Calculus Examples

${(y + z)}^{3} - 125$

Step 1

Rewrite $125$ as $5^{3}$ .

${(y + z)}^{3} - 5^{3}$

Step 2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = y + z$ and $b = 5$ .

$(y + z - 5) ({(y + z)}^{2} + (y + z) \cdot 5 + 5^{2})$

Step 3

Simplify.

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Step 3.1

Rewrite ${(y + z)}^{2}$ as $(y + z) (y + z)$ .

$(y + z - 5) ((y + z) (y + z) + (y + z) \cdot 5 + 5^{2})$

Step 3.2

Expand $(y + z) (y + z)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$(y + z - 5) (y (y + z) + z (y + z) + (y + z) \cdot 5 + 5^{2})$

Step 3.2.2

Apply the distributive property.

$(y + z - 5) (y \cdot y + y z + z (y + z) + (y + z) \cdot 5 + 5^{2})$

Step 3.2.3

Apply the distributive property.

$(y + z - 5) (y \cdot y + y z + z y + z \cdot z + (y + z) \cdot 5 + 5^{2})$

Step 3.3

Simplify and combine like terms.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $y$ by $y$ .

$(y + z - 5) (y^{2} + y z + z y + z \cdot z + (y + z) \cdot 5 + 5^{2})$

Step 3.3.1.2

Multiply $z$ by $z$ .

$(y + z - 5) (y^{2} + y z + z y + z^{2} + (y + z) \cdot 5 + 5^{2})$

Step 3.3.2

Add $y z$ and $z y$ .

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Step 3.3.2.1

Reorder $z$ and $y$ .

$(y + z - 5) (y^{2} + y z + y z + z^{2} + (y + z) \cdot 5 + 5^{2})$

Step 3.3.2.2

Add $y z$ and $y z$ .

$(y + z - 5) (y^{2} + 2 y z + z^{2} + (y + z) \cdot 5 + 5^{2})$

Step 3.4

Apply the distributive property.

$(y + z - 5) (y^{2} + 2 y z + z^{2} + y \cdot 5 + z \cdot 5 + 5^{2})$

Step 3.5

Move $5$ to the left of $y$ .

$(y + z - 5) (y^{2} + 2 y z + z^{2} + 5 \cdot y + z \cdot 5 + 5^{2})$

Step 3.6

Move $5$ to the left of $z$ .

$(y + z - 5) (y^{2} + 2 y z + z^{2} + 5 y + 5 z + 5^{2})$

Step 3.7

Raise $5$ to the power of $2$ .

$(y + z - 5) (y^{2} + 2 y z + z^{2} + 5 y + 5 z + 25)$