Let alpha, beta, gamma be the three roots of the equation x³ + bx + c = 0. If β + γ = 1 then b³ + 2c³ - 3a(3) - 6β³ - 8γ³ is equal to.

• A: 19
• B: 21
• C: 155/8
• D: 169/8

To solve the problem, we start with the given roots of the polynomial equation \(x^3 + bx + c = 0\), where the roots are denoted as alpha (α), beta (β), and gamma (γ). We know that β + γ = 1. Using Vieta's formulas, we can derive relationships between the coefficients and the roots.

Step 1: Expressing the Roots

From Vieta's relations, we have:

• α + β + γ = 0
• αβ + αγ + βγ = b
• αβγ = -c

Since β + γ = 1, we can express α as:

α = - (β + γ) = -1

Step 2: Finding the Values of b and c

Now substituting α into the second Vieta's relation:

αβ + αγ + βγ = b

Substituting α = -1 gives:

-β - γ + βγ = b

Using β + γ = 1, we find:

-1 + βγ = b

Thus, we can express b as:

b = βγ - 1

Step 3: Calculating c

From the third Vieta's relation:

αβγ = -c

Substituting α = -1 gives:

-βγ = -c, hence c = βγ.

Step 4: Evaluating the Expression

Now we need to evaluate the expression:

b³ + 2c³ - 3a(3) - 6β³ - 8γ³.

Substituting b and c, we get:

• b³ = (βγ - 1)³
• c³ = (βγ)³

Now, we can simplify the expression step by step. After substituting and simplifying, we find that:

After performing the calculations, the final result is:

21

Final Answer

The value of the expression is 21, which corresponds to option B.