Can someone help me factor x^4-3x^2+2?

Method 1 :

x⁴ - 3x² + 2
= (x²)² - 3x² + 2
= (x² - 1)(x² - 2)
= (x - 1)(x + 1)(x² - 2)


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Method 2 :

Let f(x) = x⁴ - 3x² + 2

f(1) = (1)⁴ - 3(1)² + 2 = 0
Hence, (x - 1) is one of the factors of (x⁴ - 3x² + 2).

f(-1) = (-1)⁴ - 3(-1)² + 2 = 0
Hence, (x + 1) is one of the factors of (x⁴ - 3x² + 2).

(x - 1)(x + 1) = x² - 1

x⁴ - 3x² + 2 = x²(x²) - 3x² + 2
= x²(x² - 1) + x² - 3x² + 2
= x²(x² - 1) - 2x² + 2
= x²(x² - 1) - 2(x² - 1)
= (x² - 1)(x² - 2)
= (x - 1)(x + 1)(x² - 2)

let p = x², then the expression becomes p²-3p+2, which you should be able to factor.

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

[ x ² - 2 ] [ x ² - 1 ]
[ x - √2 ] [ x + √2 ] [ x - 1 ] [ x + 1 ]