When the expression pm\(^2\) + qm + 1 is divided by (m - 1), it has a remainder is 2, and when divided by (m + 1), the remainder is 4. Find p and q respectively
pm\(^2\) + qm + 1 = (m - 1) Q(x) + 2, where Q(x) = remainder.
p(1)\(^2\) + q(1) + 1 = 2
p + q + 1 = 2
p + q = 1.....(i)
pm\(^2\) + qm + 1 = (m - 1)Q(x) + 4
p(-1)\(^2\) + q(-1) + 1 = 4
p - q + 1 = 4
p - q = 3....(ii)
p + q = 1, p - q = 3
Using elimination method, add eqn i and ii
2p = 4, p = 2
from eqn ii
p - q = 3
2 - q = 3
-q = 3 - 2 = 1
q = - 1
Therefore, p = 2 and q = -1
q = 2
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