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mathmath333
 one year ago
Find \(p^3+q^3+r^3\)
mathmath333
 one year ago
Find \(p^3+q^3+r^3\)

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mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2\(\large \color{black}{\begin{align} \text{if}\ p,q,r\ \text{are roots of }\hspace{.33em}\\~\\ x^32x^2+x1=0 \hspace{.33em}\\~\\ \text{Find}\ p^3+q^3+r^3 \end{align}}\)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.7This is a fun problem if you know vieta's formulas

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.7But here is an alternative : \[x^3 = 2x^2x+1\] Since \(p,q,r\) are roots of the polynomial, we have \[p^3 = 2p^2p+1\] \[q^3 = 2q^2q+1\] \[r^3 = 2r^2r+1\]

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.7add them up and get \[p^3+q^3+r^3 = 2(p^2+q^2+r^2)(p+q+r)+3\]

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.7From the sum of roots, \(p+q+r = 2\) product of pairs of roots, \(pq+qr+rp = 1\) using the formula \((p+q+r)^2 = p^2+q^2+r^2 +2(pq+qr+rp)\), \(p^2+q^2+r^2 = 2^22*1 = 2\) plug them in
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