Question

If x cube + 1/x cube = 110 , find the value of x+1/x

Answer 1

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

Another simple method.

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 ANSWER