Question

find the zeros of the following function.\\( f(x) = x^4 + 9x^3 + 33x^2 + 55x + 30 \\)\\( -3, -1, \frac{-5\pm\sqrt{15}i}{2} \\)\\( 3, 1, \frac{-5\pm\sqrt{15}i}{2} \\)\\( -3, -1, \frac{5\pm\sqrt{15}i}{2} \\)

Explanation:

Step1: Try Rational Roots

By Rational Root Theorem, possible roots are factors of 30 over factors of 1: ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30. Test \(x = -1\): \(f(-1)=1 - 9 + 33 - 55 + 30 = 0\). So \((x + 1)\) is a factor. Test \(x = -3\): \(f(-3)=81 - 243 + 297 - 165 + 30 = 0\). So \((x + 3)\) is a factor.

Step2: Perform Polynomial Division

Divide \(f(x)\) by \((x + 1)(x + 3)=x^2 + 4x + 3\). Using long division or synthetic division:
\(x^4 + 9x^3 + 33x^2 + 55x + 30=(x^2 + 4x + 3)(x^2 + 5x + 10)\)

Step3: Solve Quadratic

Solve \(x^2 + 5x + 10 = 0\) using quadratic formula \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = 5\), \(c = 10\).
\(x=\frac{-5\pm\sqrt{25 - 40}}{2}=\frac{-5\pm\sqrt{-15}}{2}=\frac{-5\pm\sqrt{15}i}{2}\)

Step4: Identify Zeros

From \((x + 1)\) and \((x + 3)\), roots are \(x=-1\), \(x=-3\). From quadratic, roots are \(\frac{-5\pm\sqrt{15}i}{2}\).

Answer:

\(-3\), \(-1\), \(\frac{-5\pm\sqrt{15}i}{2}\) (corresponding to the first option: \(-3\), \(-1\), \(\frac{-5\pm\sqrt{15}i}{2}\))