Find all zeros of the function and write the polynomial as a product of linear factors.
-$$f ( x ) = x ^ { 4 } + 25 x ^ { 2 } + 144$$
A) $f ( x ) = ( x + 3 i ) ( x - 3 i ) ( x + 4 i ) ( x - 4 i )$
B) $f ( x ) = ( x + i ) ( x - i ) ( x + 12 i ) ( x - 12 i )$
C) $f ( x ) = ( x + 3 i ) ^ { 2 } ( x + 4 i ) ^ { 2 }$
D) $f ( x ) = ( x + 3 + 4 i ) ^ { 2 } ( x + 3 - 4 i ) ^ { 2 }$

Question

Quizplus · Accepted Answer

To find the zeros of the function, we first recognize that the given polynomial can be factored as a quadratic in terms of $x^2$. Let $y = x^2$, then the equation becomes $y^2 + 25y + 144 = 0$. Factoring this quadratic equation, we get $(y + 9)(y + 16) = 0$, which gives us $y = -9$ and $y = -16$. Substituting back $x^2$ for $y$, we get $x^2 = -9$ and $x^2 = -16$, leading to the solutions $x = \pm 3i$ and $x = \pm 4i$. Therefore, the polynomial can be written as $f(x) = (x + 3i)(x - 3i)(x + 4i)(x - 4i)$.

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

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