Factor x^5+2x^3+x^2+2

$x^{5} + 2 x^{3} + x^{2} + 2$

Step 1

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x^{5} + 2 x^{3}) + x^{2} + 2$

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

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

Step 2

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

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

Step 3

Rewrite $1$ as $1^{3}$ .

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

Step 4

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 $b = 1$ .

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

Step 5

Factor.

Simplify.

Multiply $- 1$ by $1$ .

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

One to any power is one.

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

Remove unnecessary parentheses.

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

Factor x^5+2x^3+x^2+2

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