The distributive property of multiplication is a fundamental algebraic rule. It states that when we multiply a number by a sum or difference in parentheses, we can distribute the multiplication to each term inside. The general formula is a times the quantity b plus c equals a times b plus a times c. This property works for both addition and subtraction.
Let's work through a concrete example to see the distributive property in action. We have the equation 3 times the quantity x plus 4 equals 18. First, we apply the distributive property: 3 times x plus 3 times 4 equals 18. This simplifies to 3x plus 12 equals 18. Subtracting 12 from both sides gives us 3x equals 6. Finally, dividing by 3, we get x equals 2.
We can visualize the distributive property using area models. Imagine a rectangle with height 'a' and width 'b plus c'. We can divide this rectangle into two parts: one with area 'a times b' and another with area 'a times c'. The total area is a times the quantity b plus c, which equals ab plus ac. This visual representation helps us understand why the distributive property works.
Let's look at common mistakes when applying the distributive property. The most frequent error is forgetting to distribute to all terms. For example, 2 times the quantity x plus 3 is NOT equal to 2x plus 3. The correct answer is 2x plus 6, because we must multiply 2 by both x and 3. Another common mistake involves sign errors with subtraction. 3 times the quantity x minus 2 equals 3x minus 6, not 3x minus 2.
The distributive property has many important applications in mathematics and real life. It's essential for simplifying algebraic expressions, solving linear equations, and factoring polynomials. It's also useful for mental math calculations. For example, to calculate 5 times 23, we can think of it as 5 times 20 plus 3, which equals 100 plus 15, or 115. The distributive property appears in area calculations, cost calculations, and physics formulas. Remember the key formula: a times the quantity b plus c equals ab plus ac.