Write the polynomial $x ^ { 4 } - 5 x ^ { 3 } + 4 x ^ { 2 } + x - 15$ in completely factored form. (Hint: One factor is $x ^ { 2 } - 2 x + 3$ .)
A) $( x - 1 + \sqrt { 2 } i ) ( x + 1 + \sqrt { 2 } i ) \left( x - \frac { 3 + \sqrt { 29 } } { 2 } \right) \left( x - \frac { 3 - \sqrt { 29 } } { 2 } \right)$
B) $( x - 1 + \sqrt { 2 } i ) ( x + 1 - \sqrt { 2 } i ) \left( x - \frac { 3 + \sqrt { 29 } } { 2 } \right) \left( x - \frac { 3 - \sqrt { 29 } } { 2 } \right)$
C) $( x - 1 + \sqrt { 2 } i ) ( x - 1 + \sqrt { 2 } i ) \left( x - \frac { 3 + \sqrt { 29 } } { 2 } \right) \left( x - \frac { 3 - \sqrt { 29 } } { 2 } \right)$
D) $( x + 1 + \sqrt { 2 i } ) ( x - 1 + \sqrt { 2 i } ) \left( x - \frac { 3 + \sqrt { 29 } } { 2 } \right) \left( x - \frac { 3 - \sqrt { 29 } } { 2 } \right)$
E) $( x - 1 + \sqrt { 2 } i ) ( x - 1 - \sqrt { 2 } i ) \left( x - \frac { 3 + \sqrt { 29 } } { 2 } \right) \left( x - \frac { 3 - \sqrt { 29 } } { 2 } \right)$

Question

Quizplus · Accepted Answer

Given the hint that one factor is $x^2 - 2x + 3$, we can find its roots using the quadratic formula, which are $x = 1 \pm \sqrt{2}i$. These roots correspond to the factors $(x - 1 - \sqrt{2}i)$ and $(x - 1 + \sqrt{2}i)$. To find the other factors, we can perform polynomial division or use the fact that a polynomial of degree 4 must have four roots (counting multiplicity). The remaining roots must then be real and can be found by dividing the original polynomial by $x^2 - 2x + 3$ and solving the resulting quadratic equation, which yields $\left(x - \frac{3 + \sqrt{29}}{2}\right)$ and $\left(x - \frac{3 - \sqrt{29}}{2}\right)$. Thus, the completely factored form includes both complex and real roots as indicated in option E.

Write the Polynomial x4−5x3+4x2+x−15x ^ { 4 } - 5 x ^ { 3 } + 4 x ^ { 2 } + x - 15x4−5x3+4x2+x−15

Write the Polynomial $x ^ { 4 } - 5 x ^ { 3 } + 4 x ^ { 2 } + x - 15$