Answer:
The roots (zeros) are:
2+3i
2-3i
-3
Step-by-step explanation:
First, if a 2 + 3i is the root, according to the complex conjugate root theorem, its complex conjugate is also a root.
So, the second root is 2 - 3i.
Now, let's find the third root.
To find this root, we have to factor the equation.
So, we will first relate the first root (2+3i) with the coefficients of the equation (1, -1, 1, 39)
2+3i -> 1 -1 +1 +39
Now, we have to follow the steps to solve the equation:
1) Copy the first exponent:
2+3i -> 1 -1 +1 +39
1
2) Multiply 1 by (2+3i) and write it below -1. Then sum the result with -1.
1*(2+3i) = 2+3i
-1 + 2 + 3i = 1 +3i
2+3i -> 1 -1 +1 +39
2+3i
1 1 + 3i
3) Keep doing the same.
Multiply (1+3i) by (2+3i) and write it below +1. Then sum the result with +1.
(1+3i)*(2+3i) = 2+3i+6i+9i² = 2 - 9 + 9i = -7 + 9i
Remeber i² = -1
-7 + 9i + 1 = -6 + 9i
2+3i -> 1 -1 +1 +39
2+3i -7+9i
1 1 + 3i -6+9i
4) Keep doing the same.
Multiply (-6+9i) by (2+3i) and write it below 39. Then sum the result with 39.
(-6+9i)*(2+3i) = -12 - 18i +18i +27i²
= -12 -27 = -39
-39+39 = 0
2+3i -> 1 -1 +1 +39
2+3i -7+9i -39
1 1 + 3i -6+9i 0
Now, we can use the coefficients 1, 1 +3i, and -6+9i and do the same using the root 2-3i:
2 - 3i -> 1 1 + 3i -6+9i 0
1) Copy the first exponent:
2 - 3i -> 1 1 + 3i -6+9i 0
1
2) Multiply 1 by (2-3i) and write it below 1 + 3i. Then sum the result with 1 + 3i.
1*(2-3i) = 2 - 3i
2 - 3i + 1 +3i = 3
2 - 3i -> 1 1 + 3i -6+9i 0
2 - 3i
1 3
3) Keep doing the same. Multiply 3 by (2 - 3i) and write it below -6 + 9i. Then sum the result with -6 + 9i.
3*(2-3i) = 6 - 9i
6-9i + 6 +9i = 0
2 - 3i -> 1 1 + 3i -6+9i 0
2 - 3i 6 - 9i
1 3 0
Finally, we can use the coefficients 1 and 3 to find the third root.
1x + 3 = 0
x = -3
So, -3 is the third root.
