Complex Numbers And Quadratic Equations question 14
Question: If $ x^{2}+px+1 $ is a factor of the expression $ ax^{3}+bx+c $ , then [IIT 1980]
Options:
A) $ a^{2}+c^{2}=-ab $
B) $ a^{2}-c^{2}=-ab $
C) $ a^{2}-c^{2}=ab $
D) None of these
Show Answer
Answer:
Correct Answer: C
Solution:
Given that $ x^{2}+px+1 $ is factor of $ ax^{3}+bx+c=0 $ , then let $ ax^{3}+bx+c\equiv (x^{2}+px+1)(ax+\lambda ) $ , where $ \lambda $ is a constant. Then equating the coefficient of like powers of x on both sides, we get $ 0=ap+\lambda ,\ \ b=p\lambda +a,\ c=\lambda $
$ \Rightarrow p=-\frac{\lambda }{a}=-\frac{c}{a} $ Hence $ b=( -\frac{c}{a} )c+a $ or $ ab=a^{2}-c^{2} $ .
BETA
