Question

questions 4 through 6 refer to the following.

directions:

• unless otherwise specified, the domain of a function \\(f\\) is assumed to be the set of all real numbers \\(x\\) for which \\(f(x)\\) is a real number.
• solutions to equations must be real numbers. determine the exact value of any expression that can be obtained without a calculator.
• unless otherwise specified, combine terms using algebraic methods and rules for exponents and logarithms, where applicable. for example, \\(2x + 3x\\), \\(5^2 \cdot 5^3\\), and \\(\frac{x^5}{x^2}\\) should be rewritten in equivalent forms.
• for each part of the question, show the work that leads to your answers.

part c

the function \\(m\\) is a polynomial function of degree 4. zeros of \\(m\\) include 0, 5, and \\(3 - 2i\\). write a possible expression for \\(m(x)\\).

Explanation:

Identify the given zeros and degree

The function \(m\) is a polynomial function of degree 4.
The given zeros are \(0\), \(5\), and \(3 - 2i\).

Apply the Complex Conjugate Root Theorem

Since the polynomial has real coefficients, any non-real complex zeros must occur in conjugate pairs.
The conjugate of \(3 - 2i\) is \(3 + 2i\).
Therefore, the four zeros of the degree 4 polynomial are:
\(x_1 = 0\), \(x_2 = 5\), \(x_3 = 3 - 2i\), and \(x_4 = 3 + 2i\).

Set up the factored form of the polynomial

Using the zeros, we can write the polynomial in its factored form:

$$m(x) = a(x - 0)(x - 5)(x - (3 - 2i))(x - (3 + 2i))$$

For a possible expression, we can choose the leading coefficient \(a = 1\):

$$m(x) = x(x - 5)(x - 3 + 2i)(x - 3 - 2i)$$

Simplify the complex factors

We simplify the product of the complex conjugate factors:

$$(x - (3 - 2i))(x - (3 + 2i)) = ((x - 3) + 2i)((x - 3) - 2i)$$

Applying the difference of squares:

$$((x - 3) + 2i)((x - 3) - 2i) = (x - 3)^2 - (2i)^2$$

$$= (x^2 - 6x + 9) - (-4)$$

$$= x^2 - 6x + 13$$

Write the final polynomial expression

Substitute the simplified quadratic factor back into the expression:

$$m(x) = x(x - 5)(x^2 - 6x + 13)$$

Expanding the first two terms:

$$m(x) = (x^2 - 5x)(x^2 - 6x + 13)$$

Either the factored form or the expanded form is a valid possible expression.

Answer:

\(m(x) = x(x - 5)(x^2 - 6x + 13)\)