Simplify (3/(3x^2-147)*(3x^2+24x+21)/(6x^3+18x^2+12x))÷((x-2)/(x^2-4))

$\frac{3}{3 x^{2} - 147} \cdot \frac{3 x^{2} + 24 x + 21}{6 x^{3} + 18 x^{2} + 12 x} \div \frac{x - 2}{x^{2} - 4}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{3}{3 x^{2} - 147} \cdot \frac{3 x^{2} + 24 x + 21}{6 x^{3} + 18 x^{2} + 12 x} \frac{x^{2} - 4}{x - 2}$

Step 2

Simplify the denominator.

Factor $3$ out of $3 x^{2} - 147$ .

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

$\frac{3}{3 (x^{2}) - 147} \cdot \frac{3 x^{2} + 24 x + 21}{6 x^{3} + 18 x^{2} + 12 x} \frac{x^{2} - 4}{x - 2}$

Factor $3$ out of $- 147$ .

$\frac{3}{3 x^{2} + 3 \cdot - 49} \cdot \frac{3 x^{2} + 24 x + 21}{6 x^{3} + 18 x^{2} + 12 x} \frac{x^{2} - 4}{x - 2}$

Factor $3$ out of $3 x^{2} + 3 \cdot - 49$ .

$\frac{3}{3 (x^{2} - 49)} \cdot \frac{3 x^{2} + 24 x + 21}{6 x^{3} + 18 x^{2} + 12 x} \frac{x^{2} - 4}{x - 2}$

Rewrite $49$ as $7^{2}$ .

$\frac{3}{3 (x^{2} - 7^{2})} \cdot \frac{3 x^{2} + 24 x + 21}{6 x^{3} + 18 x^{2} + 12 x} \frac{x^{2} - 4}{x - 2}$

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 $b = 7$ .

$\frac{3}{3 (x + 7) (x - 7)} \cdot \frac{3 x^{2} + 24 x + 21}{6 x^{3} + 18 x^{2} + 12 x} \frac{x^{2} - 4}{x - 2}$

Step 3

Simplify the numerator.

Factor $3$ out of $3 x^{2} + 24 x + 21$ .

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

$\frac{3}{3 (x + 7) (x - 7)} \cdot \frac{3 (x^{2}) + 24 x + 21}{6 x^{3} + 18 x^{2} + 12 x} \frac{x^{2} - 4}{x - 2}$

Factor $3$ out of $24 x$ .

$\frac{3}{3 (x + 7) (x - 7)} \cdot \frac{3 (x^{2}) + 3 (8 x) + 21}{6 x^{3} + 18 x^{2} + 12 x} \frac{x^{2} - 4}{x - 2}$

Factor $3$ out of $21$ .

$\frac{3}{3 (x + 7) (x - 7)} \cdot \frac{3 x^{2} + 3 (8 x) + 3 \cdot 7}{6 x^{3} + 18 x^{2} + 12 x} \frac{x^{2} - 4}{x - 2}$

Factor $3$ out of $3 x^{2} + 3 (8 x)$ .

$\frac{3}{3 (x + 7) (x - 7)} \cdot \frac{3 (x^{2} + 8 x) + 3 \cdot 7}{6 x^{3} + 18 x^{2} + 12 x} \frac{x^{2} - 4}{x - 2}$

Factor $3$ out of $3 (x^{2} + 8 x) + 3 \cdot 7$ .

$\frac{3}{3 (x + 7) (x - 7)} \cdot \frac{3 (x^{2} + 8 x + 7)}{6 x^{3} + 18 x^{2} + 12 x} \frac{x^{2} - 4}{x - 2}$

Factor $x^{2} + 8 x + 7$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $7$ and whose sum is $8$ .

$1, 7$

Write the factored form using these integers.

$\frac{3}{3 (x + 7) (x - 7)} \cdot \frac{3 ((x + 1) (x + 7))}{6 x^{3} + 18 x^{2} + 12 x} \frac{x^{2} - 4}{x - 2}$

$\frac{3}{3 (x + 7) (x - 7)} \cdot \frac{3 (x + 1) (x + 7)}{6 x^{3} + 18 x^{2} + 12 x} \frac{x^{2} - 4}{x - 2}$

Step 4

Simplify the denominator.

Factor $6 x$ out of $6 x^{3} + 18 x^{2} + 12 x$ .

Factor $6 x$ out of $6 x^{3}$ .

$\frac{3}{3 (x + 7) (x - 7)} \cdot \frac{3 (x + 1) (x + 7)}{6 x (x^{2}) + 18 x^{2} + 12 x} \frac{x^{2} - 4}{x - 2}$

Factor $6 x$ out of $18 x^{2}$ .

$\frac{3}{3 (x + 7) (x - 7)} \cdot \frac{3 (x + 1) (x + 7)}{6 x (x^{2}) + 6 x (3 x) + 12 x} \frac{x^{2} - 4}{x - 2}$

Factor $6 x$ out of $12 x$ .

$\frac{3}{3 (x + 7) (x - 7)} \cdot \frac{3 (x + 1) (x + 7)}{6 x (x^{2}) + 6 x (3 x) + 6 x (2)} \frac{x^{2} - 4}{x - 2}$

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

$\frac{3}{3 (x + 7) (x - 7)} \cdot \frac{3 (x + 1) (x + 7)}{6 x (x^{2} + 3 x) + 6 x (2)} \frac{x^{2} - 4}{x - 2}$

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

$\frac{3}{3 (x + 7) (x - 7)} \cdot \frac{3 (x + 1) (x + 7)}{6 x (x^{2} + 3 x + 2)} \frac{x^{2} - 4}{x - 2}$

Factor $x^{2} + 3 x + 2$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $3$ .

$1, 2$

Write the factored form using these integers.

$\frac{3}{3 (x + 7) (x - 7)} \cdot \frac{3 (x + 1) (x + 7)}{6 x ((x + 1) (x + 2))} \frac{x^{2} - 4}{x - 2}$

$\frac{3}{3 (x + 7) (x - 7)} \cdot \frac{3 (x + 1) (x + 7)}{6 x (x + 1) (x + 2)} \frac{x^{2} - 4}{x - 2}$

Step 5

Simplify terms.

Cancel the common factor of $3$ .

Factor $3$ out of $6 x (x + 1) (x + 2)$ .

$\frac{3}{3 (x + 7) (x - 7)} \cdot \frac{3 (x + 1) (x + 7)}{3 (2 x (x + 1) (x + 2))} \frac{x^{2} - 4}{x - 2}$

Cancel the common factor.

$\frac{3}{3 (x + 7) (x - 7)} \cdot \frac{3 (x + 1) (x + 7)}{3 (2 x (x + 1) (x + 2))} \frac{x^{2} - 4}{x - 2}$

Rewrite the expression.

$\frac{1}{3 (x + 7) (x - 7)} \cdot \frac{3 (x + 1) (x + 7)}{2 x (x + 1) (x + 2)} \frac{x^{2} - 4}{x - 2}$

Cancel the common factor of $3 (x + 7)$ .

Factor $3 (x + 7)$ out of $3 (x + 1) (x + 7)$ .

$\frac{1}{3 (x + 7) (x - 7)} \cdot \frac{3 (x + 7) (x + 1)}{2 x (x + 1) (x + 2)} \frac{x^{2} - 4}{x - 2}$

Cancel the common factor.

$\frac{1}{3 (x + 7) (x - 7)} \cdot \frac{3 (x + 7) (x + 1)}{2 x (x + 1) (x + 2)} \frac{x^{2} - 4}{x - 2}$

Rewrite the expression.

$\frac{1}{x - 7} \cdot \frac{x + 1}{2 x (x + 1) (x + 2)} \frac{x^{2} - 4}{x - 2}$

Multiply $\frac{1}{x - 7}$ by $\frac{x + 1}{2 x (x + 1) (x + 2)}$ .

$\frac{x + 1}{(x - 7) (2 x (x + 1) (x + 2))} \cdot \frac{x^{2} - 4}{x - 2}$

Cancel the common factor of $x + 1$ .

Cancel the common factor.

$\frac{x + 1}{(x - 7) (2 x (x + 1) (x + 2))} \cdot \frac{x^{2} - 4}{x - 2}$

Rewrite the expression.

$\frac{1}{(x - 7) (2 x (x + 2))} \cdot \frac{x^{2} - 4}{x - 2}$

Move $2$ to the left of $x - 7$ .

$\frac{1}{2 \cdot (x - 7) x (x + 2)} \cdot \frac{x^{2} - 4}{x - 2}$

Step 6

Simplify the numerator.

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

$\frac{1}{2 (x - 7) x (x + 2)} \cdot \frac{x^{2} - 2^{2}}{x - 2}$

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 $b = 2$ .

$\frac{1}{2 (x - 7) x (x + 2)} \cdot \frac{(x + 2) (x - 2)}{x - 2}$

Step 7

Simplify terms.

Cancel the common factor of $x + 2$ .

Factor $x + 2$ out of $2 (x - 7) x (x + 2)$ .

$\frac{1}{(x + 2) (2 (x - 7) x)} \cdot \frac{(x + 2) (x - 2)}{x - 2}$

Cancel the common factor.

$\frac{1}{(x + 2) (2 (x - 7) x)} \cdot \frac{(x + 2) (x - 2)}{x - 2}$

Rewrite the expression.

$\frac{1}{2 (x - 7) x} \cdot \frac{x - 2}{x - 2}$

Multiply $\frac{1}{2 (x - 7) x}$ by $\frac{x - 2}{x - 2}$ .

$\frac{x - 2}{2 (x - 7) x (x - 2)}$

Cancel the common factor of $x - 2$ .

Cancel the common factor.

$\frac{x - 2}{2 (x - 7) x (x - 2)}$

Rewrite the expression.

$\frac{1}{2 (x - 7) x}$

Reorder factors in $\frac{1}{2 (x - 7) x}$ .

$\frac{1}{2 x (x - 7)}$

Simplify (3/(3x^2-147)*(3x^2+24x+21)/(6x^3+18x^2+12x))÷((x-2)/(x^2-4))

Related Questions

Support LCMGCF.com by disabling your adblocker.