1980 AMC 12 Problem 24

Attempt Problem 24 of the 1980 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 1980 AMC 12 solutions, or check the answer key.

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24.

For some real number

r,

the polynomial

8x^3-4x^2-42x+45

is divisible by

(x-r)^2.

Which of the following numbers is closest to

r?

1.22

1.32

1.42

1.52

1.62

A repeated root appears as a squared linear factor

Factor the cubic completely and then compare the repeated root with the listed decimals

The polynomial factors as

\begin{aligned} &8x^3-4x^2-42x+45\\ &\qquad=(2x-3)^2(2x+5). \end{aligned}

Thus the repeated root is

r=\frac{3}{2}=1.5,

and the nearest listed number is

1.52.

Therefore, the correct answer is D.

Problem 24 in other years