Certainly! Let's simplify the expression [tex]\((x + y + z)^3 - (x - y - z)^3\)[/tex].
Step-by-Step Solution:
1. Write down the given expression:
[tex]\[
(x + y + z)^3 - (x - y - z)^3
\][/tex]
2. Expand both cubes separately:
[tex]\[
(x + y + z)^3 = x^3 + y^3 + z^3 + 3x^2y + 3x^2z + 3y^2x + 3y^2z + 3z^2x + 3z^2y + 6xyz
\][/tex]
[tex]\[
(x - y - z)^3 = x^3 - y^3 - z^3 - 3x^2y - 3x^2z + 3y^2x + 3y^2z + 3z^2x + 3z^2y - 6xyz
\][/tex]
3. Subtract the second expansion from the first:
[tex]\[
(x + y + z)^3 - (x - y - z)^3 = [x^3 + y^3 + z^3 + 3x^2y + 3x^2z + 3y^2x + 3y^2z + 3z^2x + 3z^2y + 6xyz]
- [x^3 - y^3 - z^3 - 3x^2y - 3x^2z + 3y^2x + 3y^2z + 3z^2x + 3z^2y - 6xyz]
\][/tex]
4. Combine like terms:
[tex]\[
(x + y + z)^3 - (x - y - z)^3 = x^3 + y^3 + z^3 + 3x^2y + 3x^2z + 3y^2x + 3y^2z + 3z^2x + 3z^2y + 6xyz - x^3 + y^3 + z^3 + 3x^2y + 3x^2z - 3y^2x - 3y^2z - 3z^2x - 3z^2y + 6xyz
\][/tex]
5. Notice that many terms cancel each other out:
[tex]\[
= 2y^3 + 2z^3 + 6x^2y + 6x^2z - (- 6y^2x - 6y^2z - 6z^2x - 6z^2y + 12xyz)
\][/tex]
6. Rewrite the simplified expression:
[tex]\[
= (-x + y + z)^3 + (x + y + z)^3
\][/tex]
Therefore, the simplified form of [tex]\((x + y + z)^3 - (x - y - z)^3\)[/tex] is:
[tex]\[
(-x + y + z)^3 + (x + y + z)^3
\][/tex]
