The Distributive Property: Videos & Practice Problems

The distributive property is a fundamental algebraic principle that connects multiplication and addition or subtraction within parentheses. It states that when a number or variable multiplies a sum or difference inside parentheses, you can distribute the multiplication to each term individually. Mathematically, this is expressed as \(a \times (b + c) = a \times b + a \times c\). This property also applies to subtraction, so \(a \times (b - c) = a \times b - a \times c\), since subtraction can be viewed as adding a negative number.

Understanding the distributive property helps simplify expressions, especially when variables are involved and direct addition or subtraction inside parentheses is not possible. For example, consider the expression \(2 \times (4 + 5)\). Using the order of operations, you first add inside the parentheses to get \(2 \times 9 = 18\). Alternatively, applying the distributive property, you multiply 2 by each term inside the parentheses: \(2 \times 4 + 2 \times 5 = 8 + 10 = 18\). Both methods yield the same result, confirming the property’s validity.

When variables are present, such as in \(4 \times (x - 8)\), you cannot simplify inside the parentheses because \(x\) is unknown. Instead, distribute the 4 to both terms: \(4 \times x - 4 \times 8 = 4x - 32\). This process removes the parentheses and expresses the product clearly.

The distributive property also works when the multiplier is on the right side of the parentheses, thanks to the commutative property of multiplication, which states that the order of factors does not affect the product. For instance, in the expression \((4x + 2y - 7z) \times 3\), you distribute the 3 to each term inside the parentheses: \(3 \times 4x + 3 \times 2y - 3 \times 7z = 12x + 6y - 21z\).

This property is essential for simplifying algebraic expressions, solving equations, and expanding expressions involving variables. It allows for flexibility in manipulating terms and is foundational for more advanced algebraic techniques.

When simplifying an expression with a negative sign outside the parentheses, such as −(2x − 3y + 4 − z), the key is to apply the distributive property carefully. The negative sign in front of the parentheses can be understood as multiplying the entire expression inside by −1. This means each term inside the parentheses will be multiplied by −1, which changes their signs.

Using the distributive property, multiply −1 by each term inside:
\(-1 \times 2x\), \(-1 \times (-3y)\), \(-1 \times 4\), and \(-1 \times (-z)\). This results in:
\(-2x\), \(+3y\), \(-4\), and \(+z\) respectively.

Notice that when multiplying by a negative, positive terms become negative, and negative terms become positive. Specifically, subtracting a negative term is equivalent to adding the positive version of that term, which is why −(−3y) becomes +3y and −(−z) becomes +z. The simplified expression is therefore:
\(-2x + 3y - 4 + z\).

This process highlights the importance of understanding how the distributive property interacts with negative signs. Recognizing that multiplying by −1 simply reverses the sign of each term can save time and reduce errors. If unsure, performing the full multiplication step-by-step ensures accuracy. This approach is fundamental in algebra for simplifying expressions and solving equations efficiently.

The distributive property allows us to rewrite expressions by factoring out a common factor and enclosing the remaining terms in parentheses. Instead of expanding expressions by distributing multiplication over addition or subtraction, we can reverse this process to simplify expressions into a factored form. For example, the expression 91 × m + 91 × n can be rewritten by factoring out the common multiplier 91, resulting in 91 × (m + n). This works because distributing 91 back over m + n returns the original expression.

Similarly, the expression 4 × x − 4 × y can be factored by taking out the common factor 4, giving 4 × (x − y). This demonstrates how the distributive property can be applied in reverse to factor expressions, which is a foundational skill in algebra. Factoring expressions by identifying and extracting the greatest common factor simplifies calculations and prepares for solving equations.

In summary, factoring expressions involves recognizing common factors in terms and rewriting the expression as a product of that factor and a sum or difference inside parentheses. This process is the reverse application of the distributive property, which states that for any numbers a, b, and c, the equation \(a \times (b + c) = a \times b + a \times c\) holds true. Factoring leverages this property to condense expressions, making them easier to work with in algebraic manipulations.