Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary.
-$$f ( x ) = x ^ { 3 } - 7 x ^ { 2 } + 15 x - 25$$
A) $5,1 + 2 \mathrm { i } , 1 - 2 \mathrm { i }$
B) $5 , - 1 + 2 \mathrm { i } , - 1 - 2 \mathrm { i }$
C) $5,1 + \sqrt { 5 } , 1 - \sqrt { 5 }$
D) $5,1 + \sqrt { 5 } i , 1 - \sqrt { 5 } i$

Question

Quizplus · Accepted Answer

To find the complex zeros of the given polynomial $f(x) = x^3 - 7x^2 + 15x - 25$, we can start by attempting to factor it or use the Rational Root Theorem to find a real root. By inspection or trial and error, we can see that $x = 5$ is a root because $f(5) = 5^3 - 7(5)^2 + 15(5) - 25 = 125 - 175 + 75 - 25 = 0$. After finding $x = 5$ as a root, we can perform polynomial division or use synthetic division to factor the polynomial as $(x - 5)(x^2 - 2x + 5)$. The quadratic formula can be used to find the roots of the quadratic factor $x^2 - 2x + 5$, which are $x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(5)}}{2(1)} = \frac{2 \pm \sqrt{4 - 20}}{2} = \frac{2 \pm \sqrt{-16}}{2} = 1 \pm 2i$.Therefore, the complex zeros of the polynomial are $5, 1 + 2i, 1 - 2i$.

Find All Complex Zeros of the Polynomial Function f(x)=x3−7x2+15x−25f ( x ) = x ^ { 3 } - 7 x ^ { 2 } + 15 x - 25f(x)=x3−7x2+15x−25

Find All Complex Zeros of the Polynomial Function $f ( x ) = x ^ { 3 } - 7 x ^ { 2 } + 15 x - 25$