Questions: Write a polynomial f(x) that satisfies the given conditions. Express the polynomial with the lowest possible leading positive integer coefficient. Polynomial of lowest degree with lowest possible integer coefficients, and with zeros 8-6i and 0 (multiplicity 3).

Write a polynomial f(x) that satisfies the given conditions. Express the polynomial with the lowest possible leading positive integer coefficient. Polynomial of lowest degree with lowest possible integer coefficients, and with zeros 8-6i and 0 (multiplicity 3).

Transcript text: Write a polynomial $f(x)$ that satisfies the given conditions. Express the polynomial with the lowest possible leading positive integer coefficient. Polynomial of lowest degree with lowest possible integer coefficients, and with zeros $8-6 i$ and 0 (multiplicity 3 ).

Solution

Solution Steps

Step 1: Identify the Zeros

The polynomial $ f(x) $ is required to have the following zeros:

$ 0 $ with multiplicity $ 3 $
$ 8 - 6i $ (and its conjugate $ 8 + 6i $ since the coefficients must be real)

Step 2: Construct the Polynomial

Using the identified zeros, we can express the polynomial as: \[ f(x) = (x - 0)^3 \cdot (x - (8 - 6i)) \cdot (x - (8 + 6i)) \] This simplifies to: \[ f(x) = x^3 \cdot (x - (8 - 6i)) \cdot (x - (8 + 6i)) \]

Step 3: Expand the Polynomial

Upon expanding the polynomial, we obtain: \[ f(x) = x^5 - 16x^4 + 100x^3 \]

Step 4: Factor the Polynomial

The polynomial can be factored as follows: \[ f(x) = x^3 \left( x^2 - 16x + 100 \right) \]