Let $f(x) = x^4 + x^2 + x - 1$. Which of the following is true? (A) $f$ has exactly two real roots (B) $f$ has no real roots (C) $f$ has four real roots (D) $f$ has exactly two real roots, one of which is $-1$
$f(-1) = 0$. Factoring: $f(x) = (x+1)(x^3 - x^2 + 2x - 1)$. For $x < 0$, $x^3 - x^2 + 2x - 1 < 0$, so no other negative roots. $f'(x) = 4x^3 + 2x + 1$ has exactly one real root (since $f''(x) = 12x^2 + 2 > 0$), so $f$ has exactly one local minimum. $f$ has exactly two real roots. (D)
Let $f(x) = x^4 + x^2 + x - 1$. Which of the following is true?
(A) $f$ has exactly two real roots
(B) $f$ has no real roots
(C) $f$ has four real roots
(D) $f$ has exactly two real roots, one of which is $-1$