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Understanding the distributive property is essential for mastering algebra. This principle allows us to multiply a single term by each term within a parenthesis. Imagine the distributive property like a delivery truck dropping off packages to multiple houses; just as each house gets a package, each term within the parenthesis gets multiplied by the term outside.

For instance, in the binomial multiplication \( (3a-2b)(2a-7b) \), we 'distribute' the first term of the first binomial \(3a\) to both \(2a\) and \(7b\) in the second binomial, and then distribute \(2b\) in the same manner. The process looks like this: \(3a \times 2a\), \(3a \times -7b\), \(2b \times -2a\), and \(2b \times 7b\). This step unpacks the expression, setting it up for further simplification.

After using the distributive property, the next step is to combine like terms to simplify the expression. But what are 'like terms'? They are terms that have the exact same variables raised to the same powers. For instance, \(ab\) and \(5ab\) are like terms because they both contain the variable 'a' and 'b' with no exponents.

In our exercise, after distributing and multiplying, we find the terms \(6a^2\), \(21ab\), \(4ab\), and \(14b^2\). Here, \(21ab\) and \(4ab\) are like terms because they share the same variables 'a' and 'b'. We can combine them by adding their coefficients, \(21 + (-4)\), which simplifies to \(17\). Thus, we get the simplified expression \(6a^2 - 17ab + 14b^2\).

Algebraic expressions are the backbone of algebra and are composed of numbers, variables, and arithmetic operations. They can range from simple, like \(3x+ 2\), to more complex expressions, such as the result of binomial multiplication. With algebraic expressions, we can represent real-world problems in mathematical terms and find solutions by manipulating these expressions.

In the context of our example, \( (3a-2b)(2a-7b) \) starts as a product of two binomials and ends up as a simplified algebraic expression after applying the distributive property and combining like terms. Understanding how to manipulate these expressions is crucial to solving algebraic equations and will pave the way for grasping higher-level math concepts.