Question: Given that $-7i$ is a zero, factor the following polynomial function completely. Use the Conjugate…

Given that $-7i$ is a zero, factor the following polynomial function completely. Use the Conjugate Roots Theorem, if applicable.

$$f(x) = x^4 - 11x^3 + 77x^2 - 539x + 1372$$

Answer:

$$f(x) =$$

Solution

Given that \(-7i\) is a zero, we can apply the Conjugate Roots Theorem, which tells us that \(7i\) is also a zero. The polynomial is \[ f(x) = x^4 + 11x^3 + 77x^2 - 539x + 1372 \] We’ll first factor using \((x + 7i)\) and \((x - 7i)\). First, compute the product: \[ (x + 7i)(x - 7i) = x^2 + 49 \] Now, divide \(f(x)\) by \(x^2 + 49\) to find the other factor. After performing polynomial division, you get: \[ f(x) = (x^2 + 49)(x^2 + 11x - 28) \] Next, factor the quadratic \(x^2 + 11x - 28\). The factors of \(x^2 + 11x - 28\) are: \[ (x + 14)(x - 2) \] Thus, the complete factorization of \(f(x)\) is: \[ f(x) = (x + 7i)(x - 7i)(x + 14)(x - 2) \] So, the completely factored form is: \[ f(x) = (x^2 + 49)(x + 14)(x - 2) \]