Question

Which of the following six statements are true about the cubic polynomial
$P(x)=2x^3+x^2+3x-2\text{ ? }$
(i) It has exactly one positive real root.
(ii) It has either one or three negative roots.
(iii) It has a root between 0 and 1 .
(iv) It must have exactly two real roots.
(v) It has a negative root between -2 and -1 .
(vi) It has no complex roots.

• A. only (i), (iii) and (vi)
• B. only (ii), (iii) and (iv)
• C. only (i) and (iii)
• D. only (iii), (iv) and (v)

Answer 1

Answer: (C)

First we notice that the function is increasing by checking the derivative $P^{\prime }(x)=6x^2+2x+3>0$. i.e. $P(x)$ will have exactly one real root (and two complex roots) Since $P(0)=-2$ and $P(1)=4$ the root is between 0 and 1 by the Intermediate Value theorem. Thus only statement (i) and (iii) are true, the correct answer is (D) ]