Algebra Examples

$2 x^{3} + 3 x^{2} - 3 x - 2$

Step 1

Regroup terms.

$2 x^{3} - 2 + 3 x^{2} - 3 x$

Step 2

Factor $2$ out of $2 x^{3} - 2$ .

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Step 2.1

Factor $2$ out of $2 x^{3}$ .

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

Step 2.2

Factor $2$ out of $- 2$ .

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

Step 2.3

Factor $2$ out of $2 (x^{3}) + 2 (- 1)$ .

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

Step 3

Rewrite $1$ as $1^{3}$ .

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

Step 4

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 $a = x$ and $b = 1$ .

$2 ((x - 1) (x^{2} + x \cdot 1 + 1^{2})) + 3 x^{2} - 3 x$

Step 5

Factor.

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Step 5.1

Simplify.

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Step 5.1.1

Multiply $x$ by $1$ .

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

Step 5.1.2

One to any power is one.

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

Step 5.2

Remove unnecessary parentheses.

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

Step 6

Factor $3 x$ out of $3 x^{2} - 3 x$ .

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Step 6.1

Factor $3 x$ out of $3 x^{2}$ .

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

Factor $x - 1$ out of $2 (x - 1) (x^{2} + x + 1) + 3 x (x - 1)$ .

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Step 7.1

Factor $x - 1$ out of $2 (x - 1) (x^{2} + x + 1)$ .

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

Step 7.2

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

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

Step 7.3

Factor $x - 1$ out of $(x - 1) (2 (x^{2} + x + 1)) + (x - 1) (3 x)$ .

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

Step 8

Apply the distributive property.

$(x - 1) (2 x^{2} + 2 x + 2 \cdot 1 + 3 x)$

Step 9

Multiply $2$ by $1$ .

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

Step 10

Add $2 x$ and $3 x$ .

$(x - 1) (2 x^{2} + 5 x + 2)$

Step 11

Factor.

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Step 11.1

Factor by grouping.

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Step 11.1.1

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 $a \cdot c = 2 \cdot 2 = 4$ and whose sum is $b = 5$ .

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Step 11.1.1.1

Factor $5$ out of $5 x$ .

$(x - 1) (2 x^{2} + 5 (x) + 2)$

Step 11.1.1.2

Rewrite $5$ as $1$ plus $4$

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

Step 11.1.1.3

Apply the distributive property.

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

Step 11.1.1.4

Multiply $x$ by $1$ .

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

Step 11.1.2

Factor out the greatest common factor from each group.

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Step 11.1.2.1

Group the first two terms and the last two terms.

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

Step 11.1.2.2

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

$(x - 1) (x (2 x + 1) + 2 (2 x + 1))$

Step 11.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

$(x - 1) ((2 x + 1) (x + 2))$

Step 11.2

Remove unnecessary parentheses.

$(x - 1) (2 x + 1) (x + 2)$