Question

Ifx^3+1/x^3=110,then x+1/x=

Answer 1

x³ + 1/x³  = 110

(x + 1/x)³ - 3(x + 1/x) = 110

x + 1/x = k let

k³ - 3k - 110 = 0 

k(k² - 3) = 110 

k(k² - 3) = 5(5² - 3)

so k = 5

There is a alternate method too:

Given x3+ 1/x3 =110
Recall that (x+1/x)3 = x3+ 1/x3 + 3(x+1/x)
Let (x+1/x)=a
⇒ a3 = 110 + 3a
⇒a3 − 3a −110=0
Put a=5
⇒ 53 − 3(5) −110
=125 −15 −110=0
Hence (a − 5) is a factor.
On dividing (a3 − 3a −110) with (a − 5) we get the quotient as (a2+5a+22)
∴ a3 − 3a −110 = (a − 5)(a2+5a+22) = 0
Hence (a − 5) = 0 and (a2+5a+22) ≠ 0
⇒(x+1/x) − 5=0
∴ (x+1/x) = 5

Hope This Helps :)