Given that `x^2-3x+1=0,` then the value of the expression `y=x^9+x^7+x^(-9)+x^(-7)` is divisible by prime number.

Answer

6621

Explanation

The given equation `x+(1)/(x)=3`
`:.x^(2)+(1)/(x^(2))=7`
`impliesx^(4)+(1)/(x^(4))=47`
`x^(8)+(1)/(x^(8))=(47)^(2)-1`
`:.x^(8)+x^(-8)=2207" "(1)`
Now `E=x^(9)+x^(7)+x^(-9)+x^(-7)`
`=x^(8)(x+(1)/(x))+x^(-9)+x^(-7)`
`=x^(8)(x+(1)/(x))+x^(-8)(x+(1)/(x))`
`=(xx+(1)/(x))(x^(8)+x^(-8))" "(2)`
Substitute the value of `x^(8)+x^(-8)=2207` from (1) and `x+(1)/(x)=3`
`E=(3)(2207)=6621`