If p and q are the roots of the equation x² – 30x + 221 = 0, what is the value of p³ + q³?

Concept:

If α and β are the roots of the quadratic equation, ax² + bx + c = 0. Then

\(\alpha + \beta = \; - \frac{b}{a}\;and\;\alpha \times \beta = \frac{c}{a}\)

Calculation:

Given: p and q are the roots of the equation x² – 30x + 221

By comparing the given equation with the standard quadratic equation ax² + bx + c = 0, we get a = 1, b = - 30 and c = 221.

As we know that, if α and β are the roots of the quadratic equation, ax² + bx + c = 0. Then

\(\alpha + \beta = \; - \frac{b}{a}\;and\;\alpha \times \beta = \frac{c}{a}\)

\(\Rightarrow p + q = \;\;30\;and\;pq = 221\)

\(\Rightarrow {p^3} + {q^3} = \left( {p + q} \right) \times \left( {{p^2} - pq + {q^2}} \right) = \;\left( {p + q} \right) \times \left[ {{{\left( {p + q} \right)}^2} - 3pq} \right]\)\(\Rightarrow {p^3} + {q^3} = \;\left( {p + q} \right) \times \left[ {{{\left( {p + q} \right)}^2} - 3pq} \right] = 30 \times \left[ {900 - 663} \right] = 7110\)