Precalculus Examples

$4 - 14 x^{2} - 8 x^{4}$

Step 1

Factor $- 2$ out of $4 - 14 x^{2} - 8 x^{4}$ .

Tap for more steps...

Step 1.1

Reorder the expression.

Tap for more steps...

Step 1.1.1

Move $4$ .

$- 14 x^{2} - 8 x^{4} + 4$

Step 1.1.2

Reorder $- 14 x^{2}$ and $- 8 x^{4}$ .

$- 8 x^{4} - 14 x^{2} + 4$

Step 1.2

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

$- 2 (4 x^{4}) - 14 x^{2} + 4$

Step 1.3

Factor $- 2$ out of $- 14 x^{2}$ .

$- 2 (4 x^{4}) - 2 (7 x^{2}) + 4$

Step 1.4

Factor $- 2$ out of $4$ .

$- 2 (4 x^{4}) - 2 (7 x^{2}) - 2 (- 2)$

Step 1.5

Factor $- 2$ out of $- 2 (4 x^{4}) - 2 (7 x^{2})$ .

$- 2 (4 x^{4} + 7 x^{2}) - 2 (- 2)$

Step 1.6

Factor $- 2$ out of $- 2 (4 x^{4} + 7 x^{2}) - 2 (- 2)$ .

$- 2 (4 x^{4} + 7 x^{2} - 2)$

Step 2

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

$- 2 (4 {(x^{2})}^{2} + 7 x^{2} - 2)$

Step 3

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

$- 2 (4 u^{2} + 7 u - 2)$

Step 4

Factor by grouping.

Tap for more steps...

Step 4.1

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 $a \cdot c = 4 \cdot - 2 = - 8$ and whose sum is $b = 7$ .

Tap for more steps...

Step 4.1.1

Factor $7$ out of $7 u$ .

$- 2 (4 u^{2} + 7 (u) - 2)$

Step 4.1.2

Rewrite $7$ as $- 1$ plus $8$

$- 2 (4 u^{2} + (- 1 + 8) u - 2)$

Step 4.1.3

Apply the distributive property.

$- 2 (4 u^{2} - 1 u + 8 u - 2)$

Step 4.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 4.2.1

Group the first two terms and the last two terms.

$- 2 ((4 u^{2} - 1 u) + 8 u - 2)$

Step 4.2.2

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

$- 2 (u (4 u - 1) + 2 (4 u - 1))$

Step 4.3

Factor the polynomial by factoring out the greatest common factor, $4 u - 1$ .

$- 2 ((4 u - 1) (u + 2))$

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Factor.

Tap for more steps...

Step 8.1

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

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

Step 8.2

Remove unnecessary parentheses.

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