1975 AMC 12 Problem 13

Attempt Problem 13 of the 1975 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 1975 AMC 12 solutions, or check the answer key.

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13.

The equation

x^6-3x^5-6x^3-x+8=0

has

no real roots

exactly two distinct negative roots

exactly one negative root

no negative roots, but at least one positive root

none of these

Determine the sign of every term when

x\lt0

Compare the polynomial’s values at

0

and

1

For

x\lt0,

each of

x^6,-3x^5,-6x^3,-x,

and

8

is positive, so there is no negative root. At

x=0

the polynomial is

8,

while at

x=1

it is

-1.

Continuity therefore gives a positive root between

0

and

1.

Therefore, the correct answer is D.

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