MathDB

Cubic Polynomial

Source: AIME I 2007 #8

March 15, 2007

algebrapolynomialquadraticsnumber theorygreatest common divisoralgorithmleast common multiple

Problem Statement

The polynomial

P(x)

is cubic. What is the largest value of

k

for which the polynomials

Q_{1}(x) = x^{2}+(k-29)x-k

and

Q_{2}(x) = 2x^{2}+(2k-43)x+k

are both factors of

P(x)