Question

Tính giá trị đa thức \(f_{\left(x\right)}=\left(x^4-3x+1\right)^{2016}\) tại \(x=9-\frac{1}{\sqrt{\frac{9}{4}-\sqrt{5}}}+\frac{1}{\sqrt{\frac{9}{4}+\sqrt{5}}}\)

Answer 1

\(x=9-\frac{1}{\sqrt{\frac{9}{4}-\sqrt{5}}}+\frac{1}{\sqrt{\frac{9}{4}+\sqrt{5}}}\)

\(=9-\frac{2}{\sqrt{9-4\sqrt{5}}}+\frac{2}{\sqrt{9+4\sqrt{5}}}\)

\(=9-\frac{2}{\sqrt{\left(\sqrt{5}-2\right)^2}}+\frac{2}{\sqrt{\left(\sqrt{5}+2\right)^2}}\)

\(=9-\frac{2}{\sqrt{5}-2}+\frac{2}{\sqrt{5}+2}\)

\(=9-\frac{4+2\sqrt{5}-2\sqrt{5}+4}{\left(\sqrt{5}-2\right)\left(\sqrt{5}+2\right)}\)

\(=9-\frac{8}{5-4}\)

= 1

\(f\left(x\right)=\left(1^4-3+1\right)^{2016}=1\)