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, since subtraction can be viewed as adding a negative number, so \(a \times (b - c) = a \times b - a \times c\).

For example, consider the expression \$2 \times (4 + 5)\(. Using the order of operations, you first add inside the parentheses: \)4 + 5 = 9\(, then multiply: \)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 validity of the distributive property.

This property becomes especially useful when dealing with variables, where addition or subtraction inside parentheses cannot be simplified directly. For instance, in the expression \)4 \times (x - 8)\(, since \)x\( is unknown, you cannot simplify inside the parentheses first. Instead, distribute the 4: \)4 \times x - 4 \times 8 = 4x - 32\(. This allows you to rewrite expressions without parentheses, facilitating further algebraic manipulation.

The distributive property also works when the multiplier is on the right side of the parentheses, due to the commutative property of multiplication (i.e., \)a \times b = b \times a\(). For example, in \) (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 demonstrates that the distributive property applies regardless of the number of terms inside the parentheses or the position of the multiplier.

Understanding and applying the distributive property is essential for simplifying algebraic expressions, solving equations, and working with variables efficiently. It bridges the operations of multiplication and addition/subtraction, enabling the expansion and simplification of expressions in a consistent and reliable way.

When simplifying an expression with a negative sign outside the parentheses, such as −(2x − 3y + 4 − z), the negative sign acts as a multiplier of −1 applied to every term inside the parentheses. This means you distribute −1 across each term: −1 × 2x, −1 × (−3y), −1 × 4, and −1 × (−z). It is important to carefully handle the signs during this process, especially when subtraction is involved inside the parentheses.

Applying the distributive property, the multiplication results in:

\(-1 \times 2x = -2x\)
\(-1 \times (-3y) = +3y\)
\(-1 \times 4 = -4\)
\(-1 \times (-z) = +z\)

Thus, the simplified expression becomes:

\(-2x + 3y - 4 + z\)

This final expression contains the same terms as the original but with opposite signs, which is the expected outcome when multiplying by a negative number. Recognizing this pattern can save time by allowing you to directly change the signs of each term without performing each multiplication step. However, if there is any uncertainty, fully applying the distributive property ensures accuracy.

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 or factor expressions.

For example, consider the expression 91 × m + 91 × n. Both terms share the common factor 91. By factoring out 91, we rewrite the expression as 91 × (m + n). This works because distributing 91 back over m + n returns the original expression, confirming the equivalence.

Similarly, for the expression 4 × x − 4 × y, the common factor is 4. Factoring out 4 gives 4 × (x − y). Distributing 4 back over x − y yields the original expression, demonstrating the correctness of the factoring.

This process of factoring by applying the distributive property in reverse is fundamental in algebra. It helps simplify expressions and solve equations more efficiently. Recognizing common factors and rewriting expressions with parentheses enhances problem-solving skills and prepares for more advanced topics such as polynomial factoring and equation solving.