# Thread: Reducibility of a polynomial

1. ## Reducibility of a polynomial

Let f(x)=x^5-120x^4+1152x^3+341x^2-1185x-21945

The roots of this, as presented by the computer, happen to be

3.041032416, 109.44694126, 10.55305874, -1.520516208 +/- 1.983912452 i

Using only the roots, is f(x) reducible in Z[x]?

I'm a little confused how to go about this...do you just multiply together possible combinations and see if integers result, or is there a more efficient way?

Call those roots $a,\, b,\, c,\, d\pm ei$ respectively. Clearly none of the linear factors $x-a,\, x-b,\, x-c,\,x-d-ei,\,x-d+ei$ is in Z[x]. So if there is a factorisation in Z[x] it must consist of a quadratic factor and a cubic factor. The two complex linear factors must obviously belong in the same real factor, hence so must their product $(x-d-ei)(x-d+ei) = x^2 -2dx+ d^2+e^2$. But 2d is not an integer, so that quadratic expression is not in Z[x]. Notice however that a=-2d, which raises the possibility that the cubic factor $(x-a)(x^2 -2dx+ d^2+e^2)$ might be in Z[x]. If so, then the quadratic factor would be the product $(x-b)(x-c) = x^2 - (b+c)x + bc$ of the remaining two linear factors. Use your calculator to find b+c and bc, to see whether they look like integers.