1, Chứng minh các đẳng thức :
a, \(\left(x^2+y^2\right)^2-\left(2xy\right)^2=\left(x+y\right)^2\left(x-y\right)^2\)
b, \(\left(x+y\right)^3=x\left(x-3y\right)^2+y\left(y-3x\right)^2\)
2, CMR : \(\left(a+b\right)^3-\left(a-b\right)^3=2b\left(b^2+3a^2\right)\)
1.a, VT= \(\left(x^2+y^2\right)^2-\left(2xy\right)^2=\)\(\left(x^2+y^2-2xy\right)\left(x^2+y^2+2xy\right)=\left(x-y\right)^2\left(x+y\right)^2=VP.\left(đpcm\right)\)
b, VP=\(x\left(x-3y\right)^2+y\left(y-3x\right)^2\)\(=x\left(x^2-6xy+9y^2\right)+y\left(y^2-6xy+9x^2\right)\)\(=x^3-6x^2y+9xy^2+y^3-6xy^2+9x^2y\)
\(=x^3+3x^2y+3xy^2+y^3\)\(=\left(x+y\right)^3=VT\left(đpcm\right)\)
2. VT=\(\left(a+b\right)^3-\left(a-b\right)^3\)\(=\left(a+b-a+b\right)\left(a^2+2ab+b^2+a^2-b^2+a^2-2ab+b^2\right)\)
\(2b\left(b^2+3a^2\right)\)\(=VP\left(đpcm\right)\).
a) (x2 + y2)2 - (2xy)2
= [(x2 + y2) - 2xy].[(x2 + y2) + 2xy]
= [x2 + y2 - 2xy].[(x2 + y2 + 2xy]
= (x - y)2 . (x + y)2
a \(\left(x^2+y^2\right)^2-\left(2xy\right)^2=\left(x+y\right)^2\left(x-y\right)^2\)
Ta có : \(\left(x^2+y^2\right)^2-\left(2xy\right)^2=\left(x^2+y^2+2xy\right)\left(x^2+y^2-2xy\right)\)
\(=\left(x+y\right)^2\left(x-y\right)^2\)
b) \(\left(x+y\right)^3=x\left(x-3y\right)^2+y\left(y-3x\right)^2\)
ta có: \(x\left(x-3y\right)^2+y\left(y-3x\right)^2\)
\(=x\left(x^2-6xy+9y^2\right)+y\left(y^2-6xy+9x^2\right)\)
\(=x^3-6x^2y+9xy^2+y^3-6xy^2+9x^2y\)
= \(x^3+3x^2y+3xy^2+y^3\)
\(=\left(x+y\right)^3\)
2. \(\left(a+b\right)^3-\left(a-b\right)^3=2b\left(b^2+3a^2\right)\)
Ta có: \(\left(a+b\right)^3-\left(a-b\right)^3=a^3+3a^2b+3ab^2+b^3-a^3+3a^2b-3ab^2+b^3\)
= \(2b^3+6a^2b\)
\(=2b\left(b^2+3a^2\right)\)
