Question

factorise x² – 2x – 1,​

Answer 1

This cannot be factorized

Answer 2

If you start with your polymomial,

x2−2x−1x2−2x−1

completing the square gives you

x2−2x+1−1−1x2−2x+1−1−1

which simplifies to

(x−1)2−2(x−1)2−2.

Since we can think of 22 as (2–√)2(2)2, we can look at our polynomial as a difference of squares. We get

((x−1)−2–√)((x−1)+2–√)((x−1)−2)((x−1)+2)

which, if we rearrange it a bit, gives us

(x−(1+2–√))(x−(1−2–√))(x−(1+2))(x−(1−2)).

If you don’t want to use completing the square, you can use the factor theorem, which says that if cc is a zero of a polynomial, then (x−c)(x−c) is a factor. The quadratic formula will give you your zeros, 1±2–√1±2. The factor theorem then gives you the factorization we got above.

Hope that helps.