If-P-x-aX-3-bX-c-Q-X-a-0-has-the-roots-x-1-x-2-and-x-3-such-that-x-1-x-2-x-3-then-prove-that-x-1-0-if-and-only-if-b-c-

Question Number 153161 by mathdanisur last updated on 05/Sep/21

If P (x) = {aX}^{3} + bX + c \in Q [X]; (a \neq 0)

has the roots x_{1}, x_{2} and x_{3}

such that x_{1} = x_{2} x_{3}

then prove that x_{1} = 0 if and only if b = c

Answered by mr W last updated on 05/Sep/21

x_{1} + x_{2} + x_{3} = 0

x_{1} (x_{2} + x_{3}) + x_{2} x_{3} = \frac{b}{a}

x_{1} x_{2} x_{3} = - \frac{c}{a}

w i t h x_{1} = x_{2} x_{3} :

- x_{1}^{2} + x_{1} = \frac{b}{a}

x_{1}^{2} = - \frac{c}{a}

\Rightarrow x_{1} = \frac{b - c}{a}

x_{1} = 0 i f a n d o n l y i f b - x = 0 \Rightarrow b = c

Commented by mathdanisur last updated on 05/Sep/21

veri nice thank you ser