# How to divide a known root out of a polynomial?

Consider this simple 5th order polynomial:

 pol=x^5+x^4+1
Factor[pol]
(*   (1+x+x^2)(1-x+x^3)   *)
root=FindInstance[pol==0, x]
(*   -1/2 -I/2 Sqrt[3]   *)


Suppose we would like to see the 4th order polynomial with this root removed. This does not immediately seem to be easy... For instance:

FullSimplify[pol/(x-root)]
Factor[pol/(x-root)]


are both just returning the fraction of a 5th order polynomial divided by a 1st order one... How to see the 4th order result explicitly?

(PS: I know it could become a very messy expression, but then I would try one of the other roots. I just need to see the result in order to judge that.)

PolynomialQuotient is enough. I just used PolynomialQuotientRemainder to see that remainder is indeed 0.

PolynomialQuotientRemainder[x^5 + x^4 + 1, x - (-1/2 - I/2 Sqrt[3]), x]


$$\left\{x^4+\left(\frac{1}{2}-\frac{i \sqrt{3}}{2}\right) x^3-x^2+\left(\frac{1}{2}+\frac{i \sqrt{3}}{2}\right) x-\frac{i \sqrt{3}}{2}+\frac{1}{2},0\right\}$$

• Thanks, that solves it... Commented Apr 27 at 9:42

FindInstance returns a rule, not a number. Therefore you need:

root = x /. FindInstance[pol == 0, x][[1]]

1/2 (-1 - I Sqrt[3])


To divide the polynomial by the root:

red = PolynomialQuotient[pol, (x - root), x]


Further, "Factor" factors a polynomial over the integers, not the Complexes. Therefore, to get all the roots of "red":

Solve[red == 0, x] // N

{{x -> -0.5 + 0.866025 I}, {x -> -1.32472}, {x ->
0.662359 - 0.56228 I}, {x -> 0.662359 + 0.56228 I}}