Solve the polynomial equation. In order to obtain the first root, use synthetic division to test the possible rational
roots.
-$$3 x ^ { 4 } - 17 x ^ { 3 } + 61 x ^ { 2 } - 73 x + 26 = 0$$
A) $\left\{ 1 , \frac { 2 } { 3 } , 2 + 3 i , 2 - 3 i \right\}$
B) $\left\{ - 1 , - \frac { 2 } { 3 } , 3 + 2 i , 3 - 2 i \right\}$
C) $\left\{ 1 , - \frac { 2 } { 3 } , 3 + 2 i , 3 - 2 i \right\}$
D) $\left\{ - 1 , \frac { 2 } { 3 } , 2 + 3 i , 2 - 3 i \right\}$

Question

Quizplus · Accepted Answer

The possible rational roots of the polynomial are given by the factors of the constant term divided by the factors of the leading coefficient. For the given polynomial, the possible rational roots are $\pm 1, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{13}{3}, \pm 26$. Testing these using synthetic division or by direct substitution, we find that $x = 1$ and $x = \frac{2}{3}$ are roots. After finding the first root and applying synthetic division, we reduce the polynomial degree and eventually find that the other roots are complex, specifically $2 + 3i$ and $2 - 3i$, making the set of roots $\left\{ 1 , \frac { 2 } { 3 } , 2 + 3 i , 2 - 3 i \right\}$.

Solve the Polynomial Equation 3x4−17x3+61x2−73x+26=03 x ^ { 4 } - 17 x ^ { 3 } + 61 x ^ { 2 } - 73 x + 26 = 03x4−17x3+61x2−73x+26=0

Solve the Polynomial Equation $3 x ^ { 4 } - 17 x ^ { 3 } + 61 x ^ { 2 } - 73 x + 26 = 0$