3 -b^3=(a-b)(a^2 +ab +b^2)

x³-(2x-y)³=(x-2x+y)((x^2 +x(2x-y) +y^2)=

(y-x)(3x^2 -xy +y^2)

Answer 2

a) 8a³+12a²+6a+1=8a³+8a²+4a²+4a+2a+1
=8a²(a+1)+4a(a+1)+2a+1
=(a+1)(8a²+4a)+2a+1
=4a(a+1)(2a+1)+2a+1
=(2a+1)(4a(a+1)+1))
b) a³-3a²b+3ab²-b³=(a-b)(a²-2ab+b²)
c) x³-(2x-y)³  aplicam formula pentru a³-b³=(a-b)(a²+ab+b²)
si avem
=(x-2x+y)(x²+x(2x-y)+(2x-y)²)
=(y-x)(x²+2x²-xy+4x²-4xy+y²)
=(y-x)(7x²-5xy+y²)