To factor the polynomial \( f(x) = x^3 - 3i x^2 - 9x + 27i \) into linear factors, follow the steps below:
1. Identify the known root: We know that \( 3i \) is a root of \( f(x) \). If \( 3i \) is a root, \( (x - 3i) \) is one of the factors of \( f(x) \).
2. Polynomial Division: To factor \( f(x) \), we need to divide \( f(x) \) by \( (x - 3i) \) using polynomial division. This will give us the other factor, which should be a quadratic polynomial.
3. Perform polynomial division:
Divide \( f(x) = x^3 - 3i x^2 - 9x + 27i \) by \( (x - 3i) \):
- Divide the leading term of \( x^3 \) by the leading term of \( x - 3i \), which is \( x^2 \).
- Multiply \( x^2 \) by \( x - 3i \) to get \( x^3 - 3i x^2 \).
- Subtract this from \( f(x) \) to obtain the new polynomial.
\( f(x) - x^3 + 3i x^2 = -9x + 27i \). This simplifies to \( -9(x - 3i) \).
So, \( f(x) \) can be written as \( (x - 3i)(x^2 - 9) \).
4. Factor the quadratic term: The quadratic term \( x^2 - 9 \) can be factored further:
[tex]\[ x^2 - 9 = (x - 3)(x + 3) \][/tex]
5. Write the final factors: Combining all the factors, we get:
[tex]\[ f(x) = (x - 3i)(x - 3)(x + 3) \][/tex]
Thus, the fully factored form of \( f(x) = x^3 - 3i x^2 - 9x + 27i \) into linear factors is:
[tex]\[ f(x) = (x - 3i)(x - 3)(x + 3) \][/tex]
