Calculus Examples

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

Step 1

Expand $(4 - z^{2}) (z^{3} - z + 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $- 1$ by $4$ .

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

Step 2.1.2

Multiply $4$ by $1$ .

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

Step 2.1.3

Multiply $z^{2}$ by $z^{3}$ by adding the exponents.

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Step 2.1.3.1

Move $z^{3}$ .

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

Step 2.1.3.2

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

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

Step 2.1.3.3

Add $3$ and $2$ .

$4 z^{3} - 4 z + 4 - z^{5} - z^{2} (- z) - z^{2} \cdot 1$

Step 2.1.4

Rewrite using the commutative property of multiplication.

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

Step 2.1.5

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

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Step 2.1.5.1

Move $z$ .

$4 z^{3} - 4 z + 4 - z^{5} - 1 \cdot - 1 (z \cdot z^{2}) - z^{2} \cdot 1$

Step 2.1.5.2

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

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Step 2.1.5.2.1

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

$4 z^{3} - 4 z + 4 - z^{5} - 1 \cdot - 1 (z^{1} z^{2}) - z^{2} \cdot 1$

Step 2.1.5.2.2

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

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

Step 2.1.5.3

Add $1$ and $2$ .

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

Step 2.1.6

Multiply $- 1$ by $- 1$ .

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

Step 2.1.7

Multiply $z^{3}$ by $1$ .

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

Step 2.1.8

Multiply $- 1$ by $1$ .

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

Step 2.2

Add $4 z^{3}$ and $z^{3}$ .

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