Factor x^5-4x^3+8x^2-32

$x^{5} - 4 x^{3} + 8 x^{2} - 32$

Step 1

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x^{5} - 4 x^{3}) + 8 x^{2} - 32$

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

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

Step 2

Factor the polynomial by factoring out the greatest common factor, $x^{2} - 4$ .

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

Step 3

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

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

Step 4

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 $b = 2$ .

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

Step 5

Rewrite $8$ as $2^{3}$ .

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

Step 6

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 2$ .

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

Step 7

Factor.

Simplify.

Multiply $2$ by $- 1$ .

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

Raise $2$ to the power of $2$ .

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

Remove unnecessary parentheses.

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

Step 8

Combine exponents.

Raise $x + 2$ to the power of $1$ .

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

Raise $x + 2$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Factor x^5-4x^3+8x^2-32

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