The number of real solutions of x^7+5x³ +3x+1=0 is

Key Ideas to Crack the Problem

1. Odd-degree Polynomial: A polynomial of odd degree (here degree $7$) must go to $-\infty$ on one side and $+\infty$ on the other. Hence it must cross the $x$-axis at least once.

2. Monotonic Behaviour via Derivative:

• Every term in $f'(x)$ is non-negative for all real $x$.
• In fact, the constant $3$ makes $f'(x)$ strictly positive ($>0$) everywhere.
• Therefore $f(x)$ is strictly increasing. A strictly increasing function cannot turn back – it hits any $y$-value at most once.

3. Combining 1 & 2:
   • Because $f(x)$ starts from $-\infty$ (as $x \to -\infty$) and rises to $+\infty$ (as $x \to +\infty$) without ever slipping down, it must cross $0$ exactly once.

Hence, number of real roots = 1.

🚂 Imagine a Train Track!

1. Think of our equation $x^7 + 5x^3 + 3x + 1 = 0$ as a long train track.
2. The train (the graph) starts way down in the valley on the left (because $x^7$ is negative when $x$ is a big negative number).
3. It keeps climbing up and up with no dips at all because the slope (speed) is always positive.
4. Finally, when the train goes far to the right, it is very high in the sky (because $x^7$ is very positive).
5. A track that only climbs and never comes down can cross the ground level (the $x$-axis) exactly once.

So, there is only one place where the train touches the ground: one real solution.

Step-by-Step Solution

1. Write the function:
   $f(x) = x^7 + 5x^3 + 3x + 1.$

2. Compute its derivative:
   $f'(x) = 7x^6 + 15x^2 + 3.$

3. Analyse the derivative:
   • For any real $x$, $7x^6 \ge 0$, $15x^2 \ge 0$, and the constant $3 > 0$.
   • Therefore $f'(x) > 0$ for all $x \in \mathbb{R}$.
   • Hence $f(x)$ is strictly increasing everywhere.

4. Endpoint behaviour (degree consideration):
   • As $x \to -\infty$, the leading term $x^7 \to -\infty$ so $f(x) \to -\infty$.
   • As $x \to +\infty$, $x^7 \to +\infty$ so $f(x) \to +\infty$.

5. Conclusion:
   • A strictly increasing function that goes from $-\infty$ to $+\infty$ must cut the $x$-axis exactly once.

6. Final Answer:
   $\boxed{1}$