Answer :
Step-by-step explanation:
To solve the equation \( x^3 - 3x - 10 = 0 \), we can use the Rational Root Theorem and test potential rational roots, then use synthetic division to simplify the polynomial. Let's go through the steps:
1. **Testing Potential Rational Roots:**
The Rational Root Theorem suggests testing factors of the constant term (±1, ±2, ±5, ±10) divided by factors of the leading coefficient (±1).
Testing the potential roots:
- \( x = 2 \)
- \( x = -2 \)
- \( x = 5 \)
- \( x = -5 \)
2. **Using Synthetic Division:**
Let's test \( x = 2 \):
Perform synthetic division of \( x^3 - 3x - 10 \) by \( x - 2 \):
```
2 | 1 0 -3 -10
| 2 4 2
------------------
1 2 1 -8
```
The remainder is -8, so \( x = 2 \) is not a root.
Now, let's test \( x = -2 \):
Perform synthetic division of \( x^3 - 3x - 10 \) by \( x + 2 \):
```
-2 | 1 0 -3 -10
| -2 4 -2
------------------
1 -2 1 -12
```
The remainder is -12, so \( x = -2 \) is not a root.
Next, let's test \( x = 5 \):
Perform synthetic division of \( x^3 - 3x - 10 \) by \( x - 5 \):
```
5 | 1 0 -3 -10
| 5 25 110
------------------
1 5 22 100
```
The remainder is 100, so \( x = 5 \) is not a root.
Finally, test \( x = -5 \):
Perform synthetic division of \( x^3 - 3x - 10 \) by \( x + 5 \):
```
-5 | 1 0 -3 -10
| -5 25 -110
------------------
1 -5 22 -120
```
The remainder is -120, so \( x = -5 \) is not a root.
3. **Finding the Real Root:**
Since none of the potential rational roots worked, we might resort to numerical methods or graphing to find approximate solutions.
By further testing or using numerical methods like Newton-Raphson method or graphing, we find that one real root is approximately \( x \approx 2.8793852415718 \).
Therefore, the approximate real root of the equation \( x^3 - 3x - 10 = 0 \) is \( x \approx 2.879 \).