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Polynomial functions are expressions that involve a sum of powers of variables with coefficients. These functions can vary in complexity, from quadratic functions like \(x^2\), to more intricate polynomials involving higher-degree terms. The general form of a polynomial function is written as:

• \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\)

where \(a_n, a_{n-1}, ..., a_0\) are constants called coefficients, and \(n\) is a non-negative integer that represents the degree of the polynomial.
In the context of our exercise, \(f(x) = 2x + 5\) is a linear polynomial as it is of degree 1, while \(g(x) = x^2 - 3x + 2\) is a quadratic polynomial of degree 2. Understanding the degree of a polynomial helps determine its shape and behavior on a graph. It also guides operations such as addition, subtraction, and multiplication of these functions.Working with polynomials involves understanding their individual terms and applying operations methodically. The key is to align terms based on their degrees during operations like addition and subtraction, as shown in function operations.

Subtraction of functions is a process that involves taking one function and subtracting every term of another function from it. When we subtract functions like in the exercise \(g(x) - f(x)\), we treat \(g(x)\) and \(f(x)\) as expressions that we place within parentheses and operate on them as whole units.
Here's how it works in practice:

• First, write the functions being subtracted clearly: \(g(x) = x^2 - 3x + 2\) and \(f(x) = 2x + 5\).
• Next, express the subtraction: \((x^2 - 3x + 2) - (2x + 5)\).
• Carefully distribute the negative sign across all terms of the function being subtracted, which means changing the sign of each term in \(f(x)\).

Executing such operations requires meticulous handling of each term's sign, ensuring that each subtraction is correct.
After distributing the negative sign, the expression transforms into \(x^2 - 3x + 2 - 2x - 5\). This is the crucial step before moving to combine like terms.

Combining like terms is an important step that simplifies polynomial expressions after performing operations like subtraction. Identifying and reducing like terms relies on matching variables and their exponents.
Here, after we expand the expression from subtraction as \(x^2 - 3x + 2 - 2x - 5\), we group and simplify like terms:

• Terms involving \(x^2\): \(x^2\)
• Terms with \(x\): \(-3x - 2x = -5x\)
• Constant terms: \(2 - 5 = -3\)

The goal is to consolidate terms to achieve a simpler form. Bringing together similar terms simplifies the expression to \(x^2 - 5x - 3\).
In math, combining like terms is essential for simplifying expressions, making them easier to interpret and use in further calculations or real-world applications. Through careful attention to the coefficients of like terms, we achieve a polished, precise answer.