Question

find the product of (x³+1/x³) * (x+1/x)

Answer 1

Answer:

If x³+ (1/x³) =2,what is the value of x+(1/x)?

We dont know whether x is a real or imaginary number satisfying this condition so let x be a complex no in the form e^ ia=cosa +isina i.e Euler form

So x^3 +1/x^3 will be e^3ia +(1/e^3ia) which is thus cos3a +isin3a +cos3a -isin3a =2cos3a.

=>2cos3a=2, thus cos3a=1 => 3a =2pi or a=2pi/3

Thus x =cos2pi/3 + sin2pi/3 =-1/2 +( i(3)^1/2)/2

which we know is w; cube root of unity.

Now as x=w, so x+1/x is w +1/w which is w+w^2=-1. Hence x +1/x=-1

as 1+w+w^2 =0 & w^2=1/w.

Apart from this x can also be 1 so answer can also be 2.