Question

find the zeros of the following function. f(x) = x³ - 6x² + 15x - 18 3, 3 ± √15 i; -3, (3 ± √15 i)/2; 3, (3 ± √15 i)/2

Explanation:

Step1: Test possible rational roots

Using Rational Root Theorem, possible roots are factors of 18 over factors of 1, so ±1, ±2, ±3, ±6, ±9, ±18. Test \(x = 3\):
\(f(3)=3^3 - 6(3)^2+15(3)-18 = 27 - 54 + 45 - 18 = 0\). So \(x - 3\) is a factor.

Step2: Perform polynomial division or factorization

Divide \(f(x)\) by \(x - 3\) or factor. Using synthetic division with root 3:
Coefficients: 1 | -6 | 15 | -18
Bring down 1. Multiply by 3: 3. Add to -6: -3.
Multiply -3 by 3: -9. Add to 15: 6.
Multiply 6 by 3: 18. Add to -18: 0.
Quotient: \(x^2 - 3x + 6\).

Step3: Solve quadratic equation

Solve \(x^2 - 3x + 6 = 0\) using quadratic formula \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -3\), \(c = 6\):
Discriminant: \((-3)^2 - 4(1)(6)=9 - 24=-15\).
Thus, \(x=\frac{3\pm\sqrt{-15}}{2}=\frac{3\pm\sqrt{15}i}{2}\).

Step4: Combine roots

Zeros are \(x = 3\) and \(x=\frac{3\pm\sqrt{15}i}{2}\).

Answer:

\(3, \frac{3\pm\sqrt{15}i}{2}\) (corresponding to the third option: \(3, \frac{3\pm\sqrt{15}i}{2}\))