Simplify (z-1)(z^2-4z+9)

$(z - 1) (z^{2} - 4 z + 9)$

Step 1

Expand $(z - 1) (z^{2} - 4 z + 9)$ by multiplying each term in the first expression by each term in the second expression.

$z \cdot z^{2} + z (- 4 z) + z \cdot 9 - 1 z^{2} - 1 (- 4 z) - 1 \cdot 9$

Step 2

Simplify terms.

Simplify each term.

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

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

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

$z^{1} z^{2} + z (- 4 z) + z \cdot 9 - 1 z^{2} - 1 (- 4 z) - 1 \cdot 9$

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

$z^{1 + 2} + z (- 4 z) + z \cdot 9 - 1 z^{2} - 1 (- 4 z) - 1 \cdot 9$

Add $1$ and $2$ .

$z^{3} + z (- 4 z) + z \cdot 9 - 1 z^{2} - 1 (- 4 z) - 1 \cdot 9$

Rewrite using the commutative property of multiplication.

$z^{3} - 4 z \cdot z + z \cdot 9 - 1 z^{2} - 1 (- 4 z) - 1 \cdot 9$

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

Move $z$ .

$z^{3} - 4 (z \cdot z) + z \cdot 9 - 1 z^{2} - 1 (- 4 z) - 1 \cdot 9$

Multiply $z$ by $z$ .

$z^{3} - 4 z^{2} + z \cdot 9 - 1 z^{2} - 1 (- 4 z) - 1 \cdot 9$

Move $9$ to the left of $z$ .

$z^{3} - 4 z^{2} + 9 \cdot z - 1 z^{2} - 1 (- 4 z) - 1 \cdot 9$

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

$z^{3} - 4 z^{2} + 9 z - z^{2} - 1 (- 4 z) - 1 \cdot 9$

Multiply $- 4$ by $- 1$ .

$z^{3} - 4 z^{2} + 9 z - z^{2} + 4 z - 1 \cdot 9$

Multiply $- 1$ by $9$ .

$z^{3} - 4 z^{2} + 9 z - z^{2} + 4 z - 9$

Simplify by adding terms.

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

$z^{3} - 5 z^{2} + 9 z + 4 z - 9$

Add $9 z$ and $4 z$ .

$z^{3} - 5 z^{2} + 13 z - 9$