Final answer:
To factor the given polynomial with -3i as a zero, we consider that complex zeros come in conjugate pairs, so +3i is also a zero. After dividing the polynomial by (x + 3i)(x - 3i), we determine the quadratic factor and potentially apply the quadratic formula to find the remaining zeros, thus fully factoring the polynomial.
Step-by-step explanation:
To completely factor the polynomial x⁴ + 3x³ + 5x² + 27x - 36 into linear functions given that -3i is a zero, we must use the fact that complex zeros of polynomials with real coefficients come in conjugate pairs. This means that if -3i is a zero, so is its conjugate 3i. Therefore, the polynomial can be written as (x + 3i)(x - 3i) times a quadratic factor.
First, we'll find the quadratic factor by dividing the original polynomial by the product of the factors corresponding to the known zeros:
(x⁴ + 3x³ + 5x² + 27x - 36) ÷ ((x + 3i)(x - 3i))
The product (x + 3i)(x - 3i) simplifies to x² + 9, which is a difference of squares. Now, we can perform polynomial division or use a synthetic division approach to determine the quadratic factor. After factoring the quadratic expression, if possible, we can then express the original polynomial as a product of linear factors.
Let's assume the division gives us a quadratic factor, which we can represent as ax² + bx + c. We can then factor this quadratic expression further using the quadratic formula if it doesn't factor easily:
For the equation ax² + bx + c = 0,
The solution for x is given by:
x = (-b ± √(b² - 4ac)) / (2a)
Applying this to our quadratic factor yields the remaining zeros. Combining these with the known complex zeros gives us the fully factored form of the original polynomial.