Pre-Algebra Examples

$\frac{3 x^{2} + 18 x y + 27 y^{2}}{x^{2} + 6 x y + 5 y^{2}} \div \frac{8 x + 24 y}{x + y}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{3 x^{2} + 18 x y + 27 y^{2}}{x^{2} + 6 x y + 5 y^{2}} \cdot \frac{x + y}{8 x + 24 y}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $3$ out of $3 x^{2} + 18 x y + 27 y^{2}$ .

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Step 2.1.1

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

$\frac{3 (x^{2}) + 18 x y + 27 y^{2}}{x^{2} + 6 x y + 5 y^{2}} \cdot \frac{x + y}{8 x + 24 y}$

Step 2.1.2

Factor $3$ out of $18 x y$ .

$\frac{3 (x^{2}) + 3 (6 x y) + 27 y^{2}}{x^{2} + 6 x y + 5 y^{2}} \cdot \frac{x + y}{8 x + 24 y}$

Step 2.1.3

Factor $3$ out of $27 y^{2}$ .

$\frac{3 (x^{2}) + 3 (6 x y) + 3 (9 y^{2})}{x^{2} + 6 x y + 5 y^{2}} \cdot \frac{x + y}{8 x + 24 y}$

Step 2.1.4

Factor $3$ out of $3 (x^{2}) + 3 (6 x y)$ .

$\frac{3 (x^{2} + 6 x y) + 3 (9 y^{2})}{x^{2} + 6 x y + 5 y^{2}} \cdot \frac{x + y}{8 x + 24 y}$

Step 2.1.5

Factor $3$ out of $3 (x^{2} + 6 x y) + 3 (9 y^{2})$ .

$\frac{3 (x^{2} + 6 x y + 9 y^{2})}{x^{2} + 6 x y + 5 y^{2}} \cdot \frac{x + y}{8 x + 24 y}$

Step 2.2

Factor using the perfect square rule.

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Step 2.2.1

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

$\frac{3 (x^{2} + 6 x y + {(3 y)}^{2})}{x^{2} + 6 x y + 5 y^{2}} \cdot \frac{x + y}{8 x + 24 y}$

Step 2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x y = 2 \cdot x \cdot (3 y)$

Step 2.2.3

Rewrite the polynomial.

$\frac{3 (x^{2} + 2 \cdot x \cdot (3 y) + {(3 y)}^{2})}{x^{2} + 6 x y + 5 y^{2}} \cdot \frac{x + y}{8 x + 24 y}$

Step 2.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 3 y$ .

$\frac{3 {(x + 3 y)}^{2}}{x^{2} + 6 x y + 5 y^{2}} \cdot \frac{x + y}{8 x + 24 y}$

Step 3

Factor by grouping.

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Step 3.1

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 $a \cdot c = 1 \cdot 5 = 5$ and whose sum is $b = 6$ .

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Step 3.1.1

Reorder terms.

$\frac{3 {(x + 3 y)}^{2}}{x^{2} + 5 y^{2} + 6 x y} \cdot \frac{x + y}{8 x + 24 y}$

Step 3.1.2

Reorder $5 y^{2}$ and $6 x y$ .

$\frac{3 {(x + 3 y)}^{2}}{x^{2} + 6 x y + 5 y^{2}} \cdot \frac{x + y}{8 x + 24 y}$

Step 3.1.3

Factor $6$ out of $6 x y$ .

$\frac{3 {(x + 3 y)}^{2}}{x^{2} + 6 (x y) + 5 y^{2}} \cdot \frac{x + y}{8 x + 24 y}$

Step 3.1.4

Rewrite $6$ as $1$ plus $5$

$\frac{3 {(x + 3 y)}^{2}}{x^{2} + (1 + 5) (x y) + 5 y^{2}} \cdot \frac{x + y}{8 x + 24 y}$

Step 3.1.5

Apply the distributive property.

$\frac{3 {(x + 3 y)}^{2}}{x^{2} + 1 (x y) + 5 (x y) + 5 y^{2}} \cdot \frac{x + y}{8 x + 24 y}$

Step 3.1.6

Multiply $x y$ by $1$ .

$\frac{3 {(x + 3 y)}^{2}}{x^{2} + x y + 5 (x y) + 5 y^{2}} \cdot \frac{x + y}{8 x + 24 y}$

Step 3.1.7

Move parentheses.

$\frac{3 {(x + 3 y)}^{2}}{x^{2} + x y + 5 x y + 5 y^{2}} \cdot \frac{x + y}{8 x + 24 y}$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$\frac{3 {(x + 3 y)}^{2}}{(x^{2} + x y) + 5 x y + 5 y^{2}} \cdot \frac{x + y}{8 x + 24 y}$

Step 3.2.2

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

$\frac{3 {(x + 3 y)}^{2}}{x (x + y) + 5 y (x + y)} \cdot \frac{x + y}{8 x + 24 y}$

Step 3.3

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

$\frac{3 {(x + 3 y)}^{2}}{(x + y) (x + 5 y)} \cdot \frac{x + y}{8 x + 24 y}$

Step 4

Cancel the common factor of $x + y$ .

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Step 4.1

Cancel the common factor.

$\frac{3 {(x + 3 y)}^{2}}{(x + y) (x + 5 y)} \cdot \frac{x + y}{8 x + 24 y}$

Step 4.2

Rewrite the expression.

$\frac{3 {(x + 3 y)}^{2}}{x + 5 y} \cdot \frac{1}{8 x + 24 y}$

Step 5

Multiply $\frac{3 {(x + 3 y)}^{2}}{x + 5 y}$ by $\frac{1}{8 x + 24 y}$ .

$\frac{3 {(x + 3 y)}^{2}}{(x + 5 y) (8 x + 24 y)}$

Step 6

Factor $8$ out of $8 x + 24 y$ .

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Step 6.1

Factor $8$ out of $8 x$ .

$\frac{3 {(x + 3 y)}^{2}}{(x + 5 y) (8 (x) + 24 y)}$

Step 6.2

Factor $8$ out of $24 y$ .

$\frac{3 {(x + 3 y)}^{2}}{(x + 5 y) (8 (x) + 8 (3 y))}$

Step 6.3

Factor $8$ out of $8 (x) + 8 (3 y)$ .

$\frac{3 {(x + 3 y)}^{2}}{(x + 5 y) (8 (x + 3 y))}$

$\frac{3 {(x + 3 y)}^{2}}{(x + 5 y) \cdot 8 (x + 3 y)}$

Step 7

Cancel the common factor of ${(x + 3 y)}^{2}$ and $x + 3 y$ .

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Step 7.1

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

$\frac{(x + 3 y) (3 (x + 3 y))}{(x + 5 y) \cdot 8 (x + 3 y)}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $x + 3 y$ out of $(x + 5 y) \cdot 8 (x + 3 y)$ .

$\frac{(x + 3 y) (3 (x + 3 y))}{(x + 3 y) ((x + 5 y) \cdot 8)}$

Step 7.2.2

Cancel the common factor.

$\frac{(x + 3 y) (3 (x + 3 y))}{(x + 3 y) ((x + 5 y) \cdot 8)}$

Step 7.2.3

Rewrite the expression.

$\frac{3 (x + 3 y)}{(x + 5 y) \cdot 8}$

Step 8

Move $8$ to the left of $x + 5 y$ .

$\frac{3 (x + 3 y)}{8 (x + 5 y)}$

Step 9

Apply the distributive property.

$\frac{3 x + 3 (3 y)}{8 (x + 5 y)}$

Step 10

Multiply $3$ by $3$ .

$\frac{3 x + 9 y}{8 (x + 5 y)}$

Step 11

Split the fraction $\frac{3 x + 9 y}{8 (x + 5 y)}$ into two fractions.

$\frac{3 x}{8 (x + 5 y)} + \frac{9 y}{8 (x + 5 y)}$