Understanding the Distributive Law in Multiplication

How to Use the Distributive Property in Multiplication

The distributive property in mathematics is a fundamental rule involving multiplication and addition. It is one of the key characteristics of multiplication, stating that adding a set of numbers first and then multiplying the sum by another number yields the same result as multiplying each individual number by that number separately and then adding the products. This law is expressed symbolically:^[1]

A × (B + C) = A × B + A × C

Here,

• A: the first number.
• B: the second number.
• C: the third number.

The monomial expression A is distributed across each term in the binomial expression (B + C), resulting in A × B + A × C.^[1] In short, this law can be summarized as multiplying whatever is outside the parentheses by everything inside it.^[2]

Applications of the Distributive Property in Multiplication

The distributive property has numerous applications in mathematical problems. Key uses include:

Mental Math

This property greatly supports mental arithmetic, enabling children to compute the product of large numbers mentally without writing them down. For example, to calculate 53 × 4, the distributive property allows us to rewrite the problem as: 53 × 4 = 4 × (50 + 3) = 4 × 50 + 4 × 3. This simplifies the process, allowing students to easily multiply 4 × 50 and 4 × 3, find each product separately, and then add the results to reach the final answer: 200 + 12 = 212.^[3]

Simplifying Complex Mathematical Expressions

The distributive property helps break down complex expressions into simpler parts. It is particularly useful when multiplying or dividing polynomials—algebraic expressions consisting of real numbers and variables—and also when working with monomials—algebraic expressions containing only one term. The process works as follows:^[3]

• Multiplying a monomial by a polynomial: This involves three simple steps:
  • Multiply the outer term by the first term inside the parentheses.
  • Then multiply it by the second term inside the parentheses.
  • Finally, add the results. For instance, s(2s + 10) becomes 2s² + 10s.
• Finding the product of two binomials: The distributive property can also be used to multiply binomials, such as (s + t)(s + 2t) = s(s + 2t) + t(s + 2t) = s² + 2st + st + 2t² = s² + 3st + 2t².

Note: The distributive property does not apply to subtraction, addition, or division. For example, (4 + 8)/24 = 24/12 = 2, but applying distribution incorrectly gives 24/4 + 24/8 = 6 + 3 = 9, which is incorrect.^[4]

Diverse Examples of the Distributive Property in Multiplication

Below are diverse examples illustrating the use of the distributive property in multiplication:

Examples of Solving Arithmetic Problems

Example 1: Calculate the result of 3 × (2 + 4).^[4]

Solution:

• Solving without the distributive property: 3 × (2 + 4) = 3 × 6 = 18
• Solving using the distributive property: 3 × (2 + 4) = 3 × 2 + 3 × 4 = 6 + 12 = 18—the same result.

Example 2: Calculate 204 × 6 using the distributive property.

Solution: 6 × 204 = 6 × (200 + 4) = 6 × 200 + 4 × 6 = 1200 + 24 = 1224

Example 3: Sara attempted to solve a math problem as follows: 40 / 9 = (5 + 4) / 40 = 4 / 40 + 5 / 40 = 10 + 8 = 18. Is Sara’s method correct?^[5]

Solution: Sara’s approach is incorrect because the distributive property does not apply to division. The correct way is long division, yielding 40 / 9 ≈ 4.444.

Example 4: Find the result of 5 × (6 + 2 − 4).^[2]

Solution: 5 × (6 + 2 − 4) = 5 × 6 + 5 × 2 − 5 × 4 = 30 + 10 − 20 = 20.

Example 5: Find the result of 3 × (6 + 7).^[6]

Solution: 3 × (6 + 7) = 3 × 6 + 3 × 7 = 18 + 21 = 39.

Example 6: Find the result of 7 × 997 using the distributive property.^[6]

Solution: 7 × 997 = 7 × (1000 − 3) = 7000 − 21 = 6979.

Example 7: Find the result of 3 × 1309 using the distributive property.^[6]

Solution: 3 × 1309 = 3 × (1000 + 3 + 9) = 3000 + 9 + 27 = 3927.

Examples of Simplifying Algebraic Expressions

Example 1: Rewrite the following using the distributive property: 5s(3s² + 2s − 4).^[2]

Solution: 5s(3s² + 2s − 4) = 15s³ + 10s² − 20s.

Example 2: Simplify the expression using the distributive property: 4a³(3a − a²).^[7]

Solution: Applying the distributive property: 4a³(3a − a²) = 12a⁴ − 4a⁵.

Example 3: Find the product: (s + 3)(s − 2).^[7]

Solution: (s + 3)(s − 2) = s² − 2s + 3s − 6 = s² + s − 6.

Example 4: Find the product: (s² + 2)(s − 1).^[7]

Solution: (s² + 2)(s − 1) = s³ − s² + 2s − 2.

Example 5: Find the product: (4s − t + 4)(s + 2t − 3), and determine the coefficient of t in the final simplified expression.^[7]

Solution: (4s − t + 4)(s + 2t − 3) = 4s² + 8st − 12s − st − 2t² + 3t + 4s + 8t − 12. After simplification: 4s² − 2t² + 7st − 8s + 11t − 12. Thus, the coefficient of t is 11.

Example 6: Given that b + c = 15 and a − d = 4, find the value of ab − cd + ac − bd.^[7]

Solution:

• First, rearrange the expression: ab − bd + ac − cd.
• Second, factor out b from the first two terms and c from the last two: ab − bd + ac − cd = b(a − d) + c(a − d).
• Third, factor out (a − d): b(a − d) + c(a − d) = (a − d)(b + c). Substituting the given values: (a − d)(b + c) = 4 × 15 = 60.

Example 7: Simplify the expression using the distributive property: (s² + s + 1)(s² − s − 1).

Solution: (s² + s + 1)(s² − s − 1) = s⁴ − s³ − s² + s³ − s² − s + s² − s − 1 = s⁴ − s² − 2s − 1.

Example 8: Is (s² + t²)√ = (s + t)?^[8]

Solution: (s² + t²)√ ≠ (s + t); the distributive property does not apply to addition under square roots. To verify, assume s = 3, t = 4. Substituting into the right-hand side: (s² + t²)√ = (9 + 16)√ = √25 = 5. Substituting into the left-hand side: s + t = 3 + 4 = 7. Since 5 ≠ 7, the equation does not hold.

References

1. ^ ^{A B} Editors of Encyclopædia Britannica, "Distributive law", www.britannica.com, Retrieved 25-10-2017. Edited.
2. ^ ^{A B C} "ALGEBRA LAWS: Commutative, Associative, Distributive Laws", www.whatcom.edu, Retrieved 19-2-2020. Edited.
3. ^ ^{A B} Deb Russell (1-8-2017), "The Distributive Property Law", www.thoughtco.com, Retrieved 25-10-2017. Edited.
4. ^ ^{A B} "Commutative, Associative and Distributive Laws", www.mathsisfun.com, Retrieved 25-10-2017. Edited.
5. ↑ "Commutative, Associative and Distributive Laws", www.mathopolis.com, Retrieved 19-2-2020. Edited.
6. ^ ^{A B C} "The Distributive Property", www.varsitytutors.com, Retrieved 19-2-2020. Edited.
7. ^ ^{A B C D E} Ashley Toh, A Former Brilliant Member, PJ Beck, and 5 others, "Distributive Property", brilliant.org, Retrieved 19-2-2020. Edited.
8. ↑ Nihar Mahajan, A Former Brilliant Member, Zandra Vinegar, and 2 others, "common misconceptions", brilliant.org, Retrieved 20-2-2020. Edited.