Factor x^4-16x^2+x^3-16x-2x^2+32

$x^{4} - 16 x^{2} + x^{3} - 16 x - 2 x^{2} + 32$

Step 1

Regroup terms.

$x^{4} + x^{3} - 2 x^{2} - 16 x^{2} - 16 x + 32$

Step 2

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

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

$x^{2} x^{2} + x^{3} - 2 x^{2} - 16 x^{2} - 16 x + 32$

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

$x^{2} x^{2} + x^{2} x - 2 x^{2} - 16 x^{2} - 16 x + 32$

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

$x^{2} x^{2} + x^{2} x + x^{2} \cdot - 2 - 16 x^{2} - 16 x + 32$

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

$x^{2} (x^{2} + x) + x^{2} \cdot - 2 - 16 x^{2} - 16 x + 32$

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

$x^{2} (x^{2} + x - 2) - 16 x^{2} - 16 x + 32$

Step 3

Factor.

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

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 $- 2$ and whose sum is $1$ .

$- 1, 2$

Write the factored form using these integers.

$x^{2} ((x - 1) (x + 2)) - 16 x^{2} - 16 x + 32$

Remove unnecessary parentheses.

$x^{2} (x - 1) (x + 2) - 16 x^{2} - 16 x + 32$

Step 4

Factor $16$ out of $- 16 x^{2} - 16 x + 32$ .

Factor $16$ out of $- 16 x^{2}$ .

$x^{2} (x - 1) (x + 2) + 16 (- x^{2}) - 16 x + 32$

Factor $16$ out of $- 16 x$ .

$x^{2} (x - 1) (x + 2) + 16 (- x^{2}) + 16 (- x) + 32$

Factor $16$ out of $32$ .

$x^{2} (x - 1) (x + 2) + 16 (- x^{2}) + 16 (- x) + 16 (2)$

Factor $16$ out of $16 (- x^{2}) + 16 (- x)$ .

$x^{2} (x - 1) (x + 2) + 16 (- x^{2} - x) + 16 (2)$

Factor $16$ out of $16 (- x^{2} - x) + 16 (2)$ .

$x^{2} (x - 1) (x + 2) + 16 (- x^{2} - x + 2)$

Step 5

Factor.

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

Factor $- 1$ out of $- x$ .

$x^{2} (x - 1) (x + 2) + 16 (- x^{2} - (x) + 2)$

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

$x^{2} (x - 1) (x + 2) + 16 (- x^{2} + (1 - 2) x + 2)$

Apply the distributive property.

$x^{2} (x - 1) (x + 2) + 16 (- x^{2} + 1 x - 2 x + 2)$

Multiply $x$ by $1$ .

$x^{2} (x - 1) (x + 2) + 16 (- x^{2} + x - 2 x + 2)$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$x^{2} (x - 1) (x + 2) + 16 ((- x^{2} + x) - 2 x + 2)$

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

$x^{2} (x - 1) (x + 2) + 16 (x (- x + 1) + 2 (- x + 1))$

Factor the polynomial by factoring out the greatest common factor, $- x + 1$ .

$x^{2} (x - 1) (x + 2) + 16 ((- x + 1) (x + 2))$

Remove unnecessary parentheses.

$x^{2} (x - 1) (x + 2) + 16 (- x + 1) (x + 2)$

Step 6

Factor $x + 2$ out of $x^{2} (x - 1) (x + 2) + 16 (- x + 1) (x + 2)$ .

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

$(x + 2) (x^{2} (x - 1)) + 16 (- x + 1) (x + 2)$

Factor $x + 2$ out of $16 (- x + 1) (x + 2)$ .

$(x + 2) (x^{2} (x - 1)) + (x + 2) (16 (- x + 1))$

Factor $x + 2$ out of $(x + 2) (x^{2} (x - 1)) + (x + 2) (16 (- x + 1))$ .

$(x + 2) (x^{2} (x - 1) + 16 (- x + 1))$

Step 7

Apply the distributive property.

$(x + 2) (x^{2} x + x^{2} \cdot - 1 + 16 (- x + 1))$

Step 8

Multiply $x^{2}$ by $x$ by adding the exponents.

Multiply $x^{2}$ by $x$ .

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

$(x + 2) (x^{2} x^{1} + x^{2} \cdot - 1 + 16 (- x + 1))$

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

$(x + 2) (x^{2 + 1} + x^{2} \cdot - 1 + 16 (- x + 1))$

Add $2$ and $1$ .

$(x + 2) (x^{3} + x^{2} \cdot - 1 + 16 (- x + 1))$

Step 9

Move $- 1$ to the left of $x^{2}$ .

$(x + 2) (x^{3} - 1 x^{2} + 16 (- x + 1))$

Step 10

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

$(x + 2) (x^{3} - x^{2} + 16 (- x + 1))$

Step 11

Apply the distributive property.

$(x + 2) (x^{3} - x^{2} + 16 (- x) + 16 \cdot 1)$

Step 12

Multiply $- 1$ by $16$ .

$(x + 2) (x^{3} - x^{2} - 16 x + 16 \cdot 1)$

Step 13

Multiply $16$ by $1$ .

$(x + 2) (x^{3} - x^{2} - 16 x + 16)$

Step 14

Factor.

Rewrite $x^{3} - x^{2} - 16 x + 16$ in a factored form.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x + 2) ((x^{3} - x^{2}) - 16 x + 16)$

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

$(x + 2) (x^{2} (x - 1) - 16 (x - 1))$

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

$(x + 2) ((x - 1) (x^{2} - 16))$

Rewrite $16$ as $4^{2}$ .

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

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 $b = 4$ .

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

Remove unnecessary parentheses.

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

Remove unnecessary parentheses.

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

Factor x^4-16x^2+x^3-16x-2x^2+32

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