If p and q are the roots of the equation `x^(2) - 30x + 221 = 0`. What is the value of `p^(3) + q^(3)`?

Options

A. 7010
B. 7110
C. 7210
D. 7240

Answer

To find the value of \( p^3 + q^3 \) where \( p \) and \( q \) are the roots of the quadratic equation \( x^2 - 30x + 221 = 0 \), we can follow these steps:

### Step 1: Identify coefficients
The given quadratic equation is in the form \( ax^2 + bx + c = 0 \). Here, we have:
- \( a = 1 \)
- \( b = -30 \)
- \( c = 221 \)

### Step 2: Calculate the sum and product of the roots
Using Vieta's formulas:
- The sum of the roots \( p + q = -\frac{b}{a} = -\frac{-30}{1} = 30 \)
- The product of the roots \( pq = \frac{c}{a} = \frac{221}{1} = 221 \)

### Step 3: Use the identity for \( p^3 + q^3 \)
We can use the identity:
\[ p^3 + q^3 = (p + q)(p^2 - pq + q^2) \]
To use this identity, we first need to calculate \( p^2 + q^2 \).

### Step 4: Calculate \( p^2 + q^2 \)
Using the square of the sum of the roots:
\[ p^2 + q^2 = (p + q)^2 - 2pq \]
Substituting the values we found:
\[ p^2 + q^2 = (30)^2 - 2 \cdot 221 = 900 - 442 = 458 \]

### Step 5: Substitute into the identity
Now we can substitute \( p^2 + q^2 \) into the identity:
\[ p^3 + q^3 = (p + q)((p^2 + q^2) - pq) = 30 \left( 458 - 221 \right) \]
Calculating inside the parentheses:
\[ 458 - 221 = 237 \]
Now substituting back:
\[ p^3 + q^3 = 30 \cdot 237 = 7110 \]

### Final Answer
Thus, the value of \( p^3 + q^3 \) is \( \boxed{7110} \).