Question

What is the nature of the roots of the equation \[{\left( {x - 3} \right)^9} + {\left( {x - {3^2}} \right)^9} + \cdots + {\left( {x - {3^9}} \right)^9} = 0\]?
A. All the roots real
B. One real and eight imaginary roots
C. Real root necessary \[x = 3,{3^2}, \cdots ,{3^9}\]
D. None of these

Answer 1

Hint: First we will find the first-order derivative. Then check whether the function increasing or decreasing. From this, we will decide the nature of the roots of the equation.

Formula used:
Power rule: \[\dfrac{d}{{dx}}{x^n} = n{x^{n - 1}}\]
The highest power of the variable of function is the number of roots of the function.
If the first-order derivative of a function is greater than zero then the function is an increasing function.

Complete step by step solution:
Given function is \[{\left( {x - 3} \right)^9} + {\left( {x - {3^2}} \right)^9} + \cdots + {\left( {x - {3^9}} \right)^9} = 0\].
Assume that, \[f\left( x \right) = {\left( {x - 3} \right)^9} + {\left( {x - {3^2}} \right)^9} + \cdots + {\left( {x - {3^9}} \right)^9}\]
Find derivative of the function:
\[f'\left( x \right) = \dfrac{d}{{dx}}\left[ {{{\left( {x - 3} \right)}^9} + {{\left( {x - {3^2}} \right)}^9} + \cdots + {{\left( {x - {3^9}} \right)}^9}} \right]\]
\[ \Rightarrow f'\left( x \right) = 9{\left( {x - 3} \right)^8} + 9{\left( {x - {3^2}} \right)^8} + \cdots + 9{\left( {x - {3^9}} \right)^8}\]
Take common 9 from each term
\[ \Rightarrow f'\left( x \right) = 9\left[ {{{\left( {x - 3} \right)}^8} + {{\left( {x - {3^2}} \right)}^8} + \cdots + {{\left( {x - {3^9}} \right)}^8}} \right]\]
We know that, \[{\left( {x - a} \right)^n}\] is always positive if \[x,a \in R\] and \[n\] is an even number.
So, \[{\left( {x - 3} \right)^8}\] is positive. Since \[x \in R\].
So, \[{\left( {x - {3^2}} \right)^8}\] is positive. Since \[x \in R\].
Similarly, \[{\left( {x - {3^9}} \right)^8}\] is positive.
The sum of positive numbers is always a positive number.
Thus, \[9\left[ {{{\left( {x - 3} \right)}^8} + {{\left( {x - {3^2}} \right)}^8} + \cdots + {{\left( {x - {3^9}} \right)}^8}} \right] > 0\]
This implies \[f'\left( x \right) > 0\].
So, the function \[f\left( x \right)\] is increasing function.
The graph of the function \[f\left( x \right)\] always moves upward direction. Since it is an increasing graph.

Image: Graph of \[f\left( x \right)\]

The \[f\left( x \right)\] cuts \[x - \]axis at a point. So, the function has only one real root.
The degree of the function \[f\left( x \right)\] is 9. So, the number of roots of the function is 9.
Since the function has one real root so the rest are imaginary.
Hence the equation has one real root and 8 imaginary roots.
Hence option B is the correct option.

Note: Sometimes students equate each term of the equation \[{\left( {x - 3} \right)^9} + {\left( {x - {3^2}} \right)^9} + \cdots + {\left( {x - {3^9}} \right)^9} = 0\] and calculate the value of \[x\]. This is an incorrect procedure. Some terms of the equation may be positive and the rest may be negative because the power of each term is an odd number.
Here we should use the concept of increasing function.