Question

Chứng minh đẳng thức:

a/ (x+y)2−y2=x(x+2y)(x+y)2−y2=x(x+2y)

b/ (x2+y2)2−(2xy)2=(x+y)2(x−y)2(x2+y2)2−(2xy)2=(x+y)2(x−y)2

c/ (a+b+c)2+(a+b−c)2+(2a−b)2(a+b+c)2+(a+b−c)2+(2a−b)2

d/ (a+b+c)2+(a+b−c)2+2(a+b)

Answer 1

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

b, \(\left(x^2+y^2\right)^2-\left(2xy\right)^2=x^4+y^4+2x^2y^2-4x^2y^2\)

\(=x^4+y^4-2x^2y^2=\left(x^2-y^2\right)^2=\left(x^2+xy-xy-y^2\right)^2\)

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

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

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

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

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

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

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

Answer 2

Phương An

Answer 3