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The distributive property is a powerful tool in algebra that helps us manipulate and simplify expressions. It states that for any three terms, let's say, \( a \), \( b \), and \( c \), the expression \( a(b + c) \) can be expanded to \( ab + ac \). Basically, you distribute the term outside the parentheses to each term inside the parentheses.

In our exercise, where we have \((3b + c)(b + 2c)\), we apply the distributive property twice. First, to multiply \(3b\) by both \(b\) and \(2c\). Then, we do the same for \(c\). By following this property, we were able to expand the expression into four terms: \((3b)(b) + (3b)(2c) + (c)(b) + (c)(2c)\).

This fundamental principle is not only essential for expanding expressions but also useful in solving equations and simplifying many algebraic problems.

Like terms are terms that contain the same variables raised to the same power. In this context, the coefficients of these terms might differ, but their variable parts are identical. Identifying like terms is crucial because it allows us to simplify expressions by combining them.

During the last step of our example, we encountered the terms \(6bc\) and \(cb\). Even though they are written differently, they are like terms because \(cb = bc\) due to the commutative property of multiplication. We can rewrite \(cb\) as \(1bc\), aligning the coefficients directly for addition.

• Combine \(6bc\) and \(1bc\) to yield \((6 + 1)bc\), which is \(7bc\).

Recognizing and combining like terms is a key step in simplifying polynomials.

When multiplying polynomials, each term in the first polynomial is multiplied by each term in the second polynomial. This involves the use of the distributive property, as discussed earlier, and ensures that all combinations of terms are accounted for.

The process is often referred to as the "FOIL" method when dealing with binomials: First, Outer, Inner, Last. However, for more complex expressions or polynomials with more than two terms, it's easier to systematically apply the distributive property. In our exercise, where we multiplied \((3b + c)\) by \((b + 2c)\), we followed this systematic approach:

• Multiply the first terms: \((3b)(b) = 3b^2\)
• Multiply the outside terms: \((3b)(2c) = 6bc\)
• Multiply the inside terms: \((c)(b) = cb\)
• Multiply the last terms: \((c)(2c) = 2c^2\)

This ensures that we don’t miss any interaction between terms and that the final expanded result reflects all contributions from the pairwise multiplications.