Factor 8x^9-125y^3z^6

$8 x^{9} - 125 y^{3} z^{6}$

Rewrite $8 x^{9}$ as ${(2 x^{3})}^{3}$ .

${(2 x^{3})}^{3} - 125 y^{3} z^{6}$

Rewrite $125 y^{3} z^{6}$ as ${(5 y z^{2})}^{3}$ .

${(2 x^{3})}^{3} - {(5 y z^{2})}^{3}$

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 $a = 2 x^{3}$ and $b = 5 y z^{2}$ .

$(2 x^{3} - (5 y z^{2})) ({(2 x^{3})}^{2} + 2 x^{3} (5 y z^{2}) + {(5 y z^{2})}^{2})$

Simplify.

Multiply $5$ by $- 1$ .

$(2 x^{3} - 5 (y z^{2})) ({(2 x^{3})}^{2} + 2 x^{3} (5 y z^{2}) + {(5 y z^{2})}^{2})$

Remove parentheses.

$(2 x^{3} - 5 y z^{2}) ({(2 x^{3})}^{2} + 2 x^{3} (5 y z^{2}) + {(5 y z^{2})}^{2})$

Apply the product rule to $2 x^{3}$ .

$(2 x^{3} - 5 y z^{2}) (2^{2} {(x^{3})}^{2} + 2 x^{3} (5 y z^{2}) + {(5 y z^{2})}^{2})$

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

$(2 x^{3} - 5 y z^{2}) (4 {(x^{3})}^{2} + 2 x^{3} (5 y z^{2}) + {(5 y z^{2})}^{2})$

Multiply the exponents in ${(x^{3})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Multiply $3$ by $2$ .

$(2 x^{3} - 5 y z^{2}) (4 x^{6} + 2 x^{3} (5 y z^{2}) + {(5 y z^{2})}^{2})$

Multiply $5$ by $2$ .

$(2 x^{3} - 5 y z^{2}) (4 x^{6} + 10 x^{3} (y z^{2}) + {(5 y z^{2})}^{2})$

Remove parentheses.

$(2 x^{3} - 5 y z^{2}) (4 x^{6} + 10 x^{3} y z^{2} + {(5 y z^{2})}^{2})$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $5 y z^{2}$ .

$(2 x^{3} - 5 y z^{2}) (4 x^{6} + 10 x^{3} y z^{2} + {(5 y)}^{2} {(z^{2})}^{2})$

Apply the product rule to $5 y$ .

$(2 x^{3} - 5 y z^{2}) (4 x^{6} + 10 x^{3} y z^{2} + 5^{2} y^{2} {(z^{2})}^{2})$

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

$(2 x^{3} - 5 y z^{2}) (4 x^{6} + 10 x^{3} y z^{2} + 25 y^{2} {(z^{2})}^{2})$

Multiply the exponents in ${(z^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$(2 x^{3} - 5 y z^{2}) (4 x^{6} + 10 x^{3} y z^{2} + 25 y^{2} z^{2 \cdot 2})$

Multiply $2$ by $2$ .

$(2 x^{3} - 5 y z^{2}) (4 x^{6} + 10 x^{3} y z^{2} + 25 y^{2} z^{4})$

Factor 8x^9-125y^3z^6

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