#Algebra-3
If x^2 - x - 1 = 0 , then find the value of (x^8-1)/(x^3+x^5)
a)2
b)3
c)4
d)none
Solution:-
x^2 = x + 1
x^8 - 1 = (x^2)^4 - 1 = (x + 1)^4 - 1 = x^4 + 4x^3 + 6x^2 + 4x + 1 - 1 = x(x^2 + 2x + 2)(x + 2) = x(3x^2)(x^2 + 1)
x^3 + x^5 = x^3(x^2 + 1)
=> (x^8 - 1)/(x^3 + x^5) = x(3x^2)(x^2 + 1)/x^3(x^2 + 1) = 3
a)2
b)3
c)4
d)none
Solution:-
x^2 = x + 1
x^8 - 1 = (x^2)^4 - 1 = (x + 1)^4 - 1 = x^4 + 4x^3 + 6x^2 + 4x + 1 - 1 = x(x^2 + 2x + 2)(x + 2) = x(3x^2)(x^2 + 1)
x^3 + x^5 = x^3(x^2 + 1)
=> (x^8 - 1)/(x^3 + x^5) = x(3x^2)(x^2 + 1)/x^3(x^2 + 1) = 3
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