Expand the Trigonometric Expression (2z-1)(z^2-2z+1)

$(2 z - 1) (z^{2} - 2 z + 1)$

Step 1

Expand $(2 z - 1) (z^{2} - 2 z + 1)$ by multiplying each term in the first expression by each term in the second expression.

$2 z \cdot z^{2} + 2 z (- 2 z) + 2 z \cdot 1 - 1 z^{2} - 1 (- 2 z) - 1 \cdot 1$

Step 2

Simplify terms.

Simplify each term.

Multiply $z$ by $z^{2}$ by adding the exponents.

Move $z^{2}$ .

$2 (z^{2} z) + 2 z (- 2 z) + 2 z \cdot 1 - 1 z^{2} - 1 (- 2 z) - 1 \cdot 1$

Multiply $z^{2}$ by $z$ .

Raise $z$ to the power of $1$ .

$2 (z^{2} z^{1}) + 2 z (- 2 z) + 2 z \cdot 1 - 1 z^{2} - 1 (- 2 z) - 1 \cdot 1$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 z^{2 + 1} + 2 z (- 2 z) + 2 z \cdot 1 - 1 z^{2} - 1 (- 2 z) - 1 \cdot 1$

Add $2$ and $1$ .

$2 z^{3} + 2 z (- 2 z) + 2 z \cdot 1 - 1 z^{2} - 1 (- 2 z) - 1 \cdot 1$

Rewrite using the commutative property of multiplication.

$2 z^{3} + 2 \cdot - 2 z \cdot z + 2 z \cdot 1 - 1 z^{2} - 1 (- 2 z) - 1 \cdot 1$

Multiply $z$ by $z$ by adding the exponents.

Move $z$ .

$2 z^{3} + 2 \cdot - 2 (z \cdot z) + 2 z \cdot 1 - 1 z^{2} - 1 (- 2 z) - 1 \cdot 1$

Multiply $z$ by $z$ .

$2 z^{3} + 2 \cdot - 2 z^{2} + 2 z \cdot 1 - 1 z^{2} - 1 (- 2 z) - 1 \cdot 1$

Multiply $2$ by $- 2$ .

$2 z^{3} - 4 z^{2} + 2 z \cdot 1 - 1 z^{2} - 1 (- 2 z) - 1 \cdot 1$

Multiply $2$ by $1$ .

$2 z^{3} - 4 z^{2} + 2 z - 1 z^{2} - 1 (- 2 z) - 1 \cdot 1$

Rewrite $- 1 z^{2}$ as $- z^{2}$ .

$2 z^{3} - 4 z^{2} + 2 z - z^{2} - 1 (- 2 z) - 1 \cdot 1$

Multiply $- 2$ by $- 1$ .

$2 z^{3} - 4 z^{2} + 2 z - z^{2} + 2 z - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$2 z^{3} - 4 z^{2} + 2 z - z^{2} + 2 z - 1$

Simplify by adding terms.

Subtract $z^{2}$ from $- 4 z^{2}$ .

$2 z^{3} - 5 z^{2} + 2 z + 2 z - 1$

Add $2 z$ and $2 z$ .

$2 z^{3} - 5 z^{2} + 4 z - 1$