# How do you factor the expression 5x^9+10x^8-15x^7+10x^6?

Mar 2, 2016

Separate out the common factor $5 {x}^{6}$ to find:

$5 {x}^{9} + 10 {x}^{8} - 15 {x}^{7} + 10 {x}^{6} = 5 {x}^{6} \left({x}^{3} + 2 {x}^{2} - 3 x + 2\right)$

#### Explanation:

All of the individual terms are divisible by $5 {x}^{6}$, so separate out that common factor to find:

$5 {x}^{9} + 10 {x}^{8} - 15 {x}^{7} + 10 {x}^{6} = 5 {x}^{6} \left({x}^{3} + 2 {x}^{2} - 3 x + 2\right)$

The remaining cubic factor has no simpler factors with rational coefficients. In fact it has one irrational Real zero and a pair of Complex conjugate zeros. It is possible to find the zeros using Vieta's substitution or my favourite, Cardano's method, but this is probably beyond the scope of the question.