Question

Chứng minh rằng:

a) (x+y)³ = x(x-3y)² + y(y-3x)²

b) (a+b)³ + (a-b)³ = 2a(a²+3b²)

c) (a+b)³ - (a-b)³ = 2b(b² + 3a²)

d) a³+b³ = (a+b)³ - 3ab(a+b)

e) a³-b³ = (a-b)³+ 3ab(a-b)

giúp mik với mik cần gấp lắmmmm

Answer 1

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

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

b/

\(\left(a+b\right)^3+\left(a-b\right)^3=a^3+3a^2b+3ab^2+b^3+a^3-3a^2b+3ab^2-b^3\)

\(=2a^3+6ab^2=2a\left(a^2+3b^2\right)\)

c/

\(\left(a+b\right)^3-\left(a-b\right)^3=a^3+3a^2b+3ab^2+b^3-\left(a^3-3a^2b+3ab^2-b^3\right)\)

\(=6a^2b+2b^3=2b\left(b^2+3a^2\right)\)

d/

\(a^3+b^3=a^3+3a^2b+3ab^2+b^3-\left(3a^2b+3ab^2\right)\)

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

e/

\(a^3-b^3=a^3-3a^2b+3ab^2-b^3+3a^2b-3ab^2\)

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