Write as a Set of Linear Factors 3x^4-2x^3+11x^2-8x-4

$3 x^{4} - 2 x^{3} + 11 x^{2} - 8 x - 4$

Step 1

Regroup terms.

$- 2 x^{3} - 8 x + 3 x^{4} + 11 x^{2} - 4$

Step 2

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

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

$- 2 x (x^{2}) - 8 x + 3 x^{4} + 11 x^{2} - 4$

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

$- 2 x (x^{2}) - 2 x (4) + 3 x^{4} + 11 x^{2} - 4$

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

$- 2 x (x^{2} + 4) + 3 x^{4} + 11 x^{2} - 4$

Step 3

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

$- 2 x (x^{2} + 4) + 3 {(x^{2})}^{2} + 11 x^{2} - 4$

Step 4

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

$- 2 x (x^{2} + 4) + 3 u^{2} + 11 u - 4$

Step 5

Factor by grouping.

For a polynomial of the form $a x^{2} + i x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 4 = - 12$ and whose sum is $i = 11$ .

Factor $11$ out of $11 u$ .

$- 2 x (x^{2} + 4) + 3 u^{2} + 11 (u) - 4$

Rewrite $11$ as $- 1$ plus $12$

$- 2 x (x^{2} + 4) + 3 u^{2} + (- 1 + 12) u - 4$

Apply the distributive property.

$- 2 x (x^{2} + 4) + 3 u^{2} - 1 u + 12 u - 4$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$- 2 x (x^{2} + 4) + (3 u^{2} - 1 u) + 12 u - 4$

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

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

Factor the polynomial by factoring out the greatest common factor, $3 u - 1$ .

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

Step 6

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

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

Step 7

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

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

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

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

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

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

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

Step 8

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

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

Step 9

Factor by grouping.

Reorder terms.

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

For a polynomial of the form $a x^{2} + i x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 1 = - 3$ and whose sum is $i = - 2$ .

Factor $- 2$ out of $- 2 u$ .

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

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

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

Apply the distributive property.

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

Multiply $u$ by $1$ .

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

Factor the polynomial by factoring out the greatest common factor, $3 u + 1$ .

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

Step 10

Factor.

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

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

Remove unnecessary parentheses.

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

Write as a Set of Linear Factors 3x^4-2x^3+11x^2-8x-4

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