A polynomial f (x) and two of its zeros are given. Solve the equation f (x) = 0.
-$f ( x ) = 2 x ^ { 5 } + 19 x ^ { 4 } + 75 x ^ { 3 } + 90 x ^ { 2 } - 162 x - 324 ; - 3 - 3 i$ and $\frac { 3 } { 2 }$ are zeros
A) $\left\{ - 2 , - 3 , \frac { 3 } { 2 } , \pm 3 - 3 i \right\}$
B) $\left\{ - 2 , - 3 , \frac { 3 } { 2 } , - 3 \pm 3 i \right\}$
C) $\left\{ - 2 , \pm \frac { 3 } { 2 } , - 3 \pm 3 i \right\}$
D) $\left\{ - 3 , \pm \frac { 3 } { 2 } , - 3 \pm 3 i \right\}$

Question

Quizplus · Accepted Answer

Complex roots of polynomials with real coefficients come in conjugate pairs, so if $-3 - 3i$ is a root, then $-3 + 3i$ must also be a root. Given $\frac{3}{2}$ is a real root, and the polynomial is of degree 5, we need a total of 5 roots. The roots provided in option B correctly include the conjugate pair $-3 \pm 3i$ and the real root $\frac{3}{2}$, along with $-3$ and $-2$ to make up the total of 5 roots required for a 5th-degree polynomial.

A Polynomial F (X) and Two of Its Zeros Are f(x)=2x5+19x4+75x3+90x2−162x−324;−3−3if ( x ) = 2 x ^ { 5 } + 19 x ^ { 4 } + 75 x ^ { 3 } + 90 x ^ { 2 } - 162 x - 324 ; - 3 - 3 if(x)=2x5+19x4+75x3+90x2−162x−324;−3−3i

A Polynomial F (X) and Two of Its Zeros Are $f ( x ) = 2 x ^ { 5 } + 19 x ^ { 4 } + 75 x ^ { 3 } + 90 x ^ { 2 } - 162 x - 324 ; - 3 - 3 i$