A) \[{{a}^{2}}+{{c}^{2}}=-ab\]
B) \[{{a}^{2}}-{{c}^{2}}=-ab\]
C) \[{{a}^{2}}-{{c}^{2}}=ab\]
D) None of these
Correct Answer: C
Solution :
Given that \[{{x}^{2}}+px+1\]is factor of\[a{{x}^{3}}+bx+c=0\], then let\[a{{x}^{3}}+bx+c\equiv ({{x}^{2}}+px+1)(ax+\lambda )\], 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 \[b=\left( -\frac{c}{a} \right)\,c+a\] or \[ab={{a}^{2}}-{{c}^{2}}\].
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