Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary.
-$$f ( x ) = x ^ { 4 } - 5 x ^ { 2 } - 36$$
A) $- 2 i , 2 i , 3 i , - 3 i$
B) $- 2 i , 2 i , 6 , - 6$
C) $- 2 i , 2 i , 3 , - 3$
D) $- 2 i , 2 i , 6 i , - 6 i$

Question

Quizplus · Accepted Answer

To find the complex zeros of the polynomial $f(x) = x^4 - 5x^2 - 36$, we can factor it as a quadratic in terms of $x^2$. Let $y = x^2$, then the equation becomes $y^2 - 5y - 36 = 0$. Factoring or using the quadratic formula gives $y = 9$ or $y = -4$. Substituting back $x^2$ for $y$, we get $x^2 = 9$ and $x^2 = -4$. Solving for $x$, we find $x = \pm 3$ from $x^2 = 9$ and $x = \pm 2i$ from $x^2 = -4$. Therefore, the complex zeros are $-2i, 2i, 3, -3$.

Find All Complex Zeros of the Polynomial Function f(x)=x4−5x2−36f ( x ) = x ^ { 4 } - 5 x ^ { 2 } - 36f(x)=x4−5x2−36

Find All Complex Zeros of the Polynomial Function $f ( x ) = x ^ { 4 } - 5 x ^ { 2 } - 36$