Question
If $${x^2} + px + 1$$ is a factor of the expression $$a{x^3} + bx + c$$ then
A.
$${a^2} + {c^2} = - ab$$
B.
$${a^2} - {c^2} = - ab$$
C.
$${a^2} - {c^2} = ab$$
D.
None of these
Answer :
$${a^2} - {c^2} = ab$$
Solution :
It is a factor if $$a{x^3} + bx + c = \left( {{x^2} + px + 1} \right)\left( {ax + c} \right)$$
$$\eqalign{
& = a{x^3} + \left( {c + ap} \right){x^2} + \left( {pc + a} \right)x + c \cr
& \Rightarrow \,\,c + ap = 0,pc + a = b \cr
& \Rightarrow \,\,p = \frac{{ - c}}{a} = \frac{{b - a}}{c}. \cr} $$