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 zeros of -2 i and 4+6 i. f(x)=

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 zeros of -2 i and 4+6 i.

f(x)=

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 zeros of $-2 i$ and $4+6 i$. \[ f(x)= \] $\square$ \[ \square^{\square} \]

Solution

Solution Steps

Step 1: Identify the Zeros

The polynomial must have zeros at $-2i$ and $4 + 6i$. To ensure the polynomial has real coefficients, we also include the conjugate zeros $2i$ and $4 - 6i$.

Step 2: Form the Polynomial

The polynomial can be expressed as the product of its factors corresponding to the zeros: \[ f(x) = (x + 2i)(x - 2i)(x - (4 + 6i))(x - (4 - 6i)) \]

Step 3: Expand the Polynomial

Upon expanding the factors, we obtain: \[ f(x) = x^4 - 8x^3 + 56x^2 - 32x + 208 \]

Final Answer

The polynomial of the lowest degree with the lowest possible integer coefficients and the specified zeros is: \[ \boxed{f(x) = x^4 - 8x^3 + 56x^2 - 32x + 208} \]

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