In Exercises 25–32, find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. n=4; -2, 5, and 3+2i are zeros; f(1) = -96

Verified step by step guidance

1. Identify the zeros of the polynomial: -2, 5, and 3+2i. Since the coefficients are real, the complex conjugate 3-2i is also a zero.
2. Write the polynomial in factored form using the zeros: \( f(x) = a(x + 2)(x - 5)(x - (3 + 2i))(x - (3 - 2i)) \).
3. Simplify the factors involving complex numbers: \((x - (3 + 2i))(x - (3 - 2i)) = (x - 3)^2 + 4\).
4. Expand the polynomial: \( f(x) = a(x + 2)(x - 5)((x - 3)^2 + 4) \).
5. Use the condition \( f(1) = -96 \) to find the value of \( a \) by substituting \( x = 1 \) into the expanded polynomial and solving for \( a \).

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The degree of the polynomial is determined by the highest power of the variable. In this case, we are looking for a fourth-degree polynomial, which means the highest exponent of the variable will be four.

Complex Conjugate Root Theorem

The Complex Conjugate Root Theorem states that if a polynomial has real coefficients, any non-real complex roots must occur in conjugate pairs. For the given zeros, since 3+2i is a root, its conjugate 3-2i must also be a root of the polynomial. This ensures that the polynomial remains a function with real coefficients.

Finding Polynomial from Zeros

To construct a polynomial from its zeros, one can use the fact that if r is a root, then (x - r) is a factor of the polynomial. For the given zeros -2, 5, 3+2i, and 3-2i, the polynomial can be expressed as f(x) = k(x + 2)(x - 5)(x - (3 + 2i))(x - (3 - 2i)), where k is a constant determined by additional conditions, such as f(1) = -96.