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Polynomial functions are expressions that involve a sum of powers of a variable, usually denoted as \( x \) or \( y \). Each term in a polynomial is made up of a coefficient multiplied by a variable raised to a non-negative integer power. For example, in the polynomial function \( P(x) = x^2 + 4x + 5 \), the terms are \( x^2 \), \( 4x \), and \( 5 \). The coefficients here are 1, 4, and 5 respectively.

Key Points about Polynomial Functions:

• They can have one or more terms.
• The highest power of the variable is called the degree of the polynomial.
• Polynomial functions are continuous and smooth.

Recognizing a polynomial function and being able to identify its degree and coefficients is crucial for solving related problems.

Complex numbers extend the idea of real numbers by adding an imaginary component. A complex number is generally written in the form \( a + bi \), where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit satisfying \( i^2 = -1 \).

Characteristics of Complex Numbers:

• The real part (\( a \)) is the term without \( i \).
• The imaginary part (\( b \)) is the term with \( i \).

Understanding how to work with complex numbers is essential for evaluating polynomials with complex number inputs.

For instance, in the given problem, we have to evaluate \( P(x) \) at \( x = -2 - i \). We substitute \( -2 - i \) into the polynomial and simplify by applying properties of complex numbers, including \( i^2 = -1 \). Complex numbers are particularly useful in many areas of engineering and physics.

The substitution method involves replacing the variable in a polynomial function with a specific value, then simplifying to find the result. This technique is straightforward and works well with polynomial functions.

Steps to Use the Substitution Method:

• Identify the polynomial function and the value to substitute.
• Replace the variable in the polynomial with the given value.
• Perform the arithmetic operations, respecting order of operations (PEMDAS/BODMAS).

In the solution to \( P(-2 - i) \):

We substitute \( -2 - i \) for \( x \) in the polynomial function \( P(x) = x^2 + 4x + 5 \):

\ P(-2 - i) = (-2 - i)^2 + 4(-2 - i) + 5 \

By calculating each term and then combining them, we obtain the result. This process illustrates how easy it can be to evaluate a polynomial at any given point, even when dealing with complex numbers.