To simplify the expression [tex]\( x^4 - (x - z)^4 \)[/tex], let's go through the steps systematically.
1. Original Expression:
[tex]\[
x^4 - (x - z)^4
\][/tex]
2. Expand [tex]\((x - z)^4\)[/tex]:
We need to expand the term [tex]\((x - z)^4\)[/tex]. Using the binomial theorem, we get:
[tex]\[
(x - z)^4 = x^4 - 4x^3z + 6x^2z^2 - 4xz^3 + z^4
\][/tex]
3. Substitute the Expanded Form:
Substitute the expanded form of [tex]\((x - z)^4\)[/tex] back into the original expression:
[tex]\[
x^4 - [x^4 - 4x^3z + 6x^2z^2 - 4xz^3 + z^4]
\][/tex]
4. Distribute the Minus Sign:
Distribute the minus sign across the terms inside the brackets:
[tex]\[
x^4 - x^4 + 4x^3z - 6x^2z^2 + 4xz^3 - z^4
\][/tex]
5. Combine Like Terms:
Combine the like terms:
[tex]\[
x^4 - x^4 = 0
\][/tex]
Therefore, the expression simplifies to:
[tex]\[
0 + 4x^3z - 6x^2z^2 + 4xz^3 - z^4
\][/tex]
6. Final Simplified Expression:
So, the simplified form of [tex]\( x^4 - (x - z)^4 \)[/tex] is:
[tex]\[
4x^3z - 6x^2z^2 + 4xz^3 - z^4
\][/tex]
Hence, the final simplified version of the given expression [tex]\( x^4 - (x - z)^4 \)[/tex] is:
[tex]\[
4x^3z - 6x^2z^2 + 4xz^3 - z^4
\][/tex]
