<s>[INST] how to factor $f(x)=x^4-7x^2+1$ How do you factor this equation?  I have tried to use factoring method where the expanded b term factors add to B and multiply to get C.
 [/INST] It's easy to see that $x^4-7x^2+1$ has no rational roots. So if it factors (over the integers), it can only do so as the product of two quadratics.  Since the lead and constant coefficients are both $1$ and there are no terms of odd degree, the factorization must be of the form $(x^2+bx+\sigma)(x^2-bx+\sigma)$ with $\sigma=\pm1$.  We have
$$(x^2+bx+\sigma)(x^2-bx+\sigma)=x^4+(2\sigma-b^2)x^2+1$$
so we need $2\sigma-b^2=-7$, which rewrites as $b^2=7+2\sigma$.  Letting $\sigma=1$ gives $b=\pm3$, so we have the factorization
$$x^4-7x^2+1=(x^2+3x+1)(x^2-3x+1)$$
Note, the other option, $\sigma=-1$, gives another valid factorization, just not over the integers:
$$x^4-7x^2+1=(x^2+\sqrt5x-1)(x^2-\sqrt5x-1)$$
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