Which of the following six statements are tru | vedclass

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(C) Given the polynomial $P(x) = 2x^3 + x^2 + 3x - 2$.First,we check the derivative $P'(x) = 6x^2 + 2x + 3$.The discriminant of $P'(x)$ is $D = (2)^2

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only $(i)$ and $(iii)$

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(C) Given the polynomial $P(x) = 2x^3 + x^2 + 3x - 2$.First,we check the derivative $P'(x) = 6x^2 + 2x + 3$.The discriminant of $P'(x)$ is $D = (2)^2 - 4(6)(3) = 4 - 72 = -68 Since the leading coefficient is positive and the discriminant is negative,$P'(x) > 0$ for all $x \in \mathbb{R}$.This implies that $P(x)$ is a strictly increasing function.As a strictly increasing cubic polynomial,it must have exactly one real root and two complex conjugate roots.Evaluating at endpoints: $P(0) = -2$ and $P(1) = 2(1)^3 + (1)^2 + 3(1) - 2 = 4$.Since $P(0)  0$,by the Intermediate Value Theorem,there exists a root in the interval $(0, 1)$.Since there is only one real root and it is positive,statements $(i)$ and $(iii)$ are true.Statement $(ii)$ is false because there are no negative roots.Statement $(iv)$ is false because a cubic polynomial with real coefficients must have either one or three real roots.Statement $(v)$ is false because the only real root is positive.Statement $(vi)$ is false because a cubic polynomial with one real root must have two complex roots.Therefore,only $(i)$ and $(iii)$ are true.

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