If a + b + c = 7 and ab + bc + ca = -6, then the value of `a^(3) + b^(3) + c^(3) - 3abc` is:

Options

A. a) 469
B. b) 472
C. c) 463
D. d) 479

Answer

To solve the problem, we need to find the value of \( a^3 + b^3 + c^3 - 3abc \) given the equations \( a + b + c = 7 \) and \( ab + bc + ca = -6 \).

### Step 1: Use the identity for \( a^3 + b^3 + c^3 - 3abc \)

We can use the identity:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]

### Step 2: Calculate \( a^2 + b^2 + c^2 \)

We know:
\[ a + b + c = 7 \]
Now, we will square both sides:
\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + ac + bc) \]
Substituting the known values:
\[ 7^2 = a^2 + b^2 + c^2 + 2(-6) \]
\[ 49 = a^2 + b^2 + c^2 - 12 \]
Now, solving for \( a^2 + b^2 + c^2 \):
\[ a^2 + b^2 + c^2 = 49 + 12 = 61 \]

### Step 3: Substitute into the identity

Now we can substitute \( a + b + c \) and \( a^2 + b^2 + c^2 \) into the identity:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]
We already have:
- \( a + b + c = 7 \)
- \( ab + ac + bc = -6 \)
- \( a^2 + b^2 + c^2 = 61 \)

Now calculate \( a^2 + b^2 + c^2 - ab - ac - bc \):
\[ a^2 + b^2 + c^2 - ab - ac - bc = 61 - (-6) = 61 + 6 = 67 \]

### Step 4: Final calculation

Now substitute these values back into the identity:
\[ a^3 + b^3 + c^3 - 3abc = 7 \times 67 = 469 \]

Thus, the final value is:
\[ \boxed{469} \]