Question

The equation $(x-3{)}^9+(x-3^2{)}^9+..(x-3^9{)}^9=0$ has

• A. all real equation
• B. real root namely $3,3^2,.{.3}^9$
• C. one real & of imaginary roots
• D. None of these

Answer 1

The correct option is C one real & of imaginary roots

Given equation is

Let $f\left(x\right)=(x-3{)}^9+(x-3^2{)}^9+....+(x-3^9{)}^9=0$

$f^{\prime }\left(x\right)=9(x-3{)}^8+9(x-3^2{)}^8+.....+9(x-3^9{)}^8$

since $(x-3{)}^8>0$

so $f^{\prime }\left(x\right)>0$

SO $f\left(x\right)$ is increasing

So the Given equation only one and imaginary root.