Simplify (9x^2-4)/(3x^2-5x+2)*(9x^4-6x^3+4x^2)/(27x^4+8x)

$\frac{9 x^{2} - 4}{3 x^{2} - 5 x + 2} \cdot \frac{9 x^{4} - 6 x^{3} + 4 x^{2}}{27 x^{4} + 8 x}$

Step 1

Simplify the numerator.

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

$\frac{{(3 x)}^{2} - 4}{3 x^{2} - 5 x + 2} \cdot \frac{9 x^{4} - 6 x^{3} + 4 x^{2}}{27 x^{4} + 8 x}$

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

$\frac{{(3 x)}^{2} - 2^{2}}{3 x^{2} - 5 x + 2} \cdot \frac{9 x^{4} - 6 x^{3} + 4 x^{2}}{27 x^{4} + 8 x}$

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 $b = 2$ .

$\frac{(3 x + 2) (3 x - 2)}{3 x^{2} - 5 x + 2} \cdot \frac{9 x^{4} - 6 x^{3} + 4 x^{2}}{27 x^{4} + 8 x}$

Step 2

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 $a \cdot c = 3 \cdot 2 = 6$ and whose sum is $b = - 5$ .

Factor $- 5$ out of $- 5 x$ .

$\frac{(3 x + 2) (3 x - 2)}{3 x^{2} - 5 (x) + 2} \cdot \frac{9 x^{4} - 6 x^{3} + 4 x^{2}}{27 x^{4} + 8 x}$

Rewrite $- 5$ as $- 2$ plus $- 3$

$\frac{(3 x + 2) (3 x - 2)}{3 x^{2} + (- 2 - 3) x + 2} \cdot \frac{9 x^{4} - 6 x^{3} + 4 x^{2}}{27 x^{4} + 8 x}$

Apply the distributive property.

$\frac{(3 x + 2) (3 x - 2)}{3 x^{2} - 2 x - 3 x + 2} \cdot \frac{9 x^{4} - 6 x^{3} + 4 x^{2}}{27 x^{4} + 8 x}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(3 x + 2) (3 x - 2)}{(3 x^{2} - 2 x) - 3 x + 2} \cdot \frac{9 x^{4} - 6 x^{3} + 4 x^{2}}{27 x^{4} + 8 x}$

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

$\frac{(3 x + 2) (3 x - 2)}{x (3 x - 2) - (3 x - 2)} \cdot \frac{9 x^{4} - 6 x^{3} + 4 x^{2}}{27 x^{4} + 8 x}$

Factor the polynomial by factoring out the greatest common factor, $3 x - 2$ .

$\frac{(3 x + 2) (3 x - 2)}{(3 x - 2) (x - 1)} \cdot \frac{9 x^{4} - 6 x^{3} + 4 x^{2}}{27 x^{4} + 8 x}$

Step 3

Factor $x^{2}$ out of $9 x^{4} - 6 x^{3} + 4 x^{2}$ .

Factor $x^{2}$ out of $9 x^{4}$ .

$\frac{(3 x + 2) (3 x - 2)}{(3 x - 2) (x - 1)} \cdot \frac{x^{2} (9 x^{2}) - 6 x^{3} + 4 x^{2}}{27 x^{4} + 8 x}$

Factor $x^{2}$ out of $- 6 x^{3}$ .

$\frac{(3 x + 2) (3 x - 2)}{(3 x - 2) (x - 1)} \cdot \frac{x^{2} (9 x^{2}) + x^{2} (- 6 x) + 4 x^{2}}{27 x^{4} + 8 x}$

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

$\frac{(3 x + 2) (3 x - 2)}{(3 x - 2) (x - 1)} \cdot \frac{x^{2} (9 x^{2}) + x^{2} (- 6 x) + x^{2} \cdot 4}{27 x^{4} + 8 x}$

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

$\frac{(3 x + 2) (3 x - 2)}{(3 x - 2) (x - 1)} \cdot \frac{x^{2} (9 x^{2} - 6 x) + x^{2} \cdot 4}{27 x^{4} + 8 x}$

Factor $x^{2}$ out of $x^{2} (9 x^{2} - 6 x) + x^{2} \cdot 4$ .

$\frac{(3 x + 2) (3 x - 2)}{(3 x - 2) (x - 1)} \cdot \frac{x^{2} (9 x^{2} - 6 x + 4)}{27 x^{4} + 8 x}$

Step 4

Simplify the denominator.

Factor $x$ out of $27 x^{4} + 8 x$ .

Factor $x$ out of $27 x^{4}$ .

$\frac{(3 x + 2) (3 x - 2)}{(3 x - 2) (x - 1)} \cdot \frac{x^{2} (9 x^{2} - 6 x + 4)}{x (27 x^{3}) + 8 x}$

Factor $x$ out of $8 x$ .

$\frac{(3 x + 2) (3 x - 2)}{(3 x - 2) (x - 1)} \cdot \frac{x^{2} (9 x^{2} - 6 x + 4)}{x (27 x^{3}) + x \cdot 8}$

Factor $x$ out of $x (27 x^{3}) + x \cdot 8$ .

$\frac{(3 x + 2) (3 x - 2)}{(3 x - 2) (x - 1)} \cdot \frac{x^{2} (9 x^{2} - 6 x + 4)}{x (27 x^{3} + 8)}$

Rewrite $27 x^{3}$ as ${(3 x)}^{3}$ .

$\frac{(3 x + 2) (3 x - 2)}{(3 x - 2) (x - 1)} \cdot \frac{x^{2} (9 x^{2} - 6 x + 4)}{x ({(3 x)}^{3} + 8)}$

Rewrite $8$ as $2^{3}$ .

$\frac{(3 x + 2) (3 x - 2)}{(3 x - 2) (x - 1)} \cdot \frac{x^{2} (9 x^{2} - 6 x + 4)}{x ({(3 x)}^{3} + 2^{3})}$

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

$\frac{(3 x + 2) (3 x - 2)}{(3 x - 2) (x - 1)} \cdot \frac{x^{2} (9 x^{2} - 6 x + 4)}{x ((3 x + 2) ({(3 x)}^{2} - (3 x) \cdot 2 + 2^{2}))}$

Simplify.

Apply the product rule to $3 x$ .

$\frac{(3 x + 2) (3 x - 2)}{(3 x - 2) (x - 1)} \cdot \frac{x^{2} (9 x^{2} - 6 x + 4)}{x ((3 x + 2) (3^{2} x^{2} - (3 x) \cdot 2 + 2^{2}))}$

Raise $3$ to the power of $2$ .

$\frac{(3 x + 2) (3 x - 2)}{(3 x - 2) (x - 1)} \cdot \frac{x^{2} (9 x^{2} - 6 x + 4)}{x ((3 x + 2) (9 x^{2} - (3 x) \cdot 2 + 2^{2}))}$

Multiply $3$ by $- 1$ .

$\frac{(3 x + 2) (3 x - 2)}{(3 x - 2) (x - 1)} \cdot \frac{x^{2} (9 x^{2} - 6 x + 4)}{x ((3 x + 2) (9 x^{2} - 3 x \cdot 2 + 2^{2}))}$

Multiply $2$ by $- 3$ .

$\frac{(3 x + 2) (3 x - 2)}{(3 x - 2) (x - 1)} \cdot \frac{x^{2} (9 x^{2} - 6 x + 4)}{x ((3 x + 2) (9 x^{2} - 6 x + 2^{2}))}$

Raise $2$ to the power of $2$ .

$\frac{(3 x + 2) (3 x - 2)}{(3 x - 2) (x - 1)} \cdot \frac{x^{2} (9 x^{2} - 6 x + 4)}{x ((3 x + 2) (9 x^{2} - 6 x + 4))}$

$\frac{(3 x + 2) (3 x - 2)}{(3 x - 2) (x - 1)} \cdot \frac{x^{2} (9 x^{2} - 6 x + 4)}{x (3 x + 2) (9 x^{2} - 6 x + 4)}$

Step 5

Simplify terms.

Combine.

$\frac{(3 x + 2) (3 x - 2) (x^{2} (9 x^{2} - 6 x + 4))}{(3 x - 2) (x - 1) (x (3 x + 2) (9 x^{2} - 6 x + 4))}$

Cancel the common factor of $3 x + 2$ .

Cancel the common factor.

$\frac{(3 x + 2) (3 x - 2) (x^{2} (9 x^{2} - 6 x + 4))}{(3 x - 2) (x - 1) (x (3 x + 2) (9 x^{2} - 6 x + 4))}$

Rewrite the expression.

$\frac{(3 x - 2) (x^{2} (9 x^{2} - 6 x + 4))}{(3 x - 2) (x - 1) (x (9 x^{2} - 6 x + 4))}$

Cancel the common factor of $3 x - 2$ .

Cancel the common factor.

$\frac{(3 x - 2) (x^{2} (9 x^{2} - 6 x + 4))}{(3 x - 2) (x - 1) (x (9 x^{2} - 6 x + 4))}$

Rewrite the expression.

$\frac{x^{2} (9 x^{2} - 6 x + 4)}{(x - 1) (x (9 x^{2} - 6 x + 4))}$

Cancel the common factor of $x^{2}$ and $x$ .

Factor $x$ out of $x^{2} (9 x^{2} - 6 x + 4)$ .

$\frac{x (x (9 x^{2} - 6 x + 4))}{(x - 1) (x (9 x^{2} - 6 x + 4))}$

Cancel the common factors.

Factor $x$ out of $(x - 1) (x (9 x^{2} - 6 x + 4))$ .

$\frac{x (x (9 x^{2} - 6 x + 4))}{x ((x - 1) (9 x^{2} - 6 x + 4))}$

Cancel the common factor.

$\frac{x (x (9 x^{2} - 6 x + 4))}{x ((x - 1) (9 x^{2} - 6 x + 4))}$

Rewrite the expression.

$\frac{x (9 x^{2} - 6 x + 4)}{(x - 1) (9 x^{2} - 6 x + 4)}$

Cancel the common factor of $9 x^{2} - 6 x + 4$ .

Cancel the common factor.

$\frac{x (9 x^{2} - 6 x + 4)}{(x - 1) (9 x^{2} - 6 x + 4)}$

Rewrite the expression.

$\frac{x}{x - 1}$

Simplify (9x^2-4)/(3x^2-5x+2)*(9x^4-6x^3+4x^2)/(27x^4+8x)

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