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Using the identity prove: x3+y3+z33xyz=(x+y+z)(x2+y2+z2xyyzzx).

Answer
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Hint: To prove the equation we will use the identity of (a+b)3=a3+b3+3ab(a+b). We can also write the equation as a3+b3=(a+b)33ab(a+b). Put this identity equation in the LHS of the equation x3+y3+z33xyz, simplify the equation, use multiplication of polynomials. Use the identity of (x+y+z)2=x2+y2+z2+2xy+2yz+2xz to simplify it further. Arrange the terms to obtain the RHS of the equation (x+y+z)(x2+y2+z2xyyzzx).

Complete step-by-step answer:
Here we have to prove the given equation, which is x3+y3+z33xyz=(x+y+z)(x2+y2+z2xyyzzx).
We will use the identity to prove the LHS of the equation equals the RHS of the equation.
As we know the equation (a+b)3=a3+b3+3ab(a+b), we can write the equation as a3+b3=(a+b)33ab(a+b) also. We will use this identity equation to prove.
Now taking Left hand side of the equation,
LHS=x3+y3+z33xyz
Now using the identity a3+b3=(a+b)33ab(a+b), we can write as x3+y3=(x+y)33xy(x+y).
Putting this in the above equation, LHS=(x+y)33xy(x+y)+z33xyz,
Arranging the terms of equation, LHS=(x+y)3+z33xy(x+y)3xyz
Now again using the same identity for (x+y)3+z3, we can write the equation as (x+y)3+z3={(x+y)+z}33(x+y)z{(x+y)+z}.
Putting the equation of (x+y)3+z3 in our LHS equation, LHS={(x+y)+z}33(x+y)z{(x+y)+z}3xy(x+y)3xyz
So, LHS=(x+y+z)33(x+y)z(x+y+z)3xy(x+y)3xyz
Use multiplication of two polynomials to simplify the equation. For multiplication of two polynomials, we multiply each monomial of one polynomial with its sign by each monomial of another polynomial with its sign.
So, LHS=(x+y+z)33(x+y)z(x+y+z)3xyx3xyy3xyz
Taking 3xy common in last three terms above equation,
LHS=(x+y+z)33(x+y)z(x+y+z)3xy(x+y+z)
Now taking (x+y+z) common in all terms of the equation,
So, LHS=(x+y+z){(x+y+z)23(x+y)z3xy}
Now use another identity of (x+y+z)2=x2+y2+z2+2xy+2yz+2xz. Put this in the above equation.
So, LHS=(x+y+z){(x2+y2+z2+2xy+2yz+2xz)3(x+y)z3xy},
Multiply the polynomials, LHS=(x+y+z){x2+y2+z2+2xy+2yz+2xz3xz3yz3xy}
Arranging the terms, LHS=(x+y+z){x2+y2+z2+2xy3xy+2yz3yz+2xz3xz}
So, LHS=(x+y+z){x2+y2+z2xyyzxz}
So, LHS=RHS.
Hence it is proved that x3+y3+z33xyz=(x+y+z)(x2+y2+z2xyyzzx).

Note: Without using the identity we can also prove x3+y3+z33xyz=(x+y+z)(x2+y2+z2xyyzzx). For this method take first RHS=(x+y+z){x2+y2+z2xyyzxz}. After this use multiplication of polynomials to simplify the equation.
So, RHS=xx2+xy2+xz2xxyxyzxzx+yx2+yy2+yz2yxyyyzyxz+zx2+zy2+zz2zxyzyzzxz
Simplifying the equation,
RHS=x3+xy2+xz2x2yxyzx2z+yx2+y3+yz2xy2y2zxyz+zx2+zy2+z3xyzyz2xz2
Arranging the terms, RHS=(x3+y3+z3)+(xy2+xz2x2yx2z+x2y+yz2xy2y2z+x2z+zy2yz2xz2)3xyz
Simplifying, RHS=(x3+y3+z3)+(0)3xyz
So, RHS=x3+y3+z33xyz
So, RHS=LHS.
Hence it is proved that x3+y3+z33xyz=(x+y+z)(x2+y2+z2xyyzzx)