Basic Math Examples

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

Step 1

Expand $(z - y) (z^{2} + y^{2})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify each term.

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Step 2.1

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

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Step 2.1.1

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

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Step 2.1.1.1

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

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

Step 2.1.1.2

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

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

Step 2.1.2

Add $1$ and $2$ .

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

Step 2.2

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

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Step 2.2.1

Move $y^{2}$ .

$(z^{3} + z y^{2} - y z^{2} - (y^{2} y)) (z + y)$

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.3

Add $2$ and $1$ .

$(z^{3} + z y^{2} - y z^{2} - y^{3}) (z + y)$

Step 3

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

$z^{3} z + z^{3} y + z y^{2} z + z y^{2} y - y z^{2} z - y z^{2} y - y^{3} z - y^{3} y$

Step 4

Simplify terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

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Step 4.1.1.1

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

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Step 4.1.1.1.1

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

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

Step 4.1.1.1.2

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

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

Step 4.1.1.2

Add $3$ and $1$ .

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

Step 4.1.2

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

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Step 4.1.2.1

Move $z$ .

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

Step 4.1.2.2

Multiply $z$ by $z$ .

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

Step 4.1.3

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

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Step 4.1.3.1

Move $y$ .

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

Step 4.1.3.2

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

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Step 4.1.3.2.1

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

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

Step 4.1.3.2.2

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

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

Step 4.1.3.3

Add $1$ and $2$ .

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

Step 4.1.4

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

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Step 4.1.4.1

Move $z$ .

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

Step 4.1.4.2

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

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Step 4.1.4.2.1

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

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

Step 4.1.4.2.2

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

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

Step 4.1.4.3

Add $1$ and $2$ .

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

Step 4.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 4.1.5.1

Move $y$ .

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

Step 4.1.5.2

Multiply $y$ by $y$ .

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

Step 4.1.6

Multiply $y^{3}$ by $y$ by adding the exponents.

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Step 4.1.6.1

Move $y$ .

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

Step 4.1.6.2

Multiply $y$ by $y^{3}$ .

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Step 4.1.6.2.1

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

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

Step 4.1.6.2.2

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

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

Step 4.1.6.3

Add $1$ and $3$ .

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

Step 4.2

Combine the opposite terms in $z^{4} + z^{3} y + z^{2} y^{2} + z y^{3} - y z^{3} - y^{2} z^{2} - y^{3} z - y^{4}$ .

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Step 4.2.1

Reorder the factors in the terms $z^{3} y$ and $- y z^{3}$ .

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

Step 4.2.2

Subtract $z^{3} y$ from $z^{3} y$ .

$z^{4} + z^{2} y^{2} + z y^{3} + 0 - y^{2} z^{2} - y^{3} z - y^{4}$

Step 4.2.3

Add $z^{4} + z^{2} y^{2} + z y^{3}$ and $0$ .

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

Step 4.2.4

Reorder the factors in the terms $z^{2} y^{2}$ and $- y^{2} z^{2}$ .

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

Step 4.2.5

Subtract $y^{2} z^{2}$ from $y^{2} z^{2}$ .

$z^{4} + z y^{3} + 0 - y^{3} z - y^{4}$

Step 4.2.6

Add $z^{4} + z y^{3}$ and $0$ .

$z^{4} + z y^{3} - y^{3} z - y^{4}$

Step 4.2.7

Reorder the factors in the terms $z y^{3}$ and $- y^{3} z$ .

$z^{4} + y^{3} z - y^{3} z - y^{4}$

Step 4.2.8

Subtract $y^{3} z$ from $y^{3} z$ .

$z^{4} + 0 - y^{4}$

Step 4.2.9

Add $z^{4}$ and $0$ .

$z^{4} - y^{4}$