## mathmath333 one year ago Find $$p^3+q^3+r^3$$

1. mathmath333

\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}}

2. ganeshie8

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

3. ganeshie8

But here is an alternative : $x^3 = 2x^2-x+1$ Since $$p,q,r$$ are roots of the polynomial, we have $p^3 = 2p^2-p+1$ $q^3 = 2q^2-q+1$ $r^3 = 2r^2-r+1$

4. ganeshie8

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

5. ganeshie8

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

6. mathmath333

brilliant