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Polynomials are mathematical expressions that consist of variables and coefficients, forming terms through the use of addition, subtraction, and multiplication. Every term has a variable raised to a certain power, called its degree. The highest degree among the terms in a polynomial denotes the degree of the entire polynomial. For example, in the exercise provided, we have two polynomials:

• \(x^2 - 5x - 3\)
• \(6x^2 + 4x + 9\)

Each term in these polynomials is made up of a variable (in this case, \(x\)), a coefficient (such as 1 for \(x^2\), -5 for \(x\), and -3 for the constant), and a degree (2 for \(x^2\), 1 for \(x\), and 0 for constant terms). Recognizing the structure of a polynomial is key when performing operations like addition or subtraction.

To simplify polynomial operations, it is crucial to identify and group 'like terms'. Like terms are terms that share the same variable raised to the same power. This shared structure allows these terms to be added or subtracted directly. In the exercise example:

• \(x^2\) and \(6x^2\) are like terms because they are both \(x\) raised to the second power.
• \(-5x\) and \(4x\) are like terms because they both involve \(x\) to the first power.
• \(-3\) and \(9\) are constants, or like terms with a degree of zero.

When performing polynomial subtraction, aligning these like terms is essential. If polynomials are arranged vertically, it becomes easier to see which terms should be subtracted from each other. This step simplifies computing and leads to an accurate final result.

Step-by-step algebra involves systematically approaching algebraic expressions to solve them in manageable parts. Let's break down how the subtraction of polynomials is done in straightforward steps:

• **Identify Corresponding Terms:** Pinpoint like terms in each polynomial so you know which terms to subtract from each other. In this context, compare the \(x^2\), \(x\), and constants separately.
• **Subtract the x-squared Terms:** Take the first x-squared term and subtract the second: \(x^2 - 6x^2 = -5x^2\).
• **Subtract the x Terms:** Subtract the coefficients of the x terms: \(-5x - 4x = -9x\).
• **Subtract the Constants:** Finally, subtract the constants: \(-3 - 9 = -12\).

This logical method ensures that the operation is thorough and no terms are left out or mishandled. With practice, solving polynomial expressions like these becomes fast and intuitive.