if-x-2-y-2-p-x-3-y-3-q-find-x-n-y-n-in-terms-of-p-q-and-n-n-4-

Question Number 75778 by mr W last updated on 16/Dec/19

i f x^{2} + y^{2} = p, x^{3} + y^{3} = q,

f i n d x^{n} + y^{n} i n t e r m s o f p, q a n d n .

(n ⩾ 4)

Answered by mind is power last updated on 17/Dec/19

let e_{1} = x + y, e_{2} = xy

e_{k} = 0, \forall k ⩾ 3

p_{k} = x^{k} + y^{k}

\Rightarrow {\begin{cases} {2 e}_{2} = e_{1} . p_{1} - p_{2} \\ {3 e}_{3} = 0 = e_{2} p_{1} - e_{1} p_{2} + p_{3} \end{cases} \Leftrightarrow

{\begin{cases} {2 e}_{2} = p_{1}^{2} - p \\ 0 = e_{2} p_{1} - p_{1} p + q \end{cases}

\Rightarrow 0 = p_{1} (\frac{p_{1}^{2} - p}{2} - p) + q \Leftrightarrow p_{1}^{3} - {3 pp}_{1} + 2 q = 0

p_{1} Root of

X^{3} - 3 pX + 2 q cardan

p_{1} know \Rightarrow e_{2} know

{\begin{cases} x + y = p_{1} \\ xy = e_{2} \end{cases}

\Rightarrow x, y Root of X^{2} - p_{1} X + e_{2} lets say a, b

X^{n} + Y^{n} = a^{n} + b^{n}

Commented by mr W last updated on 18/Dec/19

t h a n k y o u s i r!

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