Question

Cm các hằng đẳng thức :

a. x^3+y^3= (x+y)[(x-y)^2 +xy] 

b. X^3 + y^3 - xy (x+y)=(x+y)(x-y)^2

c. (x+y)(x^2 -xy + y^2) = (x+y)^3-3xy(x+y)

Giúp mình với _( Cảm ơn rất nhiều , mình đang cần gấp ạ

Answer 1

a. Ta có : (x + y)[(x - y)² + xy]

= (x + y)(x² - 2xy + y² + xy)

= (x + y)(x² - xy + y²)

= x³ + y³ 

b. Ta có : x³ + y³ - xy(x + y) 

= x³ + y³ - x²y - xy²

=x²(x - y) + y²(y - x)

= (x - y)(x² - y²)

= (x - y)².(x + y) đpcm

c) Ta có (x + y)³ - 3xy(x + y)

= (x + y)[(x + y)² - 3xy)

= (x + y)(x² + 2xy + y² - 3xy)

= (x + y)(x² - xy + y²) (đpcm)

Answer 2

a) VP = ( x + y )( x² - 2xy + y² + xy ) = ( x + y )( x² - xy + y² ) = x³ + y³ = VT ( đpcm )

b) VP = ( x + y )( x - y )² = ( x + y )( x² - 2xy + y² ) = x³ - 2x²y + xy² + x²y - 2xy² + y³ = x³ + y³ - x²y - xy² = x³ + y³ - xy( x + y ) = VT ( đpcm )

c) VP = x³ + 3x²y + 3xy² + y³ - 3x²y - 3xy² = x³ + y³ = ( x + y )( x² - xy + y² ) = VT ( đpcm )

Answer 3

a,\(x^3+y^3=\left(x+y\right)\left[\left(x-y\right)^2+xy\right]\)

\(VP=\left(x+y\right)\left[\left(x-y\right)^2+xy\right]\)

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

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

\(=x^3+y^3=VT\)

\(\Rightarrowđpcm\)

b,\(x^3+y^3-xy\left(x+y\right)=\left(x+y\right)\left(x-y\right)^2\)

\(VT=x^3+y^3-xy\left(x+y\right)\)

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

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

\(=x^2\left(x-y\right)-y^2\left(x-y\right)\)

\(=\left(x-y\right)\left(x^2-y^2\right)\)

\(=\left(x-y\right)\left(x-y\right)\left(x+y\right)\)

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

\(\Rightarrowđpcm\)

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

\(VP=\left(x+y\right)^3-3xy\left(x+y\right)\)

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

\(=x^3+y^3\)

\(VT=\left(x+y\right)\left(x^2-xy+y^2\right)\)

\(=x^3+y^3\)

\(\Rightarrow VP=VT\left(đpcm\right)\)