Answer:
1 - 3i, 4
Explanation:
The polynomial has real coefficients, so if it has a complex root, it must have at least two complex roots which are complex conjugates.
Since 1 + 3i is a root, then 1 - 3i is also a root.
Two of the factors of the polynomial are:
x - (1 + 3i) and x - (1 - 3i)
Simplify the factors above:
x - 1 - 3i and x - 1 + 3i
Find their product:
(x - 1 - 3i)(x - 1 + 3i) =
Rewrite them showing we have the product of a sum and a difference.
= [(x - 1) - 3i][(x - 1) + 3i]
Multiply the factors above noticing they are a sum and a difference which follows the pattern (a + b)(a - b) = a² - b².
= (x - 1)² - (3i)²
= x² - 2x + 1 - 9(-1)
= x² - 2x + 10
Now we divide the original polynomial by the product we just found using long division.
x - 4
------------------------------
x² - 2x + 10 | x³ - 6x² + 18x - 40
- x³ - 2x² + 10x
------------------------
-4x² + 8x - 40
- -4x² + 8x - 40
--------------------------
0 + 0 + 0
Now we know that x³ - 6x² + 18x - 40 factors into (x² - 2x + 10)(x - 4).
(x² - 2x + 10)(x - 4) = 0
From x² - 2x + 10, we have roots x = 1 + 3i and x = 1 - 3i.
x - 4 = 0
x = 4
From x - 4, we have root x = 4.
Answer: 1 - 3i, 4