Question

find the zeros of the following function.\\( f(x) = x^3 - 10x^2 + 44x - 69 \\)\\( -3, \frac{7\pm\sqrt{43}i}{2} \\) \\( 3, \frac{7\pm 43i}{2} \\) \\( 3, \frac{7\pm\sqrt{43}i}{2} \\)

Explanation:

Step1: Test possible rational roots

By Rational Root Theorem, possible rational roots are factors of 69 over factors of 1, i.e., ±1, ±3, ±23, ±69. Test \( x = 3 \):
\( f(3)=3^3 - 10\times3^2 + 44\times3 - 69 = 27 - 90 + 132 - 69 = 0 \). So \( (x - 3) \) is a factor.

Step2: Perform polynomial division or use synthetic division

Divide \( f(x) \) by \( (x - 3) \). Using synthetic division:
Coefficients: 1 | -10 | 44 | -69
Bring down 1. Multiply by 3: 3. Add to -10: -7.
Multiply -7 by 3: -21. Add to 44: 23.
Multiply 23 by 3: 69. Add to -69: 0.
So the quotient is \( x^2 - 7x + 23 \).

Step3: Solve the quadratic equation

For \( x^2 - 7x + 23 = 0 \), use quadratic formula \( x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -7 \), \( c = 23 \).
Discriminant \( \Delta = (-7)^2 - 4\times1\times23 = 49 - 92 = -43 \).
So \( x=\frac{7\pm\sqrt{-43}}{2}=\frac{7\pm\sqrt{43}i}{2} \).

Answer:

3, \(\frac{7\pm\sqrt{43}i}{2}\) (the third option)