1965 AMC 12 Problem 19

Attempt Problem 19 of the 1965 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 1965 AMC 12 solutions, or check the answer key.

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

19.

x^4+4x^3+6px^2+4qx+r

is exactly divisible by

x^3+3x^2+9x+3,

the value of

(p+q)r

is:

-18

12

15

27

45

The quotient must be monic and linear

Subtract

(x+1)(x^3+3x^2+9x+3)

and compare coefficients

The leading terms force quotient

x+1.

Expanding,

\begin{aligned} &(x+1)(x^3+3x^2+9x+3)\\ &\quad=x^4+4x^3+12x^2+12x+3. \end{aligned}

Thus

6p=12,\ 4q=12,\ r=3,

p=2,\ q=3

and

(p+q)r=15.

Therefore, the correct answer is C.

Problem 19 in other years