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When simplifying algebraic expressions, the distributive property is a powerful tool that allows us to simplify expressions in which multiplication is distributed over addition or subtraction within parentheses. Mathematically, it's expressed as a(b + c) = ab + ac.

In the context of the exercise \( (4x - 5y) - (4x + 7y) \), applying the distributive property involves distributing the negative sign across the terms within the second parentheses. This action is similar to multiplying by -1, effectively changing each of the terms' signs. Hence, \( -4x \), which was originally positive, becomes negative, and \( -7y \), which was originally negative, becomes positive. Understanding this concept helps to avoid common mistakes when simplifying expressions involving subtraction.

For clarity, here's a breakdown:

• Before distribution: \(4x - 5y - (4x + 7y)\)
• After distribution: \(4x - 5y - 4x - 7y\)

After having applied the distributive property, the next step is to combine like terms. Terms are 'like' if they have the same variables raised to the same power. Here, our goal is to simplify the expression by combining terms involving \(x\)'s with \(x\)'s and terms with \(y\)'s with \(y\)'s.

In our problem, we have \(4x\) and \( -4x\) as like terms; when combined, they cancel each other out since they are equal and opposite. Then, we have \( -5y\) and \( -7y\). These terms are combined by adding their coefficients, which gives us \( -12y\).

To simplify, look for terms with the same variable:

• \(4x\) and \( -4x\) (like terms involving \(x\))
• \( -5y\) and \( -7y\) (like terms involving \(y\))

The simplified expression is simply \( -12y\), with all \(x\) terms eliminated due to their equal magnitude and opposite signs.

Finally, the last point of action in the given problem is to classify the polynomial by the number of terms. Polynomials can be monomials (one term), binomials (two terms), trinomials (three terms), or have multiple terms.

Considering the simplified form of our expression, \( -12y\), it consists of only one term, making it a monomial. Being familiar with these classifications is vital, as it helps in understanding the structure of equations and can simplify further operations like adding, subtracting, and factoring polynomials.

In summary, polynomials are categorized based on their term count as follows:

• Monomial: One term (e.g., \(5x^2\))
• Binomial: Two terms (e.g., \(3x + 4\))
• Trinomial: Three terms (e.g., \(x^2 - 4x + 4\))
• Polynomial with more terms (e.g., \(2x^3 - 3x^2 + x - 5\))