To expand the given expression, we need to distribute [tex]\((2z - 1)\)[/tex] through the polynomial [tex]\((z^2 - 2z + 1)\)[/tex].
Let's start by writing down the original expression:
[tex]\[
(2z - 1)(z^2 - 2z + 1)
\][/tex]
We will distribute each term in [tex]\((2z - 1)\)[/tex] to every term in [tex]\((z^2 - 2z + 1)\)[/tex].
1. First, distribute [tex]\(2z\)[/tex]:
[tex]\[
2z \cdot (z^2 - 2z + 1) = 2z \cdot z^2 - 2z \cdot 2z + 2z \cdot 1
\][/tex]
This gives us:
[tex]\[
2z^3 - 4z^2 + 2z
\][/tex]
2. Next, distribute [tex]\(-1\)[/tex]:
[tex]\[
-1 \cdot (z^2 - 2z + 1) = -1 \cdot z^2 - 1 \cdot (-2z) + (-1) \cdot 1
\][/tex]
This gives us:
[tex]\[
-z^2 + 2z - 1
\][/tex]
3. Now, combine all the terms we have obtained:
[tex]\[
2z^3 - 4z^2 + 2z - z^2 + 2z - 1
\][/tex]
4. Combine like terms:
[tex]\[
2z^3 - 4z^2 - z^2 + 2z + 2z - 1
\][/tex]
This simplifies to:
[tex]\[
2z^3 - 5z^2 + 4z - 1
\][/tex]
So, the expanded form of the expression is:
[tex]\[
(2z - 1)(z^2 - 2z + 1) = 2z^3 - 5z^2 + 4z - 1
\][/tex]
Thus, your final polynomial in standard form is:
[tex]\[
2z^3 - 5z^2 + 4z - 1
\][/tex]
