Questions: Find a polynomial (f(x)) of degree 3 with real coefficients and the following zeros. [ 3,2 i f(x)= ]

Transcript text: Find a polynomial $f(x)$ of degree 3 with real coefficients and the following zeros. \[ \begin{array}{l} 3,2 i \\ f(x)= \end{array} \]

Solution

Solution Steps

To find a polynomial $ f(x) $ of degree 3 with real coefficients and given zeros, we need to consider the following steps:

Identify the given zeros: $ 3 $ and $ 2i $.
Since the polynomial has real coefficients, the complex zero $ 2i $ must have its conjugate $ -2i $ as another zero.
Construct the polynomial by multiplying the factors corresponding to these zeros: $ (x - 3) $, $ (x - 2i) $, and $ (x + 2i) $.
Expand the product to get the polynomial in standard form.

Step 1: Identify the Zeros

The given zeros of the polynomial are $ 3 $, $ 2i $, and its conjugate $ -2i $.

Step 2: Construct the Polynomial

The polynomial can be expressed as the product of its factors corresponding to the zeros: \[ f(x) = (x - 3)(x - 2i)(x + 2i) \]

Step 3: Expand the Polynomial

First, we simplify the product of the complex factors: \[ (x - 2i)(x + 2i) = x^2 + 4 \] Now, we can multiply this result by the remaining factor: \[ f(x) = (x - 3)(x^2 + 4) \] Expanding this gives: \[ f(x) = x^3 - 3x^2 + 4x - 12 \]