Question

The number of distinct real roots of the equation: \[ x^7 - 7x - 2 = 0 \] is:

• A. 5
• B. 7
• C. 1
• D. 3

Hint:

Use Descarte’s Rule of Signs to determine the number of positive and negative real roots of a polynomial.

Answer 1

The Correct Option is D

Step 1: Understanding the function. 
Define \[ f(x) = x^7 - 7x - 2 \] We need to find the number of real roots by analyzing the function's behavior. 
Step 2: Differentiating to find critical points. 
\[ f'(x) = 7x^6 - 7 \] Setting \( f'(x) = 0 \): \[ 7(x^6 - 1) = 0 \] \[ x^6 = 1 \Rightarrow x = \pm 1 \] 
Step 3: Evaluating function at critical points. 
\[ f(-1) = (-1)^7 - 7(-1) - 2 = -1 + 7 - 2 = 4 \] \[ f(1) = 1^7 - 7(1) - 2 = 1 - 7 - 2 = -8 \] Since \( f(-1)>0 \) and \( f(1)<0 \), by the Intermediate Value Theorem, there is at least one root between \( -1 \) and \( 1 \). 
Step 4: Checking overall behavior. 
- As \( x \to \infty \), \( f(x) \to \infty \). - As \( x \to -\infty \), \( f(x) \to -\infty \). By Descarte’s Rule of Signs: - Positive roots: \( x^7 - 7x - 2 = 0 \) has 1 sign change \( (x^7, -7x) \) implying 1 positive real root. - Negative roots: Substituting \( -x \), we analyze the transformed function: \[ (-x)^7 - 7(-x) - 2 = -x^7 + 7x - 2 \] which has two sign changes, implying 2 negative real roots. 
Step 5: Conclusion. 
Thus, the total number of real roots is \( 3 \). Thus, the correct answer is (D) \( 3 \).