Question

Find the zeros of the following polynomial x^2-2x+1

Answer 1

It factors to (x - 1)(x - 1)

the zeroes are ix = 1. Duplicity 2.

Answer 2

If the polynomial is of the form x^2+bx+c (ie., no factor in front of the x^2!!) then you can look for numbers that add up to b and multiply to c.

So in this case: what two numbers, when added, give -2 and when multiplied, give 1? A little trial and error gives you -1 and -1.
So the factorization is (x-1)(x-1). This is zero for x=1. There is only one zero.