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$x^2-x-1=0$

$R(x)=\frac{{{x^{16}} - 1}}{{{x^8} + 2{x^7}}} = ?$

$R(x)=\frac{{{x^{16}} - 1}}{{{x^8} + 2{x^7}}} = \frac{{({x^2} - 1)({x^2} + 1)({x^4} + 1)({x^8} + 1)}}{{{x^7}(x + 2)}}$

$\because {x^2} - x - 1 = 0,{x^2} - 1 = x$

$R(x) = \frac{{x({x^2} + 1)({x^4} + 1)({x^8} + 1)}}{{{x^7}(x + 2)}} = \frac{{({x^2} + 1)({x^4} + 1)({x^8} + 1)}}{{{x^6}(x + 2)}}$

$\because {x^2} - x - 1 = 0,{x^2} = x + 1$

$R(x) = \frac{{(x + 1 + 1)({x^4} + 1)({x^8} + 1)}}{{{x^6}(x + 2)}} = \frac{{({x^4} + 1)({x^8} + 1)}}{{{x^6}}}$

${x^2} = x + 1,{x^4} = {(x + 1)^2} = {x^2} + 2x + 1$

${x^4} + 1 = {x^2} + 2x + 2 = {x^2} + 2(x + 1) = {x^2} + 2{x^2} = 3{x^2}$

${({x^4} + 1)^2} = 9{x^4},{x^8} + 2{x^4} + 1 = 9{x^4},{x^8} + 1 = 7{x^4}$

$R(x) = \frac{{({x^4} + 1)({x^8} + 1)}}{{{x^6}}} = \frac{{3{x^2}\times 7{x^4}}}{{{x^6}}} = 21$