Question

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

Answer 1

Separate out the common factor `5x^6` to find:

`5x^9+10x^8-15x^7+10x^6=5x^6(x^3+2x^2-3x+2)`

Explanation:

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

`5x^9+10x^8-15x^7+10x^6=5x^6(x^3+2x^2-3x+2)`

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.