mathmath333
  • mathmath333
Find \(p^3+q^3+r^3\)
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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mathmath333
  • mathmath333
\(\large \color{black}{\begin{align} \text{if}\ p,q,r\ \text{are roots of }\hspace{.33em}\\~\\ x^3-2x^2+x-1=0 \hspace{.33em}\\~\\ \text{Find}\ p^3+q^3+r^3 \end{align}}\)
ganeshie8
  • ganeshie8
This is a fun problem if you know vieta's formulas
ganeshie8
  • ganeshie8
But here is an alternative : \[x^3 = 2x^2-x+1\] Since \(p,q,r\) are roots of the polynomial, we have \[p^3 = 2p^2-p+1\] \[q^3 = 2q^2-q+1\] \[r^3 = 2r^2-r+1\]

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ganeshie8
  • ganeshie8
add them up and get \[p^3+q^3+r^3 = 2(p^2+q^2+r^2)-(p+q+r)+3\]
ganeshie8
  • 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
mathmath333
  • mathmath333
brilliant

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