1954 AHSME Problems/Problem 20

Problem 20

The equation $x^3+6x^2+11x+6=0$ has:

$\textbf{(A)}\ \text{no negative real roots}\qquad\textbf{(B)}\ \text{no positive real roots}\qquad\textbf{(C)}\ \text{no real roots}\\ \textbf{(D)}\ \text{1 positive and 2 negative roots}\qquad\textbf{(E)}\ \text{2 positive and 1 negative root}$

Solution

By the rational root theorem, $1, -1, 2, -2, 3, -3, 6, -6$ are possible rational roots. Because $x^3+6x^2+11x+6>0$ for $x>0$ , so there are no positive roots. We try $-1, -2, -3, -6$ , so $x=-1,$ x=-3, x=-2 $, so there are no positive real roots;$ \fbox{B}$

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1954 AHSME Problems/Problem 20

Problem 20

Solution