The number of real solutions of x7+5x³ +3x+1=0 is

Key ideas you need

1. Odd-degree polynomial behavior
   • When the highest power of $x$ is odd (like $7$), the ends of the graph go in opposite directions: one end down, the other end up.
2. Intermediate Value Theorem (IVT)
   • If a continuous curve goes from negative to positive, it must cross zero at least once.
3. Monotonicity via derivative
   • If the derivative $f'(x)$ is always positive, the function keeps increasing—no bends downward—so it can cross the $x$-axis only once.

Applying these ideas step by step

1. Find $f(x)$ at extreme values
   • For very large positive $x$, $x^7$ dominates and is positive ⇒ $f(x) > 0$.
   • For very large negative $x$, $x^7$ dominates and is negative ⇒ $f(x) < 0$.
   • By IVT, at least one real root exists.
2. Check the derivative
   • $f(x)=x^7+5x^3+3x+1$
   • $f'(x)=7x^6+15x^2+3$
   • Each term $7x^6$, $15x^2$, and $3$ is non-negative for every real $x$ and the constant $3$ keeps it strictly > 0.
   • Therefore, $f'(x)>0$ for all real $x$ ⇒ $f(x)$ is strictly increasing.
3. Conclusion
   • A strictly increasing curve that changes sign only once gives exactly one real solution to $f(x)=0$.

Think of the polynomial like a roller-coaster track

• A polynomial is just a fancy curved line.
• The highest power ($x^7$) decides that the left side of the track will go way down (because $x^7$ is big negative when $x$ is negative) and the right side will go way up (because $x^7$ is big positive when $x$ is positive).
• If the track starts very low on the left and ends very high on the right and keeps climbing all the time, it must cross the ground (the $x$-axis) exactly one time.
• We check if the track keeps climbing by looking at its slope (the derivative). If the slope is always positive, the track never turns back down.
• For this particular track, the slope is always positive, so there is only one place where it touches the ground. That’s why the equation has one real solution.

Step-by-step solution

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

2. Check end behavior (sign analysis)

   • As $x\to\infty$, the term $x^7\to\infty$ ⇒ $f(x)\to\infty \; (>0)$
   • As $x\to-\infty$, the term $x^7\to-\infty$ ⇒ $f(x)\to-\infty \; (<0)$
   • Hence $f(x)$ changes sign between $-\infty$ and $\infty$, guaranteeing at least one real root.

3. Find the derivative to study monotonicity $f'(x)=\frac{d}{dx}\left(x^7+5x^3+3x+1\right)=7x^6+15x^2+3$

4. Show the derivative is always positive

   • $7x^6\ge0$ for all $x$
   • $15x^2\ge0$ for all $x$
   • Constant $+3>0$
   • Therefore $f'(x)=7x^6+15x^2+3>0\quad \text{for every real }x$

5. Conclude monotonicity

   • Because $f'(x)>0$ everywhere, $f(x)$ is strictly increasing over the entire real line.

6. Determine number of real roots

   • A strictly increasing function can cross the $x$-axis at most once.
   • We already know it crosses at least once (from the sign change).
   • Hence it crosses exactly once.

7. Answer Number of real solutions = 1.