Question

Chứng minh:

a) \(\left(a-b\right)^3=-\left(b-a\right)^3\)

b) \(\left(-a-b\right)^2=\left(a+b\right)^2\)

c) \(\left(x+y\right)^3=x\left(x-3y\right)^2+y\left(y-3x\right)^2\)

d) \(\left(x+y\right)^3-\left(x-y\right)^3=2y\left(y^2+3x^2\right)\)

Answer 1

a. -(b-a)³= -b³+a³ (phá ngoặc trước có dấu trừ nên đổi dấu)

= a³ - b³ = (a-b)³

Answer 2

b)

\(\left(-a-b\right)^2=\left(-a\right)^2-2.\left(-a\right)b+b^2\\ =a^2+2ab+b^2=\left(a+b\right)^2\)

Answer 3

c) \(x\left(x-3y\right)^2+y\left(y-3x\right)^2\\ =x\left(x^2-6xy+9y^2\right)+y\left(y^2-6xy+9x^2\right)\\ =x^3-6x^2y+9xy^2+y^3-6xy^2+9x^2y\\ =x^3+3x^2y+3xy^2+y^3\\ =\left(x+y\right)^3\)

Answer 4

d) \(\left(x+y\right)^3-\left(x-y\right)^3\\ =x^3+3x^2y+3xy^2+y^3-x^3+3x^2y-3xy^2+y^3\\ =2y^3+6x^2y\\ =2y\left(y^2+3x^2\right)\)