Question

Find \(p^3+q^3+r^3\)

Answer 1

\(\large \color{black}{\begin{align} \text{if}\ p,q,r\ \text{are roots of }\hspace{.33em}\\~\\
x^3-2x^2+x-1=0 \hspace{.33em}\\~\\
\text{Find}\ p^3+q^3+r^3
\end{align}}\)

Answer 2

This is a fun problem if you know vieta's formulas

Answer 3

But here is an alternative :

\[x^3 = 2x^2-x+1\]
Since \(p,q,r\) are roots of the polynomial, we have
\[p^3 = 2p^2-p+1\]
\[q^3 = 2q^2-q+1\]
\[r^3 = 2r^2-r+1\]

Answer 4

add them up and get
\[p^3+q^3+r^3 = 2(p^2+q^2+r^2)-(p+q+r)+3\]

Answer 5

From the sum of roots, \(p+q+r = 2\)
product of pairs of roots, \(pq+qr+rp = 1\)

using the formula \((p+q+r)^2 = p^2+q^2+r^2 +2(pq+qr+rp)\),
\(p^2+q^2+r^2 = 2^2-2*1 = 2\)

plug them in

Answer 6

brilliant