How do you factor x^2 + x + 1 ?

1 Answer
May 17, 2015

x^2+x+1 has no real factors.

To ascertain this, notice that it is of the form a^2+bx+c with a = b = c = 1. This has determinant calculated as follows:

Delta = b^2-4ac = 1^2-(4xx1xx1) = 1-4 = -3

Since the determinant is negative, the polynomial has no zeros for real values of x and therefore no factors with real coefficients.

On the other hand, if you are allowed to use complex coefficients, they can be calculated from the roots of x^2+x+1=0 as follows:

x =(-b+-sqrt(Delta))/(2a) = (-1+-sqrt(-3))/2 = (-1+-isqrt(3))/2

These are the so-called primitive cube roots of unity, often denoted by the symbols omega and omega^2.

omega^3 = 1