1954 AMC 12 Problem 20

Attempt Problem 20 of the 1954 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1954 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

20.

The equation

x^3+6x^2+11x+6=0

has:

no negative real roots

no positive real roots

no real roots

1

positive and

2

negative roots

2

positive and

1

negative root

Test the small negative integers suggested by the constant term

The polynomial factors as

(x+1)(x+2)(x+3)

The polynomial factors as

\begin{gathered} x^3+6x^2+11x+6\\ =(x+1)(x+2)(x+3). \end{gathered}

Its roots are

-1,-2,-3,

so it has no positive real roots.

Thus, the correct answer is B.

Problem 20 in other years