Simplify (3x^3-4x^2y+xy^2)/(9x^2y-y^3)

$\frac{3 x^{3} - 4 x^{2} y + x y^{2}}{9 x^{2} y - y^{3}}$

Step 1

Simplify the numerator.

Factor $x$ out of $3 x^{3} - 4 x^{2} y + x y^{2}$ .

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

$\frac{x (3 x^{2}) - 4 x^{2} y + x y^{2}}{9 x^{2} y - y^{3}}$

Factor $x$ out of $- 4 x^{2} y$ .

$\frac{x (3 x^{2}) + x (- 4 x y) + x y^{2}}{9 x^{2} y - y^{3}}$

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

$\frac{x (3 x^{2}) + x (- 4 x y) + x (y^{2})}{9 x^{2} y - y^{3}}$

Factor $x$ out of $x (3 x^{2}) + x (- 4 x y)$ .

$\frac{x (3 x^{2} - 4 x y) + x (y^{2})}{9 x^{2} y - y^{3}}$

Factor $x$ out of $x (3 x^{2} - 4 x y) + x (y^{2})$ .

$\frac{x (3 x^{2} - 4 x y + y^{2})}{9 x^{2} y - y^{3}}$

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 $a \cdot c = 3 \cdot 1 = 3$ and whose sum is $b = - 4$ .

Reorder terms.

$\frac{x (3 x^{2} + y^{2} - 4 x y)}{9 x^{2} y - y^{3}}$

Reorder $y^{2}$ and $- 4 x y$ .

$\frac{x (3 x^{2} - 4 x y + y^{2})}{9 x^{2} y - y^{3}}$

Factor $- 4$ out of $- 4 x y$ .

$\frac{x (3 x^{2} - 4 (x y) + y^{2})}{9 x^{2} y - y^{3}}$

Rewrite $- 4$ as $- 1$ plus $- 3$

$\frac{x (3 x^{2} + (- 1 - 3) (x y) + y^{2})}{9 x^{2} y - y^{3}}$

Apply the distributive property.

$\frac{x (3 x^{2} - 1 (x y) - 3 (x y) + y^{2})}{9 x^{2} y - y^{3}}$

Move parentheses.

$\frac{x (3 x^{2} - 1 x y - 3 x y + y^{2})}{9 x^{2} y - y^{3}}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{x ((3 x^{2} - 1 x y) - 3 x y + y^{2})}{9 x^{2} y - y^{3}}$

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

$\frac{x (x (3 x - 1 y) - y (3 x - y))}{9 x^{2} y - y^{3}}$

Factor the polynomial by factoring out the greatest common factor, $3 x - 1 y$ .

$\frac{x ((3 x - 1 y) (x - y))}{9 x^{2} y - y^{3}}$

Rewrite $- 1 y$ as $- y$ .

$\frac{x (3 x - y) (x - y)}{9 x^{2} y - y^{3}}$

Step 2

Simplify the denominator.

Factor $y$ out of $9 x^{2} y - y^{3}$ .

Factor $y$ out of $9 x^{2} y$ .

$\frac{x (3 x - y) (x - y)}{y (9 x^{2}) - y^{3}}$

Factor $y$ out of $- y^{3}$ .

$\frac{x (3 x - y) (x - y)}{y (9 x^{2}) + y (- y^{2})}$

Factor $y$ out of $y (9 x^{2}) + y (- y^{2})$ .

$\frac{x (3 x - y) (x - y)}{y (9 x^{2} - y^{2})}$

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

$\frac{x (3 x - y) (x - y)}{y ({(3 x)}^{2} - y^{2})}$

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

$\frac{x (3 x - y) (x - y)}{y (3 x + y) (3 x - y)}$

Step 3

Cancel the common factor of $3 x - y$ .

Cancel the common factor.

$\frac{x (3 x - y) (x - y)}{y (3 x + y) (3 x - y)}$

Rewrite the expression.

$\frac{x (x - y)}{y (3 x + y)}$

Simplify (3x^3-4x^2y+xy^2)/(9x^2y-y^3)

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