Question

(x^3+x+1)/x^5,x=(√5 -1)/2

Answer 1

x=(√5 -1)/2
1/x=2/(√5 -1)
1/x=2(√5+1)/(√5 -1)(√5+1)
1/x=(√5 +1)/2

1+1/x=(√5 +3)/2
1-1/x=(√5 -1)/2

(x^3+x+1)/x^5,x=(√5 -1)/2
= 1/x^2+1/x^4+1/x^5
=1/x^2 *(1+1/x^2+1/x^3)
=1/x^2 *[(1+1/x)(1-1/x+1/x^2)+1/x^2]
=1/x^2 *[(1+1/x)(1-2/x+1/x^2)+1/x+1/x^2]
=1/x^2 *[(1+1/x)(1-1/x)^2+1/x+1/x^2]
=1/x^2 *[(1+1/x)(1-1/x)^2+1/x(1+1/x)]
=1/x^2 *{(1+1/x)(1-1/x)^2+1]}
=1/x^2 *{(√5 +3)/2*[(√5 -1)/2]^2+1}
=1/x^2 *{(√5 +3)/2*[(6-2√5)/4]+1}
=1/x^2 *{(√5 +3)/2*(3-√5)/2]+1}
=1/x^2 *{(9-5)/4+1}
=1/x^2 *2
=2/x^2
=2/[(√5 -1)/2]^2
=2/[(6-2√5)/4]
=2/[(3-√5)/2]
=2*2/(3-√5)
=4/(3-√5)
=4(3+√5)/(3-√5)(3+√5)
=4(3+√5)/4
=3+√5