first find solution of the term x^2+x+1 then put the value in the given equation

Given:
The polynomial is 3x^3 + 8x^2 + 2x + 3k.

To find:
The value of k when x^2 + x + 1 is a factor of the polynomial.

Solution:

To determine the value of k, we need to check if x^2 + x + 1 is a factor of the given polynomial. If it is, then the polynomial can be expressed as the product of (x^2 + x + 1) and another factor.

Step 1: Divide the polynomial by x^2 + x + 1 using polynomial long division.

When we divide 3x^3 + 8x^2 + 2x + 3k by x^2 + x + 1, we get:

_______________________
x^2 + x + 1 | 3x^3 + 8x^2 + 2x + 3k

Performing the division, we get:

3x + 5
_______________________
x^2 + x + 1 | 3x^3 + 8x^2 + 2x + 3k
-(3x^3 + 3x^2 + 3x)
_______________________
5x^2 - x + 3k

Step 2: Set the remainder equal to zero.

For x^2 + x + 1 to be a factor of the polynomial, the remainder (5x^2 - x + 3k) must be equal to zero. So we have:

5x^2 - x + 3k = 0

Step 3: Solve for k.

We can solve the equation 5x^2 - x + 3k = 0 to find the value of k.

Let's solve it:

5x^2 - x + 3k = 0

Using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

Here, a = 5, b = -1, and c = 3k.

Substituting the values into the formula:

x = (-(-1) ± √((-1)^2 - 4(5)(3k))) / (2(5))
x = (1 ± √(1 - 60k)) / 10

For x^2 + x + 1 to be a factor, the quadratic equation must have complex roots, meaning the discriminant (b^2 - 4ac) must be negative. In this case, the discriminant is (1 - 60k). So we have:

1 - 60k < />

Solving for k:

60k > 1

k > 1/60

Therefore, the value of k is greater than 1/60.

Conclusion:

In conclusion, the value of k for which x^2 + x + 1 is a factor of the polynomial 3x^3 + 8x^2 + 2x + 3k is any value greater than 1/60.