Question

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

Answer 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?

Answer 2

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