Algebra Examples

$27 x^{4} + 54 x^{3} - 64 x - 128$

Step 1

Factor out the greatest common factor from each group.

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Step 1.1

Group the first two terms and the last two terms.

$(27 x^{4} + 54 x^{3}) - 64 x - 128$

Step 1.2

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

$27 x^{3} (x + 2) - 64 (x + 2)$

Step 2

Factor the polynomial by factoring out the greatest common factor, $x + 2$ .

$(x + 2) (27 x^{3} - 64)$

Step 3

Rewrite $27 x^{3}$ as ${(3 x)}^{3}$ .

$(x + 2) ({(3 x)}^{3} - 64)$

Step 4

Rewrite $64$ as $4^{3}$ .

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

Step 5

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 3 x$ and $b = 4$ .

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

Step 6

Factor.

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Step 6.1

Simplify.

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Step 6.1.1

Apply the product rule to $3 x$ .

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

Step 6.1.2

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

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

Step 6.1.3

Multiply $4$ by $3$ .

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

Step 6.1.4

Raise $4$ to the power of $2$ .

$(x + 2) ((3 x - 4) (9 x^{2} + 12 x + 16))$

Step 6.2

Remove unnecessary parentheses.

$(x + 2) (3 x - 4) (9 x^{2} + 12 x + 16)$