How do you factor x squared plus x minus 1?

Answer 1

Normally when you factor a polynomial, it is to make a problem easier. This is a polynomial of the form ax2 + bx - c, where a=b=c=1. Since a=1, to factor this polynomial nicely we would have to find two factors of c=1 that add up to b=1. Since the factors of 1 are -1 and 1, the possible numbers to add to are (-1) + (-1) = -2; 1 + (-1) = 0; and 1 + 1 = 2. None of these can add up to 1 (our b), so a nice answer is impossible.

However, when you factor, you are actually just finding solutions. For example, x2-3x+2= (x-1)(x-2). If x2-3x+2=0, then either x-1=0 or x-2=0, so x=1 or x=2.

Anything to the half power, is the square-root, so we write, using the quadratic formulas (b +/- (b2 - 4ac)1/2)/(2a), the solutions of x2 + x + 1 are (1 +/- (-3)1/2)/2. Since irepresents (-1)1/2, we can factor x2+x+1 = [x - 1/2 +i(31/2)/2] [x - 1/2 -i(31/2)/2].