Simplify the following expression: (x + y + z)2 + (x + y2 + z3 )2 - ( x2 + y3 + z4 )

1. Break Down the Problem

We need to simplify the expression: $(x + y + z)^2 + (x + y^2 + z^3)^2 - (x^2 + y^3 + z^4)$

2. Relevant Concepts

We will use the binomial expansion to expand the squares in the expression.

3. Analysis and Detail

1. Expand $(x + y + z)^2$: $(x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2xz + 2yz$

2. Expand $(x + y^2 + z^3)^2$: $(x + y^2 + z^3)^2 = x^2 + y^4 + z^6 + 2xy^2 + 2xz^3 + 2y^2z^3$

3. Combine the expansions: Putting the expansions back together gives: $x^2 + y^2 + z^2 + 2xy + 2xz + 2yz + x^2 + y^4 + z^6 + 2xy^2 + 2xz^3 + 2y^2z^3$

4. Combine like terms:

   • The total $x^2$ terms: $2x^2$
   • The $y^2$ terms: $y^2$
   • The $z^2$ terms: $z^2$
   • The $y^4$ term: $y^4$
   • The $z^6$ term: $z^6$
   • Other terms: $2xy + 2xz + 2yz + 2xy^2 + 2xz^3 + 2y^2z^3$

   We get: $2x^2 + y^2 + z^2 + y^4 + z^6 + 2xy + 2xz + 2yz + 2xy^2 + 2xz^3 + 2y^2z^3$

5. Subtract $(x^2 + y^3 + z^4)$: Now, we subtract: $2x^2 + y^2 + z^2 + y^4 + z^6 + 2xy + 2xz + 2yz + 2xy^2 + 2xz^3 + 2y^2z^3 - (x^2 + y^3 + z^4)$

6. Combine again:

   • For $x^2$: $2x^2 - x^2 = x^2$
   • For $y^2$: $y^2 - y^3 = y^2 - y^3$
   • For $z^2$: $z^2 - z^4 = z^2 - z^4$
   • Remaining terms stay the same.

   The final combined expression is: $x^2 + y^2 - y^3 + z^2 - z^4 + y^4 + z^6 + 2xy + 2xz + 2yz + 2xy^2 + 2xz^3 + 2y^2z^3$

4. Verify and Summarize

After carefully expanding and combining all terms step by step, we have arrived at the simplified expression.

Final Answer

$x^2 + y^2 - y^3 + z^2 - z^4 + y^4 + z^6 + 2xy + 2xz + 2yz + 2xy^2 + 2xz^3 + 2y^2z^3$