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 given zeros of the polynomial. Since the polynomial has real coefficients and one zero is 5i (an imaginary number), its conjugate -5i must also be a zero. So the zeros are 1, 5i, and -5i.
2. Write the polynomial in factored form using the zeros: $f\left(x\right)=\; a(x\; -\; 1)(x\; -\; 5i)(x\; +\; 5i)$, where $a$ is a real number coefficient to be determined.
3. Simplify the factors involving the imaginary zeros by multiplying $(x\; -\; 5i)(x\; +\; 5i)$. Recall that $(x\; -\; bi)(x\; +\; bi)\; =\; x^2\; +\; b^2$. So, $(x\; -\; 5i)(x\; +\; 5i)\; =\; x^2\; +\; 25$.
4. Rewrite the polynomial as $f(x)\; =\; a(x\; -\; 1)(x^2\; +\; 25)$. Now, use the given condition $f(-1)\; =\; -104$ to find the value of $a$. Substitute $x\; =\; -1$ into the polynomial and set it equal to -104.
5. Solve the equation $a(-1\; -\; 1)((-1)^2\; +\; 25)\; =\; -104$ for $a$. Once you find $a$, write the final polynomial function $f(x)\; =\; a(x\; -\; 1)(x^2\; +\; 25)$.

Key Concepts

Polynomial Zeros and Complex Conjugates

For polynomials with real coefficients, non-real zeros occur in conjugate pairs. Given a zero like 5i, its conjugate -5i must also be a zero. This ensures the polynomial remains with real coefficients when expanded.

Constructing a Polynomial 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 determined by additional conditions.

Using Function Values to Find Leading Coefficient

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