Algebra Examples

$12 m^{2} n + 24 m^{3} n^{2} - 36 m^{4} n^{3}$

Step 1

Factor $- 12 m^{2} n$ out of $12 m^{2} n + 24 m^{3} n^{2} - 36 m^{4} n^{3}$ .

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Step 1.1

Reorder the expression.

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Step 1.1.1

Move $12 m^{2} n$ .

$24 m^{3} n^{2} - 36 m^{4} n^{3} + 12 m^{2} n$

Step 1.1.2

Reorder $24 m^{3} n^{2}$ and $- 36 m^{4} n^{3}$ .

$- 36 m^{4} n^{3} + 24 m^{3} n^{2} + 12 m^{2} n$

Step 1.2

Factor $- 12 m^{2} n$ out of $- 36 m^{4} n^{3}$ .

$- 12 m^{2} n (3 m^{2} n^{2}) + 24 m^{3} n^{2} + 12 m^{2} n$

Step 1.3

Factor $- 12 m^{2} n$ out of $24 m^{3} n^{2}$ .

$- 12 m^{2} n (3 m^{2} n^{2}) - 12 m^{2} n (- 2 m n) + 12 m^{2} n$

Step 1.4

Factor $- 12 m^{2} n$ out of $12 m^{2} n$ .

$- 12 m^{2} n (3 m^{2} n^{2}) - 12 m^{2} n (- 2 m n) - 12 m^{2} n (- 1)$

Step 1.5

Factor $- 12 m^{2} n$ out of $- 12 m^{2} n (3 m^{2} n^{2}) - 12 m^{2} n (- 2 m n)$ .

$- 12 m^{2} n (3 m^{2} n^{2} - 2 m n) - 12 m^{2} n (- 1)$

Step 1.6

Factor $- 12 m^{2} n$ out of $- 12 m^{2} n (3 m^{2} n^{2} - 2 m n) - 12 m^{2} n (- 1)$ .

$- 12 m^{2} n (3 m^{2} n^{2} - 2 m n - 1)$

Step 2

Rewrite $m^{2} n^{2}$ as ${(m n)}^{2}$ .

$- 12 m^{2} n (3 {(m n)}^{2} - 2 (m n) - 1)$

Step 3

Let $u = m n$ . Substitute $u$ for all occurrences of $m n$ .

$- 12 m^{2} n (3 u^{2} - 2 u - 1)$

Step 4

Factor by grouping.

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Step 4.1

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 4.1.1

Factor $- 2$ out of $- 2 u$ .

$- 12 m^{2} n (3 u^{2} - 2 (u) - 1)$

Step 4.1.2

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

$- 12 m^{2} n (3 u^{2} + (1 - 3) u - 1)$

Step 4.1.3

Apply the distributive property.

$- 12 m^{2} n (3 u^{2} + 1 u - 3 u - 1)$

Step 4.1.4

Multiply $u$ by $1$ .

$- 12 m^{2} n (3 u^{2} + u - 3 u - 1)$

Step 4.2

Factor out the greatest common factor from each group.

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Step 4.2.1

Group the first two terms and the last two terms.

$- 12 m^{2} n ((3 u^{2} + u) - 3 u - 1)$

Step 4.2.2

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

$- 12 m^{2} n (u (3 u + 1) - (3 u + 1))$

Step 4.3

Factor the polynomial by factoring out the greatest common factor, $3 u + 1$ .

$- 12 m^{2} n ((3 u + 1) (u - 1))$

Step 5

Replace all occurrences of $u$ with $m n$ .

$- 12 m^{2} n ((3 (m n) + 1) (m n - 1))$

Step 6

Factor.

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Step 6.1

Remove parentheses.

$- 12 m^{2} n ((3 m n + 1) (m n - 1))$

Step 6.2

Remove unnecessary parentheses.

$- 12 m^{2} n (3 m n + 1) (m n - 1)$