Finite Math Examples

$\frac{x + 4}{5 x + 15} \cdot \frac{x^{2} + 6 x + 9}{x^{2} - 16} \cdot \frac{15 x^{2} - 30 x}{2 x^{5}}$

Step 1

Simplify with factoring out.

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Step 1.1

Factor $15 x$ out of $15 x^{2} - 30 x$ .

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Step 1.1.1

Factor $15 x$ out of $15 x^{2}$ .

$\frac{x + 4}{5 x + 15} \cdot \frac{x^{2} + 6 x + 9}{x^{2} - 16} \cdot \frac{15 x (x) - 30 x}{2 x^{5}}$

Step 1.1.2

Factor $15 x$ out of $- 30 x$ .

$\frac{x + 4}{5 x + 15} \cdot \frac{x^{2} + 6 x + 9}{x^{2} - 16} \cdot \frac{15 x (x) + 15 x (- 2)}{2 x^{5}}$

Step 1.1.3

Factor $15 x$ out of $15 x (x) + 15 x (- 2)$ .

$\frac{x + 4}{5 x + 15} \cdot \frac{x^{2} + 6 x + 9}{x^{2} - 16} \cdot \frac{15 x (x - 2)}{2 x^{5}}$

Step 1.2

Factor $5$ out of $5 x + 15$ .

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Step 1.2.1

Factor $5$ out of $5 x$ .

$\frac{x + 4}{5 (x) + 15} \cdot \frac{x^{2} + 6 x + 9}{x^{2} - 16} \cdot \frac{15 x (x - 2)}{2 x^{5}}$

Step 1.2.2

Factor $5$ out of $15$ .

$\frac{x + 4}{5 x + 5 \cdot 3} \cdot \frac{x^{2} + 6 x + 9}{x^{2} - 16} \cdot \frac{15 x (x - 2)}{2 x^{5}}$

Step 1.2.3

Factor $5$ out of $5 x + 5 \cdot 3$ .

$\frac{x + 4}{5 (x + 3)} \cdot \frac{x^{2} + 6 x + 9}{x^{2} - 16} \cdot \frac{15 x (x - 2)}{2 x^{5}}$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $9$ as $3^{2}$ .

$\frac{x + 4}{5 (x + 3)} \cdot \frac{x^{2} + 6 x + 3^{2}}{x^{2} - 16} \cdot \frac{15 x (x - 2)}{2 x^{5}}$

Step 2.2

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

$6 x = 2 \cdot x \cdot 3$

Step 2.3

Rewrite the polynomial.

$\frac{x + 4}{5 (x + 3)} \cdot \frac{x^{2} + 2 \cdot x \cdot 3 + 3^{2}}{x^{2} - 16} \cdot \frac{15 x (x - 2)}{2 x^{5}}$

Step 2.4

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

$\frac{x + 4}{5 (x + 3)} \cdot \frac{{(x + 3)}^{2}}{x^{2} - 16} \cdot \frac{15 x (x - 2)}{2 x^{5}}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $16$ as $4^{2}$ .

$\frac{x + 4}{5 (x + 3)} \cdot \frac{{(x + 3)}^{2}}{x^{2} - 4^{2}} \cdot \frac{15 x (x - 2)}{2 x^{5}}$

Step 3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 4$ .

$\frac{x + 4}{5 (x + 3)} \cdot \frac{{(x + 3)}^{2}}{(x + 4) (x - 4)} \cdot \frac{15 x (x - 2)}{2 x^{5}}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x + 4$ .

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Step 4.1.1

Cancel the common factor.

$\frac{x + 4}{5 (x + 3)} \cdot \frac{{(x + 3)}^{2}}{(x + 4) (x - 4)} \cdot \frac{15 x (x - 2)}{2 x^{5}}$

Step 4.1.2

Rewrite the expression.

$\frac{1}{5 (x + 3)} \cdot \frac{{(x + 3)}^{2}}{x - 4} \cdot \frac{15 x (x - 2)}{2 x^{5}}$

Step 4.2

Cancel the common factor of $x + 3$ .

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Step 4.2.1

Factor $x + 3$ out of $5 (x + 3)$ .

$\frac{1}{(x + 3) \cdot 5} \cdot \frac{{(x + 3)}^{2}}{x - 4} \cdot \frac{15 x (x - 2)}{2 x^{5}}$

Step 4.2.2

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

$\frac{1}{(x + 3) \cdot 5} \cdot \frac{(x + 3) (x + 3)}{x - 4} \cdot \frac{15 x (x - 2)}{2 x^{5}}$

Step 4.2.3

Cancel the common factor.

$\frac{1}{(x + 3) \cdot 5} \cdot \frac{(x + 3) (x + 3)}{x - 4} \cdot \frac{15 x (x - 2)}{2 x^{5}}$

Step 4.2.4

Rewrite the expression.

$\frac{1}{5} \cdot \frac{x + 3}{x - 4} \cdot \frac{15 x (x - 2)}{2 x^{5}}$

Step 4.3

Multiply $\frac{1}{5}$ by $\frac{x + 3}{x - 4}$ .

$\frac{x + 3}{5 (x - 4)} \cdot \frac{15 x (x - 2)}{2 x^{5}}$

Step 4.4

Cancel the common factor of $5$ .

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Step 4.4.1

Factor $5$ out of $15 x (x - 2)$ .

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

Step 4.4.2

Cancel the common factor.

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

Step 4.4.3

Rewrite the expression.

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

Step 4.5

Multiply $\frac{x + 3}{x - 4}$ by $\frac{3 x (x - 2)}{2 x^{5}}$ .

$\frac{(x + 3) (3 x (x - 2))}{(x - 4) (2 x^{5})}$

Step 4.6

Cancel the common factor of $x$ and $x^{5}$ .

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Step 4.6.1

Factor $x$ out of $(x + 3) (3 x (x - 2))$ .

$\frac{x ((x + 3) (3 (x - 2)))}{(x - 4) (2 x^{5})}$

Step 4.6.2

Cancel the common factors.

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Step 4.6.2.1

Factor $x$ out of $(x - 4) (2 x^{5})$ .

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

Step 4.6.2.2

Cancel the common factor.

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

Step 4.6.2.3

Rewrite the expression.

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

Step 4.7

Simplify the expression.

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Step 4.7.1

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

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

Step 4.7.2

Move $2$ to the left of $x - 4$ .

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

Step 4.7.3

Reorder factors in $\frac{3 (x + 3) (x - 2)}{2 (x - 4) x^{4}}$ .

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