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=3; 1 and 5i are zeros; f(-1) = -104

Verified step by step guidance

1. Identify the zeros of the polynomial. Since 1 and 5i are zeros and the coefficients are real, the conjugate of 5i, which is -5i, must also be a zero.
2. Write the polynomial in factored form using the zeros: $f(x)=a(x-1)(x-5i)(x+5i)$.
3. Simplify the factors involving complex numbers: $(x-5i)(x+5i)=x^2+25$.
4. Substitute back into the polynomial: $f(x)=a(x-1)(x^2+25)$.
5. Use the condition $f(-1)=-104$ to find the value of $a$. Substitute $x=-1$ into the polynomial and solve 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 dealing with a third-degree polynomial, which can have up to three real or complex roots.

Complex Zeros and Conjugate Pairs

In polynomial functions with real coefficients, complex zeros must occur in conjugate pairs. This means that if 5i is a zero, its conjugate -5i must also be a zero. Therefore, the polynomial will have the zeros 1, 5i, and -5i, which are essential for constructing the polynomial function.

Evaluating Polynomial Functions

To find a specific polynomial function that meets given conditions, we can use the zeros to construct the polynomial and then apply the conditions, such as f(-1) = -104. This involves substituting -1 into the polynomial and solving for any unknown coefficients to ensure the function meets the specified value.