Solve the polynomial equation. In order to obtain the first root, use synthetic division to test the possible rational
roots.
-$$x ^ { 3 } - 5 x ^ { 2 } + 6 x - 2 = 0$$
A) $\{ 1,2 + \sqrt { 2 } , 2 - \sqrt { 2 } \}$
B) $\{ 1 , - 1,2 \}$
C) $\{ - 1,4 + \sqrt { 2 } , 4 - \sqrt { 2 } \}$
D) $\{ 1,4 + \sqrt { 2 } , 4 - \sqrt { 2 } \}$

Question

Quizplus · Accepted Answer

The possible rational roots of the polynomial $x^3 - 5x^2 + 6x - 2 = 0$ are $\pm1, \pm2$. Using synthetic division to test these roots, we find that $x = 1$ is a root because substituting $x = 1$ into the polynomial gives $1 - 5 + 6 - 2 = 0$. After dividing the polynomial by $x - 1$, the quotient is $x^2 - 4x + 2$, which can be solved using the quadratic formula to find the other two roots, $2 + \sqrt{2}$ and $2 - \sqrt{2}$. Therefore, the roots of the polynomial are $\{1, 2 + \sqrt{2}, 2 - \sqrt{2}\}$.

Solve the Polynomial Equation x3−5x2+6x−2=0x ^ { 3 } - 5 x ^ { 2 } + 6 x - 2 = 0x3−5x2+6x−2=0

Solve the Polynomial Equation $x ^ { 3 } - 5 x ^ { 2 } + 6 x - 2 = 0$