Factor (p-2)^3-125q^3

${(p - 2)}^{3} - 125 q^{3}$

Step 1

Rewrite $125 q^{3}$ as ${(5 q)}^{3}$ .

${(p - 2)}^{3} - {(5 q)}^{3}$

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 $a = p - 2$ and $b = 5 q$ .

$(p - 2 - (5 q)) ({(p - 2)}^{2} + (p - 2) (5 q) + {(5 q)}^{2})$

Step 3

Simplify.

Multiply $5$ by $- 1$ .

$(p - 2 - 5 q) ({(p - 2)}^{2} + (p - 2) (5 q) + {(5 q)}^{2})$

Rewrite ${(p - 2)}^{2}$ as $(p - 2) (p - 2)$ .

$(p - 2 - 5 q) ((p - 2) (p - 2) + (p - 2) (5 q) + {(5 q)}^{2})$

Expand $(p - 2) (p - 2)$ using the FOIL Method.

Apply the distributive property.

$(p - 2 - 5 q) (p (p - 2) - 2 (p - 2) + (p - 2) (5 q) + {(5 q)}^{2})$

Apply the distributive property.

$(p - 2 - 5 q) (p \cdot p + p \cdot - 2 - 2 (p - 2) + (p - 2) (5 q) + {(5 q)}^{2})$

Apply the distributive property.

$(p - 2 - 5 q) (p \cdot p + p \cdot - 2 - 2 p - 2 \cdot - 2 + (p - 2) (5 q) + {(5 q)}^{2})$

Simplify and combine like terms.

Simplify each term.

Multiply $p$ by $p$ .

$(p - 2 - 5 q) (p^{2} + p \cdot - 2 - 2 p - 2 \cdot - 2 + (p - 2) (5 q) + {(5 q)}^{2})$

Move $- 2$ to the left of $p$ .

$(p - 2 - 5 q) (p^{2} - 2 p - 2 p - 2 \cdot - 2 + (p - 2) (5 q) + {(5 q)}^{2})$

Multiply $- 2$ by $- 2$ .

$(p - 2 - 5 q) (p^{2} - 2 p - 2 p + 4 + (p - 2) (5 q) + {(5 q)}^{2})$

Subtract $2 p$ from $- 2 p$ .

$(p - 2 - 5 q) (p^{2} - 4 p + 4 + (p - 2) (5 q) + {(5 q)}^{2})$

Apply the distributive property.

$(p - 2 - 5 q) (p^{2} - 4 p + 4 + p (5 q) - 2 (5 q) + {(5 q)}^{2})$

Rewrite using the commutative property of multiplication.

$(p - 2 - 5 q) (p^{2} - 4 p + 4 + 5 p q - 2 (5 q) + {(5 q)}^{2})$

Multiply $5$ by $- 2$ .

$(p - 2 - 5 q) (p^{2} - 4 p + 4 + 5 p q - 10 q + {(5 q)}^{2})$

Apply the product rule to $5 q$ .

$(p - 2 - 5 q) (p^{2} - 4 p + 4 + 5 p q - 10 q + 5^{2} q^{2})$

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

$(p - 2 - 5 q) (p^{2} - 4 p + 4 + 5 p q - 10 q + 25 q^{2})$

Factor (p-2)^3-125q^3

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