Algebra Examples

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

Step 1

Regroup terms.

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

Step 2

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

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

Step 3

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 $b = 2$ .

$P (x) = (x - 2) (x^{2} + x \cdot 2 + 2^{2}) + 3 x^{2} - 6 x$

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

$P (x) = (x - 2) (x^{2} + 2 x + 4) + 3 x^{2} - 6 x$

Step 5

Factor $3 x$ out of $3 x^{2} - 6 x$ .

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Step 5.1

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

$P (x) = (x - 2) (x^{2} + 2 x + 4) + 3 x (x) - 6 x$

Step 5.2

Factor $3 x$ out of $- 6 x$ .

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

Step 5.3

Factor $3 x$ out of $3 x (x) + 3 x (- 2)$ .

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

Step 6

Factor $x - 2$ out of $(x - 2) (x^{2} + 2 x + 4) + 3 x (x - 2)$ .

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Step 6.1

Factor $x - 2$ out of $3 x (x - 2)$ .

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

Step 6.2

Factor $x - 2$ out of $(x - 2) (x^{2} + 2 x + 4) + (x - 2) (3 x)$ .

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

Step 7

Add $2 x$ and $3 x$ .

$P (x) = (x - 2) (x^{2} + 5 x + 4)$

Step 8

Factor.

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Step 8.1

Factor $x^{2} + 5 x + 4$ using the AC method.

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Step 8.1.1

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 $4$ and whose sum is $5$ .

$1, 4$

Step 8.1.2

Write the factored form using these integers.

$P (x) = (x - 2) ((x + 1) (x + 4))$

Step 8.2

Remove unnecessary parentheses.

$P (x) = (x - 2) (x + 1) (x + 4)$