To find the complex solutions of the equation 2x^3 - 3x^2 + 32x + 17 = 0, we can use the method of factoring or synthetic division.
First, let's check if there are any rational solutions by applying the Rational Root Theorem. The Rational Root Theorem states that if a rational number p/q is a solution to the polynomial equation, then p must be a factor of the constant term (17 in this case), and q must be a factor of the leading coefficient (2 in this case).
By examining the factors of 17 (1, -1, 17, -17) and the factors of 2 (1, 2), we see that there are no rational solutions to the equation.
Now, let's proceed with factoring or synthetic division.
Factoring a cubic polynomial can be challenging, especially in cases where there are no obvious factors. Synthetic division, on the other hand, can help us simplify the equation by dividing it by a potential root and finding a quadratic equation to solve.
Let's try the synthetic division method with a possible rational root, such as x = 1.
1 | 2 -3 32 17
| 2 -1 31
-------------------
2 -1 31 48
The result of the synthetic division gives us a quadratic equation: 2x^2 - x + 31 = 0.
Now, let's solve the quadratic equation using the quadratic formula:
x = (-(-1) ± √((-1)^2 - 4(2)(31))) / (2(2))
= (1 ± √(1 - 248)) / 4
= (1 ± √(-247)) / 4
Since we have a negative value inside the square root, we know that the solutions will be complex. To simplify further, we can rewrite √(-247) as √(247)i, where i is the imaginary unit (√(-1)).
Therefore, the solutions to the quadratic equation are:
x = (1 + √(247)i) / 4
x = (1 - √(247)i) / 4
Combining these solutions with the potential root x = 1, we have three complex solutions:
1, (1 + √(247)i) / 4, (1 - √(247)i) / 4