Find a rational zero of the polynomial function and use it to find all the zeros of the function.
-$$f ( x ) = 2 x ^ { 4 } - 15 x ^ { 3 } + 45 x ^ { 2 } - 45 x + 13$$
A) $\left\{ 1 , \frac { 1 } { 2 } , 3 + 2 i , 3 - 2 i \right\}$
B) $\left\{ - 1 , - \frac { 1 } { 2 } , 2 + 3 i , 2 - 3 i \right\}$
C) $\left\{ 1 , - \frac { 1 } { 2 } , 2 + 3 i , 2 - 3 i \right\}$
D) $\left\{ - 1 , \frac { 1 } { 2 } , 3 + 2 i , 3 - 2 i \right\}$

Question

Quizplus · Accepted Answer

By Rational Root Theorem, the possible rational zeros are $\pm 1, \pm \frac{1}{2}$. By synthetic division, we find that $x = \frac{1}{2}$ is a zero. Using this zero, we can factor the polynomial as $$f(x) = 2(x - \frac{1}{2})(x^3 - 6x^2 + 18x - 13)$$The cubic factor can be further factored as $$x^3 - 6x^2 + 18x - 13 = (x - 3)^2 + 4$$Therefore, the zeros of the polynomial are $x = \frac{1}{2}, 3 \pm 2i$.

Find a Rational Zero of the Polynomial Function and Use f(x)=2x4−15x3+45x2−45x+13f ( x ) = 2 x ^ { 4 } - 15 x ^ { 3 } + 45 x ^ { 2 } - 45 x + 13f(x)=2x4−15x3+45x2−45x+13

Find a Rational Zero of the Polynomial Function and Use $f ( x ) = 2 x ^ { 4 } - 15 x ^ { 3 } + 45 x ^ { 2 } - 45 x + 13$