Use all available methods (in particular, the Conjugate Roots Theorm if applicable) to factor the polynomial function completely, making use of the given zero, f(x)=x^4+x^3+4x^2+196x-2205; -7i is a zero.
To factor the polynomial function using the Conjugate Roots Theorem, we first need to find all the other zeros.
Given that -7i is one zero, we know that its conjugate, 7i, must also be a zero. This is because complex zeros occur in conjugate pairs.
To find the other zeros, we can use polynomial division (synthetic or long division) to divide the given polynomial by the factors (x - (-7i)) and (x - 7i).
Let's perform synthetic division to find the factor (x - (-7i)):
-7i | 1 1 4 196 -2205
7i |_________________________
| 1 1 4 196 -2205
-7i 42i -280i 84i
|_________________________
1 -7i 46i -84i -2121
The remainder is -2121, which means (x - (-7i)) is a factor of the polynomial. Thus, the equation (x + 7i) = 0 gives us another zero.
Now let's perform synthetic division to find the factor (x - 7i):
7i | 1 -7i 46i -84i -2121
|___________________________
| 1 -7i 46i -84i -2121
7i 0 0 0 0
|___________________________
1 0 46i -84i -2121
The remainder is 0, which means (x - 7i) is also a factor of the polynomial.
Now we can rewrite the polynomial factored by the given zeros:
f(x) = (x - (-7i))(x - 7i)(a quadratic expression)
Expanding the first two factors gives:
f(x) = (x + 7i)(x - 7i)(a quadratic expression)
Using the difference of squares, we can further simplify this equation:
f(x) = (x^2 - (7i)^2)(a quadratic expression)
f(x) = (x^2 - 49i^2)(a quadratic expression)
Since i^2 is equal to -1, we can substitute -1 for i^2:
f(x) = (x^2 + 49)(a quadratic expression)
Therefore, the polynomial function f(x) = x^4 + x^3 + 4x^2 + 196x - 2205 factors completely as:
f(x) = (x + 7i)(x - 7i)(x^2 + 49)(a quadratic expression)