f4 a.maths

Question

It is given x+(1/x)=1. Find the value of the following expressions

(a) (x^3)+(1/x^3),

(b)(x^5)+(1/x^5)

Answer 1

Note that x can be complex numbers. So the questions are valid provided that x is not real.

(a) (x^3) + (1/x^3):
(x + 1/x)^3 = x^3 + 3x + 3/x + 1/(x^3)
= x^3 + 1/(x^3) + 3(x + 1/x) = 1^3 = 1
=> x^3 + 1/(x^3) = 1-3*(x + 1/x)
= 1-3 = -2.

(b) (x^5) + (1/x^5):
(x + 1/x)^5 = x^5 + 5x^3 + 10x + 10/x + 5/(x^3) + 1/(x^5)
= x^5 + 1/(x^5) + 5(x^3 + 1/(x^3)) + 10(x + 1/x)
= x^5 + 1/(x^5) + 5(-2) + 10(1)
= x^5 + 1/(x^5)
Hence x^5 + 1/(x^5) = (x + 1/x)^5 = 1

Answer 2

It is impossible for x+(1/x)=1
If x is positive
If x<1, then 1/x>1, so x+1/x>1
if x>1, then 1/x<1, so x+1/x>1
If x is negative,
x+1/x<0
Please check the question.