Algebra Examples

$y = 2 x^{4} + 8 x^{3} + 4 x^{2} - 8 x - 6$

Step 1

Factor $2$ out of $2 x^{4} + 8 x^{3} + 4 x^{2} - 8 x - 6$ .

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Step 1.1

Factor $2$ out of $2 x^{4}$ .

$y = 2 (x^{4}) + 8 x^{3} + 4 x^{2} - 8 x - 6$

Step 1.2

Factor $2$ out of $8 x^{3}$ .

$y = 2 (x^{4}) + 2 (4 x^{3}) + 4 x^{2} - 8 x - 6$

Step 1.3

Factor $2$ out of $4 x^{2}$ .

$y = 2 (x^{4}) + 2 (4 x^{3}) + 2 (2 x^{2}) - 8 x - 6$

Step 1.4

Factor $2$ out of $- 8 x$ .

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

Step 1.5

Factor $2$ out of $- 6$ .

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

Step 1.6

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

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

Step 1.7

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

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

Step 1.8

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

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

Step 1.9

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

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

Step 2

Regroup terms.

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

Step 3

Factor $4 x$ out of $4 x^{3} - 4 x$ .

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Step 3.1

Factor $4 x$ out of $4 x^{3}$ .

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

Step 3.2

Factor $4 x$ out of $- 4 x$ .

$y = 2 (4 x (x^{2}) + 4 x (- 1) + x^{4} + 2 x^{2} - 3)$

Step 3.3

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

$y = 2 (4 x (x^{2} - 1) + x^{4} + 2 x^{2} - 3)$

Step 4

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

$y = 2 (4 x (x^{2} - 1^{2}) + x^{4} + 2 x^{2} - 3)$

Step 5

Factor.

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Step 5.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$y = 2 (4 x ((x + 1) (x - 1)) + x^{4} + 2 x^{2} - 3)$

Step 5.2

Remove unnecessary parentheses.

$y = 2 (4 x (x + 1) (x - 1) + x^{4} + 2 x^{2} - 3)$

Step 6

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$y = 2 (4 x (x + 1) (x - 1) + {(x^{2})}^{2} + 2 x^{2} - 3)$

Step 7

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$y = 2 (4 x (x + 1) (x - 1) + u^{2} + 2 u - 3)$

Step 8

Factor $u^{2} + 2 u - 3$ using the AC method.

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Step 8.1

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 $- 3$ and whose sum is $2$ .

$- 1, 3$

Step 8.2

Write the factored form using these integers.

$y = 2 (4 x (x + 1) (x - 1) + (u - 1) (u + 3))$

Step 9

Replace all occurrences of $u$ with $x^{2}$ .

$y = 2 (4 x (x + 1) (x - 1) + (x^{2} - 1) (x^{2} + 3))$

Step 10

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

$y = 2 (4 x (x + 1) (x - 1) + (x^{2} - 1^{2}) (x^{2} + 3))$

Step 11

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$y = 2 (4 x (x + 1) (x - 1) + (x + 1) (x - 1) (x^{2} + 3))$

Step 12

Factor $(x + 1) (x - 1)$ out of $4 x (x + 1) (x - 1) + (x + 1) (x - 1) (x^{2} + 3)$ .

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Step 12.1

Factor $(x + 1) (x - 1)$ out of $4 x (x + 1) (x - 1)$ .

$y = 2 ((x + 1) (x - 1) (4 x) + (x + 1) (x - 1) (x^{2} + 3))$

Step 12.2

Factor $(x + 1) (x - 1)$ out of $(x + 1) (x - 1) (4 x) + (x + 1) (x - 1) (x^{2} + 3)$ .

$y = 2 ((x + 1) (x - 1) (4 x + x^{2} + 3))$

Step 13

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$y = 2 ((x + 1) (x - 1) (4 u + u^{2} + 3))$

Step 14

Factor $4 u + u^{2} + 3$ using the AC method.

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Step 14.1

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 $3$ and whose sum is $4$ .

$1, 3$

Step 14.2

Write the factored form using these integers.

$y = 2 ((x + 1) (x - 1) ((u + 1) (u + 3)))$

Step 15

Factor.

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Step 15.1

Replace all occurrences of $u$ with $x$ .

$y = 2 ((x + 1) (x - 1) ((x + 1) (x + 3)))$

Step 15.2

Remove unnecessary parentheses.

$y = 2 ((x + 1) (x - 1) (x + 1) (x + 3))$

Step 16

Factor.

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Step 16.1

Combine exponents.

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Step 16.1.1

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

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

Step 16.1.2

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

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

Step 16.1.3

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

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

Step 16.1.4

Add $1$ and $1$ .

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

Step 16.2

Remove unnecessary parentheses.

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