To factor the polynomial equation x³ + 3x² + 9x + 5, we can first try to find rational roots using the Rational Root Theorem. The possible rational roots can be the factors of the constant term (5) divided by the factors of the leading coefficient (1). The possible rational roots are ±1, ±5.
We can start testing these values by substituting them into the polynomial:
- For x = 1: 1³ + 3(1)² + 9(1) + 5 = 1 + 3 + 9 + 5 = 18 (not a root)
- For x = -1: (-1)³ + 3(-1)² + 9(-1) + 5 = -1 + 3 – 9 + 5 = -2 (not a root)
- For x = 5: 5³ + 3(5)² + 9(5) + 5 = 125 + 75 + 45 + 5 = 250 (not a root)
- For x = -5: (-5)³ + 3(-5)² + 9(-5) + 5 = -125 + 75 – 45 + 5 = -90 (not a root)
Since none of these values are roots, it’s likely that we cannot factor this polynomial using simple rational roots. So, we can use synthetic division or polynomial long division to check if it can be expressed as a product of polynomials of lower degree.
After various methods, if the polynomial does not neatly factor, we can conclude that either it is irreducible over the rational numbers, or we may look for numerical methods or approximations for roots if necessary.
In conclusion, the polynomial x³ + 3x² + 9x + 5 cannot be factored easily into rational factors, and its roots might need to be found using numerical methods.