Q. If at least one root of the equation $x^{3} + ax^{2} + bx + c = 0$ remains unchanged, when $a,\ b\ and\ c$ are decreased by one, then which one of the following is always a root of the given equation?
A) 1
B) $- 1$
C) $\omega,$ an imaginary cube root of unity
D) $i$
Explanation
(c)
Equations $x^{3} + ax^{2} + bx + c = 0$
and $x^{3} + (a - 1)x^{2} + (b - 1)x + (c - 1) = 0$ have at least one common root, let common root be $\alpha$.
$\therefore\ \alpha^{3} + a\alpha^{2} + b\alpha + c = 0$
and $\alpha^{3} + a\alpha^{2} + b\alpha + c - \alpha^{2} - \alpha - 1 = 0$
$\Rightarrow \ \alpha^{2} + \alpha + 1 = 0$
$\Rightarrow \ \alpha = \omega,\omega^{2}$(where $\omega\ and{\ \omega}^{2}$ are the cube roots of unity)
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.