Algebra Examples

$- x^{3} + 6 x^{2} - 12 x + 8$

Step 1

Regroup terms.

$- x^{3} + 8 + 6 x^{2} - 12 x$

Step 2

Factor $- 1$ out of $- x^{3} + 8$ .

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Step 2.1

Factor $- 1$ out of $- x^{3}$ .

$- (x^{3}) + 8 + 6 x^{2} - 12 x$

Step 2.2

Rewrite $8$ as $- 1 (- 8)$ .

$- (x^{3}) - 1 (- 8) + 6 x^{2} - 12 x$

Step 2.3

Factor $- 1$ out of $- (x^{3}) - 1 (- 8)$ .

$- (x^{3} - 8) + 6 x^{2} - 12 x$

Step 3

Rewrite $8$ as $2^{3}$ .

$- (x^{3} - 2^{3}) + 6 x^{2} - 12 x$

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 $b = 2$ .

$- ((x - 2) (x^{2} + x \cdot 2 + 2^{2})) + 6 x^{2} - 12 x$

Step 5

Factor.

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Step 5.1

Simplify.

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Step 5.1.1

Move $2$ to the left of $x$ .

$- ((x - 2) (x^{2} + 2 x + 2^{2})) + 6 x^{2} - 12 x$

Step 5.1.2

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

$- ((x - 2) (x^{2} + 2 x + 4)) + 6 x^{2} - 12 x$

Step 5.2

Remove unnecessary parentheses.

$- (x - 2) (x^{2} + 2 x + 4) + 6 x^{2} - 12 x$

Step 6

Factor $6 x$ out of $6 x^{2} - 12 x$ .

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Step 6.1

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

$- (x - 2) (x^{2} + 2 x + 4) + 6 x (x) - 12 x$

Step 6.2

Factor $6 x$ out of $- 12 x$ .

$- (x - 2) (x^{2} + 2 x + 4) + 6 x (x) + 6 x (- 2)$

Step 6.3

Factor $6 x$ out of $6 x (x) + 6 x (- 2)$ .

$- (x - 2) (x^{2} + 2 x + 4) + 6 x (x - 2)$

Step 7

Factor $x - 2$ out of $- (x - 2) (x^{2} + 2 x + 4) + 6 x (x - 2)$ .

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Step 7.1

Factor $x - 2$ out of $- (x - 2) (x^{2} + 2 x + 4)$ .

$(x - 2) (- 1 (x^{2} + 2 x + 4)) + 6 x (x - 2)$

Step 7.2

Factor $x - 2$ out of $6 x (x - 2)$ .

$(x - 2) (- 1 (x^{2} + 2 x + 4)) + (x - 2) (6 x)$

Step 7.3

Factor $x - 2$ out of $(x - 2) (- 1 (x^{2} + 2 x + 4)) + (x - 2) (6 x)$ .

$(x - 2) (- 1 (x^{2} + 2 x + 4) + 6 x)$

Step 8

Apply the distributive property.

$(x - 2) (- 1 x^{2} - 1 (2 x) - 1 \cdot 4 + 6 x)$

Step 9

Simplify.

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Step 9.1

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$(x - 2) (- x^{2} - 1 (2 x) - 1 \cdot 4 + 6 x)$

Step 9.2

Multiply $2$ by $- 1$ .

$(x - 2) (- x^{2} - 2 x - 1 \cdot 4 + 6 x)$

Step 9.3

Multiply $- 1$ by $4$ .

$(x - 2) (- x^{2} - 2 x - 4 + 6 x)$

Step 10

Add $- 2 x$ and $6 x$ .

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

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 $a \cdot c = - 1 \cdot - 4 = 4$ and whose sum is $b = 4$ .

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Step 11.1.1.1

Factor $4$ out of $4 x$ .

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

Step 11.1.1.2

Rewrite $4$ as $2$ plus $2$

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

Step 11.1.1.3

Apply the distributive property.

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

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.

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

Step 11.1.2.2

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

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

Step 11.1.3

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

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

Step 11.2

Remove unnecessary parentheses.

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

Step 12

Combine exponents.

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Step 12.1

Factor $- 1$ out of $x$ .

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

Step 12.2

Rewrite $- 2$ as $- 1 (2)$ .

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

Step 12.3

Factor $- 1$ out of $- 1 (- x) - 1 (2)$ .

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

Step 12.4

Raise $- x + 2$ to the power of $1$ .

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

Step 12.5

Raise $- x + 2$ to the power of $1$ .

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

Step 12.6

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

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

Step 12.7

Add $1$ and $1$ .

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

Step 12.8

Factor $- 1$ out of $- x$ .

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

Step 12.9

Rewrite $2$ as $- 1 (- 2)$ .

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

Step 12.10

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

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

Step 12.11

Rewrite $- (x - 2)$ as $- 1 (x - 2)$ .

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

Step 12.12

Apply the product rule to $- 1 (x - 2)$ .

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

Step 12.13

Raise $- 1$ to the power of $2$ .

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

Step 12.14

Multiply ${(x - 2)}^{2}$ by $1$ .

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

Step 12.15

Raise $x - 2$ to the power of $1$ .

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

Step 12.16

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

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

Step 12.17

Add $1$ and $2$ .

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