Question

1. 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;

• 3 and 3 + 4i are zeros;

f (- 1) = 64
f(x) =
(type an expression using x as the variable. simplify your answer.)

Explanation:

Step1: Identify Conjugate Zero

For a polynomial with real coefficients, if \(3 + 4i\) is a zero, then its conjugate \(3 - 4i\) is also a zero.

Step2: Form Polynomial Factors

The polynomial of degree 3 with zeros \(-3\), \(3 + 4i\), and \(3 - 4i\) can be written as \(f(x)=a(x + 3)(x-(3 + 4i))(x-(3 - 4i))\).
First, simplify \((x-(3 + 4i))(x-(3 - 4i))\):

$$
LATEXBLOCK0
$$

So now \(f(x)=a(x + 3)(x^2 - 6x + 25)\).

Step3: Find the Leading Coefficient \(a\)

Use the given condition \(f(-1)=64\). Substitute \(x = -1\) into \(f(x)\):

$$
LATEXBLOCK1
$$

Step4: Expand the Polynomial

Now substitute \(a = 1\) into \(f(x)\) and expand:

$$
LATEXBLOCK2
$$

Answer:

\(x^3 - 3x^2 + 7x + 75\)