MathDB

Karina has a polynomial

p_1(x) = x^2 + x + k

, where

k

is an integer. Noticing that

p_1

has integer roots, she forms a new polynomial

p_2(x) = x^2 + a_1x + b_1

, where

a_1

and

b_1

are the roots of

p_1

and

a_1 \ge b_1

. The polynomial

p_2

also has integer roots, so she forms a new polynomial

p_3(x) = x^2 + a_2x + b_2

, where

a_2

and

b_2

are the roots of

p_2

and

a_2 \ge b_2

. She continues this process until she reaches

p_7(x)

and finds that it does not have integer roots. What is the largest possible value of

k

6