Here are the essential concepts you must grasp in order to answer the question correctly.
Complex Conjugate Root Theorem
This theorem states that if a polynomial has real coefficients, then any non-real complex roots must occur in conjugate pairs. For example, if 'i' is a root, then '-i' must also be a root. In this case, since 'i' and '3i' are given as zeros, their conjugates '-i' and '-3i' must also be included as zeros of the polynomial.
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Polynomial Function Construction
To construct a polynomial function from its roots, one can use the fact that if 'r' is a root, then '(x - r)' is a factor of the polynomial. For the given roots 'i', '-i', '3i', and '-3i', the polynomial can be expressed as the product of its factors: (x - i)(x + i)(x - 3i)(x + 3i), which simplifies to a polynomial of degree 4.
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Evaluating Polynomial Functions
Evaluating a polynomial function at a specific point involves substituting the value into the polynomial expression. In this case, we need to ensure that the polynomial satisfies the condition f(-1) = 20. This requires adjusting the polynomial's leading coefficient or constant term after constructing it to meet the specified function value at x = -1.
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