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write a polynomial function with integer coefficients that has zeros of 0, -7, 2 3i, 2-3i. Show all work.

User R Ubben
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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)

User Muhammad Shah
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