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The distributive property is a fundamental algebraic principle used especially in polynomial multiplication. It allows one to multiply a single term across a sum or a difference within parentheses. When you have an expression like \(a (b + c)\), you apply the distributive property by multiplying \(a\) with \(b\) and \(c\) separately. This results in the expression \(ab + ac\).

In the example exercise, we use this property to multiply each term in the first binomial \(3x - 9\) by each term in the second binomial \(2x + 1\). This operation generates individual products:

• expressing \(3x \cdot 2x = 6x^2\)
• expressing \(3x \cdot 1 = 3x\)
• expressing \(-9 \cdot 2x = -18x\)
• expressing \(-9 \cdot 1 = -9\)

The final outcome after applying the distributive property is the polynomial \(6x^2 + 3x - 18x - 9\). This is the crucial first step in solving and simplifying polynomial expressions.

Combining like terms is a technique used to simplify expressions by consolidating terms that have the same variable components. In a polynomial expression, like terms contain the same variable raised to the same power.

From our previous steps, we obtain the expression \(6x^2 + 3x - 18x - 9\). Here, the like terms involving the variable \(x\) are \(3x\) and \(-18x\). When you combine them, you simply add or subtract the coefficients. The process is as follows:

• expressing \(3x\) and \(-18x\)
• Combine: \(3x - 18x = -15x\)
• After combining, our expression becomes: \(6x^2 - 15x - 9\)

This step is essential to arrive at the simplest form of the polynomial. It not only makes the expression easier to handle but also readies it for further algebraic manipulation if needed.

Once all possible combinations of like terms have been made, the final step is to simplify the expression. A simplified expression is one that is concise and easy to interpret, containing no further possible reductions.

Taking our combined expression \(6x^2 - 15x - 9\), there are no more like terms to combine. The expression now consists of terms each with different powers: \(6x^2\) for the quadratic term, \(-15x\) for the linear term, and \(-9\) as the constant.

At this stage:

• Verify each term is fully simplified and distinct
• Ensure no further distributive actions or combinations can be made
• Look for any opportunities to factor further, though in this case, none are applicable beyond the simplest form

By following these steps, you preserve the integrity of the expression while ensuring it is in its most manageable form. This thorough simplification is key to effectively communicating and solving polynomial equations.