To factor the cubic equation x³ + 3x² + 9x + 5, we first look for factors using the Rational Root Theorem. This theorem suggests checking for possible rational roots among the factors of the constant term and the leading coefficient. Here, the constant term is 5 and the leading coefficient is 1. The possible rational roots to test are: ±1, ±5.
1. **Testing x = -1:**
Substitute -1 into the equation:
-1³ + 3(-1)² + 9(-1) + 5 = -1 + 3 - 9 + 5 = -2. Not a root.
2. **Testing x = 1:**
Substitute 1 into the equation:
1³ + 3(1)² + 9(1) + 5 = 1 + 3 + 9 + 5 = 18. Not a root.
3. **Testing x = -5:**
Substitute -5 into the equation:
(-5)³ + 3(-5)² + 9(-5) + 5 = -125 + 75 - 45 + 5 = -90. Not a root.
4. **Testing x = 5:**
Substitute 5 into the equation:
(5)³ + 3(5)² + 9(5) + 5 = 125 + 75 + 45 + 5 = 250. Not a root.
Since none of the rational roots worked, we can apply synthetic division or general polynomial factoring techniques. However, cubic equations can also have complex roots. Next, we could use numerical methods or graphing to approximate where the roots lie. Let’s make our lives simpler by factoring it by grouping:
Grouping:
We group the first two terms and the last two terms:
- (x³ + 3x²) + (9x + 5)
Now, we can factor out what we can visibly see:
x²(x + 3) + 1(9x + 5) = x²(x + 3) + 1(9x + 5)
Final Result:
The polynomial does not factor nicely into rational numbers, indicating it may hold complex roots. It can be simplified further but will require numerical methods or advanced factoring (like by using cubic formulas) to express the final form.
In Conclusion:
The equation x³ + 3x² + 9x + 5 stands best left in its original states without easy rational factorization.