The roots of the polynomial equation x3 + 2x2 + x – 18 = 0 can be found using various methods such as factoring, synthetic division, or numerical methods.
To find the roots, we can start by trying to factor the polynomial or use the Rational Root Theorem to find possible rational roots. Testing for possible integer roots, we find that x = 2 is a root of the equation.
After finding that x = 2 is a root, we can perform synthetic division to simplify the polynomial:
- Divide the polynomial by (x – 2).
This simplifies to:
- x3 + 2x2 + x – 18 = (x – 2)(x2 + 4x + 9)
Now, we need to find the roots of the quadratic equation x2 + 4x + 9 = 0 using the quadratic formula:
- x = [-b ± √(b² – 4ac)] / 2a
- Here, a = 1, b = 4, and c = 9.
Calculating the discriminant:
- b² – 4ac = 4² – 4(1)(9) = 16 – 36 = -20
Since the discriminant is negative, the quadratic equation has complex roots:
- x = [-4 ± √(-20)] / 2(1)
- x = -2 ± i√5
In summary, the roots of the polynomial equation x3 + 2x2 + x – 18 = 0 are:
- x = 2
- x = -2 + i√5
- x = -2 – i√5