Calculus Examples

$\frac{x^{2} + 8 x y - 20 y^{2}}{x + 3 y} \div \frac{3 x + 30 y}{x - y}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} + 8 x y - 20 y^{2}}{x + 3 y} \cdot \frac{x - y}{3 x + 30 y}$

Step 2

Factor by grouping.

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Step 2.1

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 $a \cdot c = 1 \cdot - 20 = - 20$ and whose sum is $b = 8$ .

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Step 2.1.1

Reorder terms.

$\frac{x^{2} - 20 y^{2} + 8 x y}{x + 3 y} \cdot \frac{x - y}{3 x + 30 y}$

Step 2.1.2

Reorder $- 20 y^{2}$ and $8 x y$ .

$\frac{x^{2} + 8 x y - 20 y^{2}}{x + 3 y} \cdot \frac{x - y}{3 x + 30 y}$

Step 2.1.3

Factor $8$ out of $8 x y$ .

$\frac{x^{2} + 8 (x y) - 20 y^{2}}{x + 3 y} \cdot \frac{x - y}{3 x + 30 y}$

Step 2.1.4

Rewrite $8$ as $- 2$ plus $10$

$\frac{x^{2} + (- 2 + 10) (x y) - 20 y^{2}}{x + 3 y} \cdot \frac{x - y}{3 x + 30 y}$

Step 2.1.5

Apply the distributive property.

$\frac{x^{2} - 2 (x y) + 10 (x y) - 20 y^{2}}{x + 3 y} \cdot \frac{x - y}{3 x + 30 y}$

Step 2.1.6

Move parentheses.

$\frac{x^{2} - 2 x y + 10 x y - 20 y^{2}}{x + 3 y} \cdot \frac{x - y}{3 x + 30 y}$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$\frac{(x^{2} - 2 x y) + 10 x y - 20 y^{2}}{x + 3 y} \cdot \frac{x - y}{3 x + 30 y}$

Step 2.2.2

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

$\frac{x (x - 2 y) + 10 y (x - 2 y)}{x + 3 y} \cdot \frac{x - y}{3 x + 30 y}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $x - 2 y$ .

$\frac{(x - 2 y) (x + 10 y)}{x + 3 y} \cdot \frac{x - y}{3 x + 30 y}$

Step 3

Factor $3$ out of $3 x + 30 y$ .

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Step 3.1

Factor $3$ out of $3 x$ .

$\frac{(x - 2 y) (x + 10 y)}{x + 3 y} \cdot \frac{x - y}{3 (x) + 30 y}$

Step 3.2

Factor $3$ out of $30 y$ .

$\frac{(x - 2 y) (x + 10 y)}{x + 3 y} \cdot \frac{x - y}{3 (x) + 3 (10 y)}$

Step 3.3

Factor $3$ out of $3 (x) + 3 (10 y)$ .

$\frac{(x - 2 y) (x + 10 y)}{x + 3 y} \cdot \frac{x - y}{3 (x + 10 y)}$

Step 4

Cancel the common factor of $x + 10 y$ .

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Step 4.1

Factor $x + 10 y$ out of $(x - 2 y) (x + 10 y)$ .

$\frac{(x + 10 y) (x - 2 y)}{x + 3 y} \cdot \frac{x - y}{3 (x + 10 y)}$

Step 4.2

Factor $x + 10 y$ out of $3 (x + 10 y)$ .

$\frac{(x + 10 y) (x - 2 y)}{x + 3 y} \cdot \frac{x - y}{(x + 10 y) \cdot 3}$

Step 4.3

Cancel the common factor.

$\frac{(x + 10 y) (x - 2 y)}{x + 3 y} \cdot \frac{x - y}{(x + 10 y) \cdot 3}$

Step 4.4

Rewrite the expression.

$\frac{x - 2 y}{x + 3 y} \cdot \frac{x - y}{3}$

Step 5

Multiply $\frac{x - 2 y}{x + 3 y}$ by $\frac{x - y}{3}$ .

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

Step 6

Move $3$ to the left of $x + 3 y$ .

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

Step 7

Expand $(x - 2 y) (x - y)$ using the FOIL Method.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Apply the distributive property.

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

Step 8

Simplify and combine like terms.

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Step 8.1

Simplify each term.

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Step 8.1.1

Multiply $x$ by $x$ .

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

Step 8.1.2

Rewrite using the commutative property of multiplication.

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

Step 8.1.3

Rewrite using the commutative property of multiplication.

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

Step 8.1.4

Multiply $y$ by $y$ by adding the exponents.

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Step 8.1.4.1

Move $y$ .

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

Step 8.1.4.2

Multiply $y$ by $y$ .

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

Step 8.1.5

Multiply $- 2$ by $- 1$ .

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

Step 8.2

Subtract $2 y x$ from $- x y$ .

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Step 8.2.1

Move $y$ .

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

Step 8.2.2

Subtract $2 x y$ from $- x y$ .

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

Step 9

Split the fraction $\frac{x^{2} - 3 x y + 2 y^{2}}{3 (x + 3 y)}$ into two fractions.

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

Step 10

Split the fraction $\frac{x^{2} - 3 x y}{3 (x + 3 y)}$ into two fractions.

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

Step 11

Cancel the common factor of $- 3$ and $3$ .

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Step 11.1

Factor $3$ out of $- 3 x y$ .

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

Step 11.2

Cancel the common factors.

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Step 11.2.1

Cancel the common factor.

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

Step 11.2.2

Rewrite the expression.

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

Step 12

Move the negative in front of the fraction.

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