Algebra Examples

$216 + 8 m^{3} + 216 m + 72 m^{2}$

Step 1

Factor $8$ out of $216 + 8 m^{3} + 216 m + 72 m^{2}$ .

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Step 1.1

Factor $8$ out of $216$ .

$8 \cdot 27 + 8 m^{3} + 216 m + 72 m^{2}$

Step 1.2

Factor $8$ out of $216 m$ .

$8 \cdot 27 + 8 m^{3} + 8 (27 m) + 72 m^{2}$

Step 1.3

Factor $8$ out of $72 m^{2}$ .

$8 \cdot 27 + 8 m^{3} + 8 (27 m) + 8 (9 m^{2})$

Step 1.4

Factor $8$ out of $8 \cdot 27 + 8 m^{3}$ .

$8 \cdot (27 + m^{3}) + 8 (27 m) + 8 (9 m^{2})$

Step 1.5

Factor $8$ out of $8 \cdot (27 + m^{3}) + 8 (27 m)$ .

$8 \cdot (27 + m^{3} + 27 m) + 8 (9 m^{2})$

Step 1.6

Factor $8$ out of $8 \cdot (27 + m^{3} + 27 m) + 8 (9 m^{2})$ .

$8 (27 + m^{3} + 27 m + 9 m^{2})$

Step 2

Rewrite $27$ as $3^{3}$ .

$8 (3^{3} + m^{3} + 27 m + 9 m^{2})$

Step 3

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = 3$ and $b = m$ .

$8 ((3 + m) (3^{2} - 3 m + m^{2}) + 27 m + 9 m^{2})$

Step 4

Raise $3$ to the power of $2$ .

$8 ((3 + m) (9 - 3 m + m^{2}) + 27 m + 9 m^{2})$

Step 5

Factor $9 m$ out of $27 m + 9 m^{2}$ .

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Step 5.1

Factor $9 m$ out of $27 m$ .

$8 ((3 + m) (9 - 3 m + m^{2}) + 9 m (3) + 9 m^{2})$

Step 5.2

Factor $9 m$ out of $9 m^{2}$ .

$8 ((3 + m) (9 - 3 m + m^{2}) + 9 m (3) + 9 m (m))$

Step 5.3

Factor $9 m$ out of $9 m (3) + 9 m (m)$ .

$8 ((3 + m) (9 - 3 m + m^{2}) + 9 m (3 + m))$

Step 6

Factor $3 + m$ out of $(3 + m) (9 - 3 m + m^{2}) + 9 m (3 + m)$ .

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Step 6.1

Factor $3 + m$ out of $9 m (3 + m)$ .

$8 ((3 + m) (9 - 3 m + m^{2}) + (3 + m) (9 m))$

Step 6.2

Factor $3 + m$ out of $(3 + m) (9 - 3 m + m^{2}) + (3 + m) (9 m)$ .

$8 ((3 + m) (9 - 3 m + m^{2} + 9 m))$

Step 7

Add $- 3 m$ and $9 m$ .

$8 ((3 + m) (9 + m^{2} + 6 m))$

Step 8

Factor.

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Step 8.1

Factor using the perfect square rule.

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Step 8.1.1

Rearrange terms.

$8 ((3 + m) (9 + 6 m + m^{2}))$

Step 8.1.2

Rewrite $9$ as $3^{2}$ .

$8 ((3 + m) (3^{2} + 6 m + m^{2}))$

Step 8.1.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 m = 2 \cdot 3 \cdot m$

Step 8.1.4

Rewrite the polynomial.

$8 ((3 + m) (3^{2} + 2 \cdot 3 \cdot m + m^{2}))$

Step 8.1.5

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 3$ and $b = m$ .

$8 ((3 + m) {(3 + m)}^{2})$

Step 8.2

Remove unnecessary parentheses.

$8 (3 + m) {(3 + m)}^{2}$

Step 9

Multiply $3 + m$ by ${(3 + m)}^{2}$ by adding the exponents.

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Step 9.1

Move ${(3 + m)}^{2}$ .

$8 ({(3 + m)}^{2} (3 + m))$

Step 9.2

Multiply ${(3 + m)}^{2}$ by $3 + m$ .

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Step 9.2.1

Raise $3 + m$ to the power of $1$ .

$8 ({(3 + m)}^{2} {(3 + m)}^{1})$

Step 9.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$8 {(3 + m)}^{2 + 1}$

Step 9.3

Add $2$ and $1$ .

$8 {(3 + m)}^{3}$