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Question Number 57791 by Tawa1 last updated on 12/Apr/19

$$\mathrm{If}\mathrm{a}+\mathrm{b}+\mathrm{c}=1$$$${\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2=2$$$${\mathrm{a}}^3+{\mathrm{b}}^3+{\mathrm{c}}^3=3$$$$\mathrm{then}{\mathrm{a}}^5+{\mathrm{b}}^5+{\mathrm{c}}^5=?$$

Answered by naka3546 last updated on 12/Apr/19

$$6$$

Commented by Tawa1 last updated on 12/Apr/19

$$\mathrm{Please}\mathrm{workings}$$

Answered by MJS last updated on 12/Apr/19

$$\det \left[\begin{array}{cccc}a& b& c& 1\\ a^2& b^2& c^2& 2\\ a^3& b^3& c^3& 3\\ a^5& b^5& c^5& x\end{array}\right]=0$$$$\Rightarrow x=a^{2}bc+{ab}^{2}c+{abc}^2-4abc-2a^{2}b-2a^{2}c-2b^{2}c-2{ab}^2-2{ac}^2-2{bc}^2+3a^2+3b^2+3c^{2w}+3ab+3ac+3bc$$$$\left(1\right)c=1-a-b$$$$\left(2\right)b=\frac{-a+1+\sqrt{-3a^2+2a+3}}{2}$$$$\Rightarrow x=3a^3-3a^2-\frac{3}{2}a+\frac{11}{2}$$$$\left(3\right)3a^3-3a^2-\frac{3}{2}a-\frac{1}{2}=0$$$$\Rightarrow x=6$$

Commented by Tawa1 last updated on 12/Apr/19

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}$$

Commented by MJS last updated on 12/Apr/19

$$\mathrm{there}\mathrm{should}\mathrm{be}\mathrm{an}\mathrm{easier}\mathrm{method}\mathrm{but}\mathrm{I}\mathrm{cannot}$$$$\mathrm{remember}.$$$$\mathrm{anyway}\mathrm{this}\mathrm{is}\mathrm{better}\mathrm{than}\mathrm{solving}\mathrm{the}{3}^{\mathrm{rd}}$$$$\mathrm{degree}\mathrm{polynome}$$

Commented by Tawa1 last updated on 12/Apr/19

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}$$

Answered by naka3546 last updated on 12/Apr/19

Commented by Tawa1 last updated on 12/Apr/19

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.\mathrm{But}\mathrm{image}\mathrm{too}\mathrm{small}$$

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