To find the zeros of the polynomial function f(x) = x³ + 4x² + 5x – 20, we need to determine the values of x for which f(x) = 0. This involves either factoring the polynomial or using numerical methods such as synthetic division or the Rational Root Theorem to identify possible rational zeros.
By testing possible rational roots, we can find that x = 2 is a root. To confirm this, we can use synthetic division to divide the polynomial by (x – 2).
After doing the synthetic division, we obtain a quadratic equation:
x² + 6x + 10 = 0
Next, we can use the quadratic formula to find the remaining zeros:
x = [-b ± √(b² – 4ac)] / 2a
Here, a = 1, b = 6, and c = 10. Plugging these values into the quadratic formula gives us:
x = [-6 ± √(36 – 40)] / 2
Since the discriminant (36 – 40) is negative, the quadratic has no real solutions, but it has two complex solutions:
x = -3 ± i
Thus, the zeros of f(x) = x³ + 4x² + 5x – 20 are:
- x = 2 (real zero)
- x = -3 + i (complex zero)
- x = -3 – i (complex zero)
In summary, the zeros of the polynomial are one real and two complex, represented by 2, -3 + i, and -3 – i.