Question

For $x^2+2x+5$ to be a factor of $x^4+px^2+q,$ the values of $p$ and $q$ must be, respectively:

$-2,$ $5$

$5,$ $25$

$10,$ $20$

$6,$ $25$

$14,$ $25$

Answer: D

Concepts:polynomial algebraic manipulation factor

Difficulty rating: 1690

Small Hint:

Work modulo $x^2+2x+5,$ so $x^2=-2x-5$

Big Hint:

Reduce $x^4$ to a linear expression and make both remainder coefficients zero

Solution:

Modulo $x^2+2x+5,$ we have $x^2=-2x-5.$ Then $x^3=10-x,\qquad x^4=12x+5.$ Thus $\begin{gathered} x^4+px^2+q\\ {}\equiv(12-2p)x+(5-5p+q). \end{gathered}$ Both coefficients vanish when $p=6$ and $q=25.$

Therefore, the correct answer is D.

1960 AMC 12 Problem 31

Answer: D

Difficulty rating: 1690

Small Hint:

Big Hint:

Solution:

Problem 31 in Other Years