Algebra Examples

$\frac{2 m^{6} + 6 m}{3 m^{14} + 12 m^{9} + 9 m^{4}}$

Step 1

Factor $2 m$ out of $2 m^{6} + 6 m$ .

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Step 1.1

Factor $2 m$ out of $2 m^{6}$ .

$\frac{2 m (m^{5}) + 6 m}{3 m^{14} + 12 m^{9} + 9 m^{4}}$

Step 1.2

Factor $2 m$ out of $6 m$ .

$\frac{2 m (m^{5}) + 2 m (3)}{3 m^{14} + 12 m^{9} + 9 m^{4}}$

Step 1.3

Factor $2 m$ out of $2 m (m^{5}) + 2 m (3)$ .

$\frac{2 m (m^{5} + 3)}{3 m^{14} + 12 m^{9} + 9 m^{4}}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $3 m^{4}$ out of $3 m^{14} + 12 m^{9} + 9 m^{4}$ .

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Step 2.1.1

Factor $3 m^{4}$ out of $3 m^{14}$ .

$\frac{2 m (m^{5} + 3)}{3 m^{4} (m^{10}) + 12 m^{9} + 9 m^{4}}$

Step 2.1.2

Factor $3 m^{4}$ out of $12 m^{9}$ .

$\frac{2 m (m^{5} + 3)}{3 m^{4} (m^{10}) + 3 m^{4} (4 m^{5}) + 9 m^{4}}$

Step 2.1.3

Factor $3 m^{4}$ out of $9 m^{4}$ .

$\frac{2 m (m^{5} + 3)}{3 m^{4} (m^{10}) + 3 m^{4} (4 m^{5}) + 3 m^{4} (3)}$

Step 2.1.4

Factor $3 m^{4}$ out of $3 m^{4} (m^{10}) + 3 m^{4} (4 m^{5})$ .

$\frac{2 m (m^{5} + 3)}{3 m^{4} (m^{10} + 4 m^{5}) + 3 m^{4} (3)}$

Step 2.1.5

Factor $3 m^{4}$ out of $3 m^{4} (m^{10} + 4 m^{5}) + 3 m^{4} (3)$ .

$\frac{2 m (m^{5} + 3)}{3 m^{4} (m^{10} + 4 m^{5} + 3)}$

Step 2.2

Rewrite $m^{10}$ as ${(m^{5})}^{2}$ .

$\frac{2 m (m^{5} + 3)}{3 m^{4} ({(m^{5})}^{2} + 4 m^{5} + 3)}$

Step 2.3

Let $u = m^{5}$ . Substitute $u$ for all occurrences of $m^{5}$ .

$\frac{2 m (m^{5} + 3)}{3 m^{4} (u^{2} + 4 u + 3)}$

Step 2.4

Factor $u^{2} + 4 u + 3$ using the AC method.

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Step 2.4.1

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 $3$ and whose sum is $4$ .

$1, 3$

Step 2.4.2

Write the factored form using these integers.

$\frac{2 m (m^{5} + 3)}{3 m^{4} ((u + 1) (u + 3))}$

Step 2.5

Replace all occurrences of $u$ with $m^{5}$ .

$\frac{2 m (m^{5} + 3)}{3 m^{4} (m^{5} + 1) (m^{5} + 3)}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

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

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Step 3.1.1

Factor $m$ out of $2 m (m^{5} + 3)$ .

$\frac{m (2 (m^{5} + 3))}{3 m^{4} (m^{5} + 1) (m^{5} + 3)}$

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $m$ out of $3 m^{4} (m^{5} + 1) (m^{5} + 3)$ .

$\frac{m (2 (m^{5} + 3))}{m (3 m^{3} (m^{5} + 1) (m^{5} + 3))}$

Step 3.1.2.2

Cancel the common factor.

$\frac{m (2 (m^{5} + 3))}{m (3 m^{3} (m^{5} + 1) (m^{5} + 3))}$

Step 3.1.2.3

Rewrite the expression.

$\frac{2 (m^{5} + 3)}{3 m^{3} (m^{5} + 1) (m^{5} + 3)}$

Step 3.2

Cancel the common factor of $m^{5} + 3$ .

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Step 3.2.1

Cancel the common factor.

$\frac{2 (m^{5} + 3)}{3 m^{3} (m^{5} + 1) (m^{5} + 3)}$

Step 3.2.2

Rewrite the expression.

$\frac{2}{3 m^{3} (m^{5} + 1)}$