Find all complex zeros of the polynomial function, and write the polynomial in completely factored form.
-$$f ( x ) = x ^ { 4 } + 25 x ^ { 2 } + 144$$
A) $f ( x ) = ( x + 3 + 4 i ) ^ { 2 } ( x + 3 - 4 i ) ^ { 2 }$
B) $f ( x ) = ( x + 3 i ) ^ { 2 } ( x + 4 i ) ^ { 2 }$
C) $f ( x ) = ( x + 3 i ) ( x - 3 i ) ( x + 4 i ) ( x - 4 i )$
D) $f ( x ) = ( x + i ) ( x - i ) ( x + 12 i ) ( x - 12 i )$

Question

Quizplus · Accepted Answer

To find the complex zeros of the polynomial $f(x) = x^4 + 25x^2 + 144$, we can first treat it as a quadratic in form by substituting $y = x^2$, giving us $y^2 + 25y + 144$. Factoring this quadratic equation, we get $(y + 9)(y + 16) = 0$, which gives us the roots $y = -9$ and $y = -16$. Substituting back $x^2$ for $y$, we have $x^2 = -9$ and $x^2 = -16$, leading to the complex roots $x = \pm 3i$ and $x = \pm 4i$. Therefore, the polynomial can be factored as $f(x) = (x + 3i)(x - 3i)(x + 4i)(x - 4i)$.

Find All Complex Zeros of the Polynomial Function, and Write f(x)=x4+25x2+144f ( x ) = x ^ { 4 } + 25 x ^ { 2 } + 144f(x)=x4+25x2+144

Find All Complex Zeros of the Polynomial Function, and Write $f ( x ) = x ^ { 4 } + 25 x ^ { 2 } + 144$