Question

# Ifax3 + bx+c has a factor of the form x2 + px + 1, show that a--ca = ab.

Solution

By actual division, we get,
ax? +bx+c
R(x)
-= ax - ap+
x2 + px +1
Remainder polynomial, R(x)=(b-a +ap2)x+c+ap
If ax3 + bx+chas a factor of the form x2 +px + 1, then R(x)
must be identically zero, if b-a+ap2 = 0 and c+ap =0
Eliminating p from these equations, we get
2
b-a+a
or a? - c2 = ab