Find conditions for $a$ and $b$ such that $P(x)=x^4-(a+b)x^3+(ab+2)x^2-(a+b)x+1$ has only real roots.

Question

I need to find conditions for a and b such that $$P(x)=x^4-(a+b)x^3+(ab+2)x^2-(a+b)x+1$$ has only real roots. Any hints on how I should do that?

Answer 1

HINT :

$$x^4-(a+b)x^3+(ab+2)x^2-(a+b)x+1=(x^2-ax+1)(x^2-bx+1)$$

Answer 2

Hint: You have a palindromic polynomial, hence $P(x)=0$ is equivalent to:

$$ \left(x^2+\frac{1}{x^2}\right)-(a+b)\left(x+\frac{1}{x}\right)+(ab+2) = 0\tag{1} $$ or, by setting $z=x+\frac{1}{x}$, $$ z^2-(a+b)z+ab = (z-a)(z-b) = 0.\tag{2} $$ Consider now that the range of $f(x)=x+\frac{1}{x}$ is $\mathbb{R}\setminus(-2,2)$.