Question

$P\left(z\right)=z^4+a{z}^3+b{z}^2+cz+d$ is a polynomial where a, b, c and d are real numbers. Find $a+b+c+d$ if two zeros of polynomial P are the following complex numbers: $2-i$ and $1-i$.

• A. $2$
• B. $-1$
• C. $1$
• D. $0$

Answer 1

The correct option is C $1$

Since roots occurs in conjugate so roots are $2-i,2+i,1-i,1+i$

Now sum of roots one at a time $2-i+2+i+1-i+1+i=6⟹a=-6$

Sum of roots two at a time $(2+i)(2-i)+(2+i)(1+i)+(2+i)(1-i)+(2-i)(1+i)+(2-i)(1-i)+(1+i)(1-i)$

solving this we get $b=15$

Sum of roots three at a time $(2+i)(2-i)(1+i)+(2+i)(1-i)(2-i)+(2-i)(1-i)(1+i)+(2+i)(1+i)(1-i)$

solving this we get $-c=18⟹c=-18$

Products of roots $(2-i)(2+i)(1-i)(1+i)10⟹d=10$

So, $a+b+c+d=-6+15-18+10=1$