Pre-Algebra Examples

$\frac{3 x^{2} - x^{3}}{3 x^{2} - 5 x - 12}$

Step 1

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

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Step 1.1

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

$\frac{x^{2} \cdot 3 - x^{3}}{3 x^{2} - 5 x - 12}$

Step 1.2

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

$\frac{x^{2} \cdot 3 + x^{2} (- x)}{3 x^{2} - 5 x - 12}$

Step 1.3

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

$\frac{x^{2} (3 - x)}{3 x^{2} - 5 x - 12}$

Step 2

Factor by grouping.

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Step 2.1

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 $a \cdot c = 3 \cdot - 12 = - 36$ and whose sum is $b = - 5$ .

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Step 2.1.1

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

$\frac{x^{2} (3 - x)}{3 x^{2} - 5 (x) - 12}$

Step 2.1.2

Rewrite $- 5$ as $4$ plus $- 9$

$\frac{x^{2} (3 - x)}{3 x^{2} + (4 - 9) x - 12}$

Step 2.1.3

Apply the distributive property.

$\frac{x^{2} (3 - x)}{3 x^{2} + 4 x - 9 x - 12}$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$\frac{x^{2} (3 - x)}{(3 x^{2} + 4 x) - 9 x - 12}$

Step 2.2.2

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

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

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $3 x + 4$ .

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

Step 3

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

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Step 3.1

Rewrite $3$ as $- 1 (- 3)$ .

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

Step 3.2

Factor $- 1$ out of $- x$ .

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

Step 3.3

Factor $- 1$ out of $- 1 (- 3) - (x)$ .

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

Step 3.4

Reorder terms.

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

Step 3.5

Cancel the common factor.

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

Step 3.6

Rewrite the expression.

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

Step 4

Simplify the expression.

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Step 4.1

Move $- 1$ to the left of $x^{2}$ .

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

Step 4.2

Move the negative in front of the fraction.

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