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The distributive property is a fundamental concept in algebra that allows us to simplify expressions and solve equations. It states that for any numbers or algebraic expressions, if you multiply a sum by a number, you can distribute the multiplication to each term inside the sum. For example, in the expression \[(a + b)c\], you would distribute \(c\) to both \(a\) and \(b\), resulting in \((a \cdot c + b \cdot c)\). In our exercise, we used the distributive property to expand \((d + 3)(d + 6)\). This involves multiplying each term in the first binomial by each term in the second binomial. Remember that this step helps break down more complex multiplication into simpler, more manageable parts.

The FOIL method is a specific application of the distributive property used for multiplying two binomials. FOIL is an acronym that stands for: First, Outer, Inner, and Last, representing the terms we need to multiply together. Let's see how this works with our original expression:

• First: Multiply the first terms in each binomial: \(d \cdot d = d^2\)


• Outer: Multiply the outer terms in the binomials: \(d \cdot 6 = 6d\)


• Inner: Multiply the inner terms: \(3 \cdot d = 3d\)


• Last: Multiply the last terms: \(3 \cdot 6 = 18\)

Combine all these products to get \(d^2 + 6d + 3d + 18\). The FOIL method breaks down the multiplication process into four simpler steps, making it more intuitive to expand binomials efficiently.

Once we have applied the distributive property or FOIL method, we often end up with an expression containing several terms. The final step in simplifying an expanded expression is combining like terms. Like terms are terms that contain the same variable raised to the same power. For instance, in the expression \(d^2 + 6d + 3d + 18\), the terms \(6d\) and \(3d\) are like terms because both contain the variable \(d\) raised to the first power. We combine them by adding their coefficients:

• \(6d + 3d = 9d\)

So, the final simplified expression is \(d^2 + 9d + 18\). Combining like terms is crucial for producing a cleaner, more understandable result.