Question

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;
4 and 2i are zeros;
f (1)=30

f(x) = \square
(type an expression using x as the variable. simplify your answer.)

Explanation:

Step1: Determine all zeros

Since the polynomial has real coefficients and \(2i\) is a zero, its conjugate \(-2i\) must also be a zero. The zeros are \(4\), \(2i\), and \(-2i\).

Step2: Write the polynomial in factored form

A polynomial with these zeros can be written as \(f(x)=a(x - 4)(x - 2i)(x + 2i)\), where \(a\) is a real coefficient.

Step3: Simplify the complex factors

First, simplify \((x - 2i)(x + 2i)\) using the difference of squares formula \((a - b)(a + b)=a^2 - b^2\). Here, \(a = x\) and \(b = 2i\), so \((x - 2i)(x + 2i)=x^2-(2i)^2=x^2 - 4i^2\). Since \(i^2=-1\), this becomes \(x^2+4\).

Step4: Multiply the remaining factors

Now, \(f(x)=a(x - 4)(x^2 + 4)\). Expand \((x - 4)(x^2 + 4)\):

$$
LATEXBLOCK0
$$

So, \(f(x)=a(x^3-4x^2+4x - 16)\).

Step5: Determine the value of \(a\)

We know that \(f(1) = 30\). Substitute \(x = 1\) into the function:

$$
LATEXBLOCK1
$$

Solve for \(a\): \(a=\frac{30}{-15}=-2\).

Step6: Write the final polynomial

Substitute \(a=-2\) into \(f(x)=a(x^3-4x^2+4x - 16)\):

$$
LATEXBLOCK2
$$

Answer:

\(-2x^3 + 8x^2 - 8x + 32\)