Question

a, b, c, are real numbers in the polynomial , $p\left(z\right)=2z^4+a{z}^3+b{z}^2+cz+3.$ If two roots of the equation $p\left(z\right)=0$ are 2 and i. Then which of the following are false .

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

Answer 1

The correct option is D a=-11

If $i$ is a root of $p\left(z\right)=0$ then $-i$ must also be a root of $p\left(z\right)=0$

Let the other root be $k$

Product of roots is $\frac{3}{2}⟹2\left(i\right)(-i)k=\frac{3}{2}⟹k=\frac{3}{4}$

Sum of roots is $2+i-i+k=-\frac{a}{2}⟹a=-\frac{11}{2}$

$\frac{b}{2}=2\left(i\right)+2(-i)+2\left(k\right)+i(-i)+i\left(\frac{3}{4}\right)-i\left(\frac{3}{4}\right)⟹b=5$

$-\frac{c}{2}=2\left(i\right)(-i)+2\left(i\right)\left(k\right)+2(-i)\left(k\right)+i(-i)\left(k\right)⟹c=-\frac{11}{2}$