Chứng minh: a) (x+y+z)^2=x^2+y^2+z^2+2xy+2xz+2yz b) (x+y+z)^3=x^3+y^3+z^3+3.(x+y).(y+z).(z+x) c) (x^3+x^2y+xy^2+y^3).(x-y)=x^4-y^4

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1. a, (x + y + z)^2

   = (x+y)^2 + 2(x+y)z + z^2

   = x^2 + 2xy + y^2 + 2xz + 2y + z^2

   = x^2+y^2+z^2+2xy+2xz+2yz (đpcm)

   b, (x + y + z)^3

   =(x + y)^3 + 3(x + y)^2.z +3(x + y).z^2 + y^3

   =x^3 + 3x^2.y + 3xy^2 + y^3 + 3x^2.z + 6xyz + 3y^2.z +3xz^2 + 3yz^2 + y^3

   = x^3 + y^3 + z^3 + (3x^2.y + 3xy^2 + 3x^2.z + 6xyz + 3y^2.z +3xz^2 + 3yz^2)

   =

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

   = x^4 – x^3y + x^3y -x^2y^2 +x^2y^2 -xy^3 + xy^3 – y^4

   =x^4 – y^4

   (Phần ở trong ảnh bạn thay A, B, C thành x, y , z nha

    

    

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2. Giải thích các bước giải:

   a) $(x+y+z)^2$

   $ = (x+y+z).(x+y+z)$

   $ = x^2+xy+xz+xy+y^2+yz+zx+yz+z^2$

   $ = x^2+y^2+z^2+2xy+2yz+2zx$

   b) $(x+y+z)^3$

   $ = (x+y+z)^2.(x+y+z)$

   $= (x^2+y^2+z^2+2xy+2yz+2zx).(x+y+z)$

   $ = x^3+y^3+z^3+x^2.(y+z)+y^2.(x+z)+z^2.(y+x) +(x+y+z).(2xy+2yz+2zx)$

   $ = x^3+y^3+z^3 + 3.(x^2y+xy^2+y^2z+yz^2+z^2x+zx^2+2xyz)$

   $ = x^3+y^3+z^3+3.[xy.(x+y) + zx.(x+y) + z^2.(x+y) + yz.(x+y)]$

   $ = x^3+y^3+z^3+3.(x+y).(xy+zx+z^2+yz)$

   $ = x^3+y^3+z^3+3.(x+y).[z.(y+z)+x.(y+z)]$

   $ = x^3+y^3+z^3.(x+y).(y+z).(z+x)$

   $ = x^3+y^3+z^3 + 3.(x+y).(y+z).(z+x)$

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

   $ =x^4+x^3y+x^2y^2+y^3x-x^3y-x^2y^2-xy^3-y^4$

   $= x^4-y^4$

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