Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary.
-$$f ( x ) = x ^ { 4 } - 32 x ^ { 2 } - 144$$
A) $- 2 i , 2 i , 6 i , - 6 i$
B) $- 2 \mathrm { i } , 2 \mathrm { i } , 12 , - 12$
C) $- 2 i , 2 i , 6 , - 6$
D) $- 2 i , 2 i , 12 i , - 12 i$

Question

Quizplus · Accepted Answer

To find the complex zeros of the given polynomial $f(x) = x^4 - 32x^2 - 144$, we can factor it as follows: First, notice that the polynomial can be treated as a quadratic in form by substituting $y = x^2$, giving us $y^2 - 32y - 144$. Factoring or using the quadratic formula on this equation, we get:$$y^2 - 32y - 144 = 0$$Solving for $y$, we find the roots to be $y = 36$ and $y = -4$, by either factoring directly (if recognizable) or applying the quadratic formula where $a = 1$, $b = -32$, and $c = -144$.Returning to the original variable $x$, we substitute back $x^2 = y$ to find the roots of the original polynomial:1. For $x^2 = 36$, we get $x = \pm 6$.2. For $x^2 = -4$, we get $x = \pm 2i$.Thus, the exact complex zeros of the polynomial are $-2i$, $2i$, $6$, and $-6$.

Find All Complex Zeros of the Polynomial Function f(x)=x4−32x2−144f ( x ) = x ^ { 4 } - 32 x ^ { 2 } - 144f(x)=x4−32x2−144

Find All Complex Zeros of the Polynomial Function $f ( x ) = x ^ { 4 } - 32 x ^ { 2 } - 144$