Determine the linear factors for x4-2x^2+1

Didn't you mean x^4-2x^2+1?

Choose "p" to represent x^2. Then p^2 - 2p + 1 = (p-1) (p-1)

These results are equivalent to (x^2-1) (x^2+1).

x^2-1 can be factored: (x-1) (x+1)

x^2-1 (again) can be factored: (x-1) (x+1)

Then the factored expression is (x-1) ^2 * (x+1) ^2.

Check this by multiplication. Do you obtain x^4 - 2x^2 + 1?

Use a substitution: let b = x^2

Then b^2 = x^4.

Now you need to factor

b^2 - 2b + 1 =

(b - 1) ^2

Since b = x^2, now we replace b with x^2 and factor again.

(x^2 - 1) ^2 is the square of a difference of squares.

= (x^2 - 1) (x^2 - 1)

= (x + 1) (x - 1) (x + 1) (x - 1)

= (x + 1) ^2 (x - 1) ^2