Answer:
The polynomial function with integer coefficients that has zeros of 0, -7, 2+3i, and 2-3i is:
f(x) = x(x + 7)(x^2 - 4x + 13)
Explanation:
To find a polynomial function with integer coefficients that has zeros of 0, -7, 2+3i, and 2-3i, we can use the concept of complex conjugate roots.
Since 2+3i and 2-3i are complex conjugate roots, their product will result in a quadratic factor with real coefficients. The quadratic factor can be found by using the formula (x - (2+3i))(x - (2-3i)) = (x - 2 - 3i)(x - 2 + 3i).
Now, let's construct the polynomial function step by step:
1. Start with the factor (x - 0) for the zero 0: f(x) = x
2. Add the factor (x - (-7)) for the zero -7: f(x) = x(x + 7)
3. Add the quadratic factor (x - 2 - 3i)(x - 2 + 3i) for the complex conjugate zeros 2+3i and 2-3i:
f(x) = x(x + 7)(x - 2 - 3i)(x - 2 + 3i)
Simplifying further:
f(x) = x(x + 7)((x - 2)^2 - (3i)^2)
f(x) = x(x + 7)((x - 2)^2 - 9i^2)
f(x) = x(x + 7)((x - 2)^2 + 9)
Expanding the squared term:
f(x) = x(x + 7)(x^2 - 4x + 4 + 9)
f(x) = x(x + 7)(x^2 - 4x + 13)