The number of distinct real roots of the equation $x^7 – 7x – 2 = 0$ is

Question

If $50 \left( \frac{2x}{1 + 3i} + \frac{y}{1 - 2i} \right) = 31 + 17i$ where $x, y \in R$ & $i = \sqrt{-1}$ then value of $10(x + 3y)$ is ________

Answer 1

To determine the number of distinct real roots of the equation $x^7 - 7x - 2 = 0$, we need to analyze the function $f(x) = x^7 - 7x - 2$.

First, let's consider the nature of the polynomial:
- The function $f(x) = x^7 - 7x - 2$ is a polynomial of degree 7.
- Since $x^7$ is the highest power term, which is raised to an odd degree, the function will exhibit the following end behavior:
  - As $x \to \infty$, $f(x) \to \infty$
  - As $x \to -\infty$, $f(x) \to -\infty$
Next, analyze the derivative $f'(x)$ to check the turning points where the slope of the curve changes:
- Differentiating $f(x)$, we get: f'(x) = 7x^6 - 7
- Set the derivative to zero to find the critical points: 7x^6 - 7 = 0 \Rightarrow 7(x^6 - 1) = 0\Rightarrow x^6 = 1
- This gives $x = 1$ and $x = -1$ because these are the two real solutions of $x^6 = 1$.
Analyze the function values at critical points and other specific points to determine root behavior:
- Evaluate the function at the critical points and some additional points to determine changes in sign:
  - $f(-1) = (-1)^7 - 7(-1) - 2 = -1 + 7 - 2 = 4$
  - $f(0) = 0^7 - 7 \cdot 0 - 2 = -2$
  - $f(1) = 1^7 - 7 \cdot 1 - 2 = 1 - 7 - 2 = -8$
  - $f(2) = 2^7 - 7 \cdot 2 - 2 = 128 - 14 - 2 = 112$
- Notice the sign changes from positive to negative and vice versa over the intervals between these points, which implies sign changes and hence real roots between:
  - Between $(-1, 0)$: $f(x)$ changes from positive to negative, indicating a root.
  - Between $ (0, 1)$: $ f(x) $ changes from negative to negative with a decrease, which aligns with critical point behavior.
  - Between $ (1, 2) $: $ f(x) $ changes from negative to positive, indicating another root.
Conclusion:
- We observe that there are three sign changes, indicating three distinct real roots.

Based on the analysis, the equation $x^7 - 7x - 2 = 0$ has 3 distinct real roots. Thus, the correct answer is 3.

Answer 2

To determine the number of distinct real roots of the equation $x^7 - 7x - 2 = 0$, we need to analyze the function $f(x) = x^7 - 7x - 2$.

First, let's consider the nature of the polynomial:
- The function $f(x) = x^7 - 7x - 2$ is a polynomial of degree 7.
- Since $x^7$ is the highest power term, which is raised to an odd degree, the function will exhibit the following end behavior:
  - As $x \to \infty$, $f(x) \to \infty$
  - As $x \to -\infty$, $f(x) \to -\infty$
Next, analyze the derivative $f'(x)$ to check the turning points where the slope of the curve changes:
- Differentiating $f(x)$, we get: f'(x) = 7x^6 - 7
- Set the derivative to zero to find the critical points: 7x^6 - 7 = 0 \Rightarrow 7(x^6 - 1) = 0\Rightarrow x^6 = 1
- This gives $x = 1$ and $x = -1$ because these are the two real solutions of $x^6 = 1$.
Analyze the function values at critical points and other specific points to determine root behavior:
- Evaluate the function at the critical points and some additional points to determine changes in sign:
  - $f(-1) = (-1)^7 - 7(-1) - 2 = -1 + 7 - 2 = 4$
  - $f(0) = 0^7 - 7 \cdot 0 - 2 = -2$
  - $f(1) = 1^7 - 7 \cdot 1 - 2 = 1 - 7 - 2 = -8$
  - $f(2) = 2^7 - 7 \cdot 2 - 2 = 128 - 14 - 2 = 112$
- Notice the sign changes from positive to negative and vice versa over the intervals between these points, which implies sign changes and hence real roots between:
  - Between $(-1, 0)$: $f(x)$ changes from positive to negative, indicating a root.
  - Between $ (0, 1)$: $ f(x) $ changes from negative to negative with a decrease, which aligns with critical point behavior.
  - Between $ (1, 2) $: $ f(x) $ changes from negative to positive, indicating another root.
Conclusion:
- We observe that there are three sign changes, indicating three distinct real roots.

Based on the analysis, the equation $x^7 - 7x - 2 = 0$ has 3 distinct real roots. Thus, the correct answer is 3.

The number of distinct real roots of the equation \(x^7 – 7x – 2 = 0\) is

The Correct Option is D

Solution and Explanation

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