Questions: Given that -4 i is a zero, factor the following polynomial function completely. Use the Conjugate Roots Theorem, if applicable. f(x)=x^4+x^3-14 x^2+16 x-480

Given that -4 i is a zero, factor the following polynomial function completely. Use the Conjugate Roots Theorem, if applicable.

f(x)=x^4+x^3-14 x^2+16 x-480

Transcript text: Given that $-4 i$ is a zero, factor the following polynomial function completely. Use the Conjugate Roots Theorem, if applicable. \[ f(x)=x^{4}+x^{3}-14 x^{2}+16 x-480 \]

Solution

Solution Steps

Step 1: Identify the Given Zero and Its Conjugate

We are given that $ -4i $ is a zero of the polynomial $ f(x) = x^4 + x^3 - 14x^2 + 16x - 480 $. By the Conjugate Roots Theorem, the conjugate $ 4i $ is also a zero.

Step 2: Form the Quadratic Factor

Using the zeros $ -4i $ and $ 4i $, we can form the quadratic factor: \[ (x - (-4i))(x - 4i) = (x + 4i)(x - 4i) = x^2 + 16 \]

Step 3: Divide the Original Polynomial

Next, we divide the original polynomial $ f(x) $ by the quadratic factor $ x^2 + 16 $. The result of this division gives us the remaining quadratic factor: \[ \frac{f(x)}{x^2 + 16} = x^2 + x - 30 \]

Step 4: Combine All Factors

Now we can express the complete factorization of the polynomial: \[ f(x) = (x^2 + 16)(x^2 + x - 30) \]