The Distributive Property: Videos & Practice Problems

The distributive property is a fundamental algebraic principle that connects multiplication and addition (or subtraction) by allowing multiplication to be distributed across terms inside parentheses. Specifically, for any numbers or variables a, b, and c, the expression a × (b + c) can be expanded to a × b + a × c. This property also applies to subtraction, since subtraction can be viewed as adding a negative number, so a × (b − c) = a × b − a × c. Understanding this property is essential because it enables simplification of expressions where direct addition or subtraction inside parentheses is not possible, especially when variables are involved.

For example, consider the expression 2 × (4 + 5). Using the order of operations, you would first add 4 and 5 to get 9, then multiply by 2 to get 18. Alternatively, applying the distributive property, you multiply 2 by 4 and 2 by 5 separately, resulting in 2 × 4 + 2 × 5 = 8 + 10 = 18. Both methods yield the same result, confirming the validity of the distributive property.

When variables are present, the distributive property becomes especially useful. For instance, in the expression 4 × (x − 8), since x is unknown, you cannot simplify inside the parentheses. Instead, distribute the 4 to both terms: 4 × x − 4 × 8 = 4x − 32. This process removes the parentheses and simplifies the expression.

More complex expressions with multiple terms inside parentheses also follow the distributive property. Take (4x + 2y − 7z) × 3 as an example. Even though the multiplier 3 is on the right side, multiplication is commutative, so the distributive property still applies. Distribute 3 to each term: 3 × 4x + 3 × 2y − 3 × 7z = 12x + 6y − 21z. This demonstrates that the distributive property works regardless of the number of terms inside the parentheses or the position of the multiplier.

In summary, the distributive property is a powerful tool in algebra that facilitates the expansion and simplification of expressions involving multiplication over addition or subtraction. It ensures that multiplication can be applied to each term inside parentheses individually, which is crucial for working with variables and complex expressions. Mastery of this property enhances problem-solving skills and lays the groundwork for more advanced algebraic concepts.

When simplifying an expression with a negative sign outside the parentheses, such as −(2x − 3y + 4 − z), it is important to understand that the negative sign represents multiplication by −1. This means you are distributing −1 to each term inside the parentheses. The distributive property allows you to multiply −1 by each term individually: \(-1 \times 2x\), \(-1 \times (-3y)\), \(-1 \times 4\), and \(-1 \times (-z)\).

Applying this, the multiplication results in:

\(-1 \times 2x = -2x\)

\(-1 \times (-3y) = +3y\)

\(-1 \times 4 = -4\)

\(-1 \times (-z) = +z\)

Notice that multiplying by a negative reverses the signs of each term inside the parentheses. Specifically, subtracting a negative term becomes addition, and adding a negative term becomes subtraction. Therefore, the simplified expression is:

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

This process highlights the importance of carefully handling signs when distributing a negative value. Recognizing that multiplying by −1 simply changes the sign of each term can save time and reduce errors. However, if uncertain, fully applying the distributive property step-by-step 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 the parentheses yields the original expression:
\$91 \times m + 91 \times n = 91 \times (m + n)\(.

Similarly, for the expression 4 × x − 4 × y, the common factor is 4. Factoring out 4 gives us 4 × (x − y). Distributing 4 back over the parentheses confirms the equivalence:
\)4 \times x - 4 \times y = 4 \times (x - y)$.

This process of factoring by applying the distributive property in reverse is fundamental in algebra. It helps simplify expressions and solve equations more efficiently by identifying and extracting common factors. Mastering this skill lays the groundwork for more advanced factoring techniques and algebraic manipulations.