Use the given root to find the solution set of the polynomial equation.

x^3 + 8x^2 -18x +20= 0 ; 1+i

If 1+i is a root, so is 1-i

So, (x-(1+i))(x-(1-i)) divides the cubic polynomial

(x^2-2x+2)(x+10) = 0
so, the roots are 1+i, 1-i, -10

To find the solution set of the given polynomial equation using the given root 1+i, we can use complex conjugate roots theorem.

Step 1: Find the other complex conjugate root.

Since 1+i is a root, its complex conjugate is 1-i.

Step 2: Write the factors of the equation using the roots.

The factors of the polynomial equation can be written as:
(x - (1+i))(x - (1-i))

Step 3: Expand the factors.

(x - (1+i))(x - (1-i)) = x^2 - x - ix + 1 - i - ix + i - i^2
(x - (1+i))(x - (1-i)) = x^2 - 2x + 2

Step 4: Divide the original polynomial equation by the factor.

Using long division or synthetic division, we divide the original polynomial:
(x^3 + 8x^2 - 18x + 20) / (x^2 - 2x + 2)

Step 5: Find the third root.

Solve the quadratic equation obtained in the previous step to find the third root:
x^2 - 2x + 2 = 0

Using the quadratic formula, we can solve for x:
x = (2 ± √(-4))/2
x = 1 ± i

Therefore, the solution set of the polynomial equation x^3 + 8x^2 - 18x + 20 = 0, given the root 1+i, is {1+i, 1-i, 1+i}.

To find the solution set of a polynomial equation using a specific root, you can use the method of synthetic division. Here are the steps to find the solution set of the given polynomial equation using the root 1+i:

Step 1: Write down the equation and the given root:
x^3 + 8x^2 - 18x + 20 = 0
Root: 1+i

Step 2: To use synthetic division, convert the given root into a binomial factor by changing its sign:
Root: 1+i
Binomial factor: (x - (1+i))

Step 3: Perform synthetic division using the binomial factor:
Using synthetic division, perform the following calculations:

| 1 8 -18 20
1+i | 1 8 -18 20
| - 1-i (1+i)(3-5i) -

The result after synthetic division will be the coefficients of the reduced polynomial:

1 7-i (1+i)(3-5i) (1+i)(-2+4i) + 20

Step 4: Write down the reduced polynomial equation:
The reduced polynomial equation is:
x^2 + (7 - i)x + (1+i)(3-5i)(-2+4i) + 20 = 0

Step 5: Solve the reduced polynomial equation:
To solve the reduced polynomial equation, you can use any appropriate method, such as factoring, completing the square, or using the quadratic formula.

Once you solve the equation, you will find the solution set for the original polynomial equation x^3 + 8x^2 - 18x + 20 = 0 using the given root 1+i.