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Question Number 154931 by qaz last updated on 23/Sep/21

$$\mathrm{f}\left(\mathrm{x}\right)={({2\mathrm{x}}^5+{2\mathrm{x}}^4-{53\mathrm{x}}^3-57\mathrm{x}+54)}^{2011}$$$$\mathrm{f}\left(\frac{\sqrt{111}-1}{2}\right)=?$$

Answered by MJS_new last updated on 23/Sep/21

$$-1$$

Commented by qaz last updated on 23/Sep/21

$$\mathrm{I}\mathrm{have}\mathrm{no}\mathrm{answer}$$

Answered by mindispower last updated on 23/Sep/21

$$x=\frac{\sqrt{111}-1}{2}$$$$\Rightarrow {(2x+1)}^2=111$$$$4x^2+4x-100=0$$$$x^2+x=25,x^2=25-x$$$$x^3=25x-x^2=26x-25$$$$x^4=26x^2-25x=26(25-x)-25x$$$$=-51x+650$$$$x^5=-51x^2+650x......$$$$$$$$$$

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