Factor 180x^2y-108xy^2-75x^3

$180 x^{2} y - 108 x y^{2} - 75 x^{3}$

Step 1

Factor $3 x$ out of $180 x^{2} y - 108 x y^{2} - 75 x^{3}$ .

Factor $3 x$ out of $180 x^{2} y$ .

$3 x (60 x y) - 108 x y^{2} - 75 x^{3}$

Factor $3 x$ out of $- 108 x y^{2}$ .

$3 x (60 x y) + 3 x (- 36 y^{2}) - 75 x^{3}$

Factor $3 x$ out of $- 75 x^{3}$ .

$3 x (60 x y) + 3 x (- 36 y^{2}) + 3 x (- 25 x^{2})$

Factor $3 x$ out of $3 x (60 x y) + 3 x (- 36 y^{2})$ .

$3 x (60 x y - 36 y^{2}) + 3 x (- 25 x^{2})$

Factor $3 x$ out of $3 x (60 x y - 36 y^{2}) + 3 x (- 25 x^{2})$ .

$3 x (60 x y - 36 y^{2} - 25 x^{2})$

Step 2

Factor.

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 25 \cdot - 36 = 900$ and whose sum is $b = 60$ .

Reorder terms.

$3 x (- 25 x^{2} - 36 y^{2} + 60 x y)$

Reorder $- 36 y^{2}$ and $60 x y$ .

$3 x (- 25 x^{2} + 60 x y - 36 y^{2})$

Factor $60$ out of $60 x y$ .

$3 x (- 25 x^{2} + 60 (x y) - 36 y^{2})$

Rewrite $60$ as $30$ plus $30$

$3 x (- 25 x^{2} + (30 + 30) (x y) - 36 y^{2})$

Apply the distributive property.

$3 x (- 25 x^{2} + 30 (x y) + 30 (x y) - 36 y^{2})$

Remove unnecessary parentheses.

$3 x (- 25 x^{2} + 30 x y + 30 (x y) - 36 y^{2})$

Remove unnecessary parentheses.

$3 x (- 25 x^{2} + 30 x y + 30 x y - 36 y^{2})$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$3 x ((- 25 x^{2} + 30 x y) + 30 x y - 36 y^{2})$

Factor out the greatest common factor (GCF) from each group.

$3 x (5 x (- 5 x + 6 y) - 6 y (- 5 x + 6 y))$

Factor the polynomial by factoring out the greatest common factor, $- 5 x + 6 y$ .

$3 x ((- 5 x + 6 y) (5 x - 6 y))$

Remove unnecessary parentheses.

$3 x (- 5 x + 6 y) (5 x - 6 y)$

Step 3

Combine exponents.

Factor $- 1$ out of $- 5 x$ .

$3 x (- (5 x) + 6 y) (5 x - 6 y)$

Factor $- 1$ out of $6 y$ .

$3 x (- (5 x) - (- 6 y)) (5 x - 6 y)$

Factor $- 1$ out of $- (5 x) - (- 6 y)$ .

$3 x (- (5 x - 6 y)) (5 x - 6 y)$

Rewrite $- (5 x - 6 y)$ as $- 1 (5 x - 6 y)$ .

$3 x (- 1 (5 x - 6 y)) (5 x - 6 y)$

Remove parentheses.

$3 x \cdot - 1 (5 x - 6 y) (5 x - 6 y)$

Raise $5 x - 6 y$ to the power of $1$ .

$3 x \cdot - 1 ({(5 x - 6 y)}^{1} (5 x - 6 y))$

Raise $5 x - 6 y$ to the power of $1$ .

$3 x \cdot - 1 ({(5 x - 6 y)}^{1} {(5 x - 6 y)}^{1})$

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

$3 x \cdot - 1 {(5 x - 6 y)}^{1 + 1}$

Add $1$ and $1$ .

$3 x \cdot - 1 {(5 x - 6 y)}^{2}$

Multiply $- 1$ by $3$ .

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

Factor 180x^2y-108xy^2-75x^3

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