Answered

Expand. Your answer should be a polynomial in standard form.

[tex](2z - 1)\left(z^2 - 2z + 1\right) = \square[/tex]



Answer :

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]