1954 AMC 12 Problem 41

Attempt Problem 41 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.

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41.

The sum of all the roots of

4x^3-8x^2-63x-9=0

is:

8

2

-8

-2

0

Answer: B

Concepts:Vieta’s Formulas polynomial

Difficulty rating: 1260

Small hint:

Use Vieta’s formula for the sum of the roots of a cubic

Big hint:

Negate the coefficient of

x^2

and divide by the coefficient of

x^3

Solution:

For

ax^3+bx^2+\cdots=0,

the sum of the roots is

-\frac{b}{a}.

Here it is

-\frac{-8}{4}=2.

Thus, the correct answer is B.

Problem 41 in other years