2017/004) If $x^2+px+1$ ia a factor of $ax^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

Solution
we have other factor linear and hence $dx + e$
so $ax^3+ bx + c = (x^2+px+1)(dx + e)$
comparing coefficient of $x^2$ we have d = a and constant term gives c =e
so  $ax^3+ bx + c = (x^2+px+1)(ax + c) =  ax^3 + x^2(c + ap) + x(a + cp) + x$
comparing coefficients of $x^2$ on both sided $c+ap=0\cdots(1)$
comparing coefficient of x both sides $ a+cp = b$ or  $a^2 + acp = ab$ or $a^2-c^2 = ab$(using from (1)) Hence C