Thread: 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?

Originally Posted by twittytwitter

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 respectively. Clearly none of the linear factors 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 . 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 might be in Z[x]. If so, then the quadratic factor would be the product of the remaining two linear factors. Use your calculator to find b+c and bc, to see whether they look like integers.