Simplify (3x^5-18x^4-21x^3)/(2x^6-98x^4)

$\frac{3 x^{5} - 18 x^{4} - 21 x^{3}}{2 x^{6} - 98 x^{4}}$

Simplify the numerator.

Factor $3 x^{3}$ out of $3 x^{5} - 18 x^{4} - 21 x^{3}$ .

Factor $3 x^{3}$ out of $3 x^{5}$ .

$\frac{3 x^{3} (x^{2}) - 18 x^{4} - 21 x^{3}}{2 x^{6} - 98 x^{4}}$

Factor $3 x^{3}$ out of $- 18 x^{4}$ .

$\frac{3 x^{3} (x^{2}) + 3 x^{3} (- 6 x) - 21 x^{3}}{2 x^{6} - 98 x^{4}}$

Factor $3 x^{3}$ out of $- 21 x^{3}$ .

$\frac{3 x^{3} (x^{2}) + 3 x^{3} (- 6 x) + 3 x^{3} (- 7)}{2 x^{6} - 98 x^{4}}$

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

$\frac{3 x^{3} (x^{2} - 6 x) + 3 x^{3} (- 7)}{2 x^{6} - 98 x^{4}}$

Factor $3 x^{3}$ out of $3 x^{3} (x^{2} - 6 x) + 3 x^{3} (- 7)$ .

$\frac{3 x^{3} (x^{2} - 6 x - 7)}{2 x^{6} - 98 x^{4}}$

Factor $x^{2} - 6 x - 7$ 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 $- 6$ .

$- 7, 1$

Write the factored form using these integers.

$\frac{3 x^{3} ((x - 7) (x + 1))}{2 x^{6} - 98 x^{4}}$

$\frac{3 x^{3} (x - 7) (x + 1)}{2 x^{6} - 98 x^{4}}$

Simplify the denominator.

Factor $2 x^{4}$ out of $2 x^{6} - 98 x^{4}$ .

Factor $2 x^{4}$ out of $2 x^{6}$ .

$\frac{3 x^{3} (x - 7) (x + 1)}{2 x^{4} (x^{2}) - 98 x^{4}}$

Factor $2 x^{4}$ out of $- 98 x^{4}$ .

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

Factor $2 x^{4}$ out of $2 x^{4} (x^{2}) + 2 x^{4} (- 49)$ .

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

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

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

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

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

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x^{3}$ and $x^{4}$ .

Factor $x^{3}$ out of $3 x^{3} (x - 7) (x + 1)$ .

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

Cancel the common factors.

Factor $x^{3}$ out of $2 x^{4} (x + 7) (x - 7)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 7$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

Simplify (3x^5-18x^4-21x^3)/(2x^6-98x^4)

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