Simplify (3x^3-3x^2+7x-7)/(27x^4-147)

$\frac{3 x^{3} - 3 x^{2} + 7 x - 7}{27 x^{4} - 147}$

Simplify the numerator.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(3 x^{3} - 3 x^{2}) + 7 x - 7}{27 x^{4} - 147}$

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

$\frac{3 x^{2} (x - 1) + 7 (x - 1)}{27 x^{4} - 147}$

Factor the polynomial by factoring out the greatest common factor, $x - 1$ .

$\frac{(x - 1) (3 x^{2} + 7)}{27 x^{4} - 147}$

Simplify the denominator.

Factor $3$ out of $27 x^{4} - 147$ .

Factor $3$ out of $27 x^{4}$ .

$\frac{(x - 1) (3 x^{2} + 7)}{3 (9 x^{4}) - 147}$

Factor $3$ out of $- 147$ .

$\frac{(x - 1) (3 x^{2} + 7)}{3 (9 x^{4}) + 3 (- 49)}$

Factor $3$ out of $3 (9 x^{4}) + 3 (- 49)$ .

$\frac{(x - 1) (3 x^{2} + 7)}{3 (9 x^{4} - 49)}$

Rewrite $9 x^{4}$ as ${(3 x^{2})}^{2}$ .

$\frac{(x - 1) (3 x^{2} + 7)}{3 ({(3 x^{2})}^{2} - 49)}$

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

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 x^{2}$ and $b = 7$ .

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

Cancel the common factor of $3 x^{2} + 7$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (3x^3-3x^2+7x-7)/(27x^4-147)

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