Question

a, b, c are real numbers in the polynomial , P(z) = $2z^4+a{z}^3+b{z}^2+cz+3$. If two roots of the equations P(z) = 0 are 2 and i. Then which of the following are true.

• A. a = $-\frac{11}{2}$
• B. b = 5
• C. c = $-\frac{11}{2}$
• D. a = -11

Answer 1

The correct option is A a = $-\frac{11}{2}$

If the coefficients of a polynomial are all real numbers, then the complex roots of the polynomial must occur in conjugate pairs. As one of the roots is given to $i$, one of the other root must be $-i$.
Let the fourth root be $\alpha$
Using Vieta's formulae for sum and product of roots,
$\left(2\right)\left(i\right)(-i)\left(\alpha \right)=\frac{3}{2}⟹\alpha =\frac{3}{4}$
$2+i+(-i)+\alpha =-\frac{a}{2}⟹2+\frac{3}{4}=-\frac{a}{2}⟹\alpha =-\frac{11}{2}$