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

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Identify the given zeros of the polynomial. Since the polynomial has real coefficients and 5i is a zero, its complex conjugate -5i must also be a zero. The zeros are therefore 1, 5i, and -5i.

Write the polynomial in factored form using the zeros: $f(x) = a(x - 1)(x - 5i)(x + 5i)$, where $a$ is a real number coefficient to be determined.

Simplify the factors involving complex zeros by multiplying $(x - 5i)(x + 5i)$, which equals $x^2 - (5i)^2 = x^2 - (-25) = x^2 + 25$.

Rewrite the polynomial as $f(x) = a(x - 1)(x^2 + 25)$.

Use the given function value $f(-1) = -104$ to find $a$ by substituting $x = -1$ into the polynomial and solving for $a$: $f(-1) = a(-1 - 1)((-1)^2 + 25) = a(-2)(1 + 25) = a(-2)(26) = -52a$. Set this equal to $-104$ and solve for $a$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Complex Conjugate Root Theorem

For polynomials with real coefficients, non-real complex zeros always come in conjugate pairs. Since 5i is a zero, its conjugate -5i must also be a zero. This ensures the polynomial has real coefficients and helps determine all roots.

Constructing Polynomials from Zeros

A polynomial can be formed by multiplying factors corresponding to its zeros. For zeros r, s, and t, the polynomial is f(x) = a(x - r)(x - s)(x - t), where a is a leading coefficient. Identifying all zeros allows building the polynomial expression.

Using Function Values to Find Leading Coefficient

Given a specific function value like f(-1) = -104, substitute x = -1 into the polynomial expression and solve for the leading coefficient a. This step ensures the polynomial satisfies all given conditions, including the value at a particular point.

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