Question

Factorise: $x^4-3x^2+2$

Answer 1

Let $x^2=p$, then the equation $x^4-3x^2+2$ becomes $p^2-3p+2$

Consider the equation $p^2-3p+2$ and factorise it as follows:

$p^2-3p+2=p^2-2p-p+2=p(p-2)-1(p-2)=(p-1)(p-2)$

Now, substitute the value of $p$ as $p=x^2$:

$(p-1)(p-2)=(x^2-1)(x^2-2)=(x+1)(x-1)(x+2)(x-2)$ (Using identity $a^2-b^2=(a+b)(a-b)$)

Hence, $x^4-3x^2+2=(x^2-1)(x^2-2)=(x+1)(x-1)(x+\sqrt{2})(x-\sqrt{2})$