Algebra Examples

$4 a x^{3} + 8 a^{2} x^{2} - 12 a^{3} x$

Step 1

Factor $4 a x$ out of $4 a x^{3} + 8 a^{2} x^{2} - 12 a^{3} x$ .

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Step 1.1

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

$4 a x (x^{2}) + 8 a^{2} x^{2} - 12 a^{3} x$

Step 1.2

Factor $4 a x$ out of $8 a^{2} x^{2}$ .

$4 a x (x^{2}) + 4 a x (2 a x) - 12 a^{3} x$

Step 1.3

Factor $4 a x$ out of $- 12 a^{3} x$ .

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

Step 1.4

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

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

Step 1.5

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

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

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 $a \cdot c = - 3 \cdot 1 = - 3$ and whose sum is $b = 2$ .

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Step 2.1.1

Reorder terms.

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

Step 2.1.2

Reorder $x^{2}$ and $2 a x$ .

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

Step 2.1.3

Factor $2$ out of $2 a x$ .

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

Step 2.1.4

Rewrite $2$ as $- 1$ plus $3$

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

Step 2.1.5

Apply the distributive property.

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

Step 2.1.6

Remove unnecessary parentheses.

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

Step 2.1.7

Remove unnecessary parentheses.

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

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.

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

Step 2.2.2

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

$4 a x (a (- 3 a - 1 x) - x (- 3 a - x))$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $- 3 a - 1 x$ .

$4 a x ((- 3 a - 1 x) (a - x))$

Step 3

Factor.

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Step 3.1

Rewrite $- 1 x$ as $- x$ .

$4 a x ((- 3 a - x) (a - x))$

Step 3.2

Remove unnecessary parentheses.

$4 a x (- 3 a - x) (a - x)$

Step 4

Factor.

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Step 4.1

Factor $- 1$ out of $- 3 a - x$ .

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Step 4.1.1

Factor $- 1$ out of $- 3 a$ .

$4 a x (- (3 a) - x) (a - x)$

Step 4.1.2

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

$4 a x (- (3 a) - (x)) (a - x)$

Step 4.1.3

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

$4 a x (- (3 a + x)) (a - x)$

Step 4.2

Remove unnecessary parentheses.

$4 a x \cdot - 1 (3 a + x) (a - x)$

Step 5

Factor.

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Step 5.1

Combine exponents.

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Step 5.1.1

Factor out negative.

$- ((4 a x (3 a + x)) (a - x))$

Step 5.1.2

Multiply $4$ by $- 1$ .

$- 4 ((a x (3 a + x)) (a - x))$

Step 5.2

Remove unnecessary parentheses.

$- 4 a x (3 a + x) (a - x)$