How do you factor x^6 + 2x^5 + 3x^4 + 6x^3?

Nov 15, 2015

Separate out the common factor ${x}^{3}$ then factor by grouping to find:

${x}^{6} + 2 {x}^{5} + 3 {x}^{4} + 6 {x}^{3} = {x}^{3} \left({x}^{2} + 3\right) \left(x + 2\right)$

Explanation:

${x}^{6} + 2 {x}^{5} + 3 {x}^{4} + 6 {x}^{3}$

$= {x}^{3} \left({x}^{3} + 2 {x}^{2} + 3 x + 6\right)$

$= {x}^{3} \left(\left({x}^{3} + 2 {x}^{2}\right) + \left(3 x + 6\right)\right)$

$= {x}^{3} \left({x}^{2} \left(x + 2\right) + 3 \left(x + 2\right)\right)$

$= {x}^{3} \left({x}^{2} + 3\right) \left(x + 2\right)$

This has no simpler factors with Real coefficients since ${x}^{2} + 3 \ge 3 > 0$ for all $x \in \mathbb{R}$