Factor 4x^4-6x^2+2

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

Step 1

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

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

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

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

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

Factor $2$ out of $2$ .

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

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

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

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

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

Step 2

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

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

Step 3

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

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

Step 4

Factor by grouping.

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 $a \cdot c = 2 \cdot 1 = 2$ and whose sum is $b = - 3$ .

Factor $- 3$ out of $- 3 u$ .

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

Rewrite $- 3$ as $- 1$ plus $- 2$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

$2 (u (2 u - 1) - (2 u - 1))$

Factor the polynomial by factoring out the greatest common factor, $2 u - 1$ .

$2 ((2 u - 1) (u - 1))$

Step 5

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

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

Step 6

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

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

Step 7

Factor.

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 $b = 1$ .

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

Remove unnecessary parentheses.

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

Remove unnecessary parentheses.

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

Factor 4x^4-6x^2+2

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