For the polynomial, one zero is given. Find all others.
-$$P ( x ) = x ^ { 4 } - 10 x ^ { 3 } + 23 x ^ { 2 } - 10 x + 22 ; i$$
A) $- \mathrm { i } , 5 + \sqrt { 3 } , 5 - \sqrt { 3 }$
B) $- \mathrm { i } , 1 + \sqrt { 3 } , 1 - \sqrt { 3 }$
C) $- i , 5 + 2 \sqrt { 3 } , 5 - 2 \sqrt { 3 }$
D) $- i , - 5 + \sqrt { 3 } , - 5 - \sqrt { 3 }$

Question

Quizplus · Accepted Answer

Complex roots of polynomials with real coefficients come in conjugate pairs, so if $i$ is a root, $-i$ must also be a root. The remaining roots can be found by polynomial division or using the fact that the sum and product of the roots are related to the coefficients of the polynomial. The correct set of roots that includes $-i$ and satisfies the polynomial given is $-i, 5 + \sqrt{3}, 5 - \sqrt{3}$.

For the Polynomial, One Zero Is Given P(x)=x4−10x3+23x2−10x+22;iP ( x ) = x ^ { 4 } - 10 x ^ { 3 } + 23 x ^ { 2 } - 10 x + 22 ; iP(x)=x4−10x3+23x2−10x+22;i

For the Polynomial, One Zero Is Given $P ( x ) = x ^ { 4 } - 10 x ^ { 3 } + 23 x ^ { 2 } - 10 x + 22 ; i$