Distributive Property

Definition

The distributive property is a fundamental algebraic rule that states the product of a number and a sum is equal to the sum of the individual products. It allows for the simplification of expressions by distributing a factor across multiple terms within a parenthesis or other grouping symbol.

5 Must Know Facts For Your Next Test

1. The distributive property allows you to multiply a number or variable with a sum, by multiplying each term in the sum individually and then adding the results.
2. The distributive property is essential for simplifying algebraic expressions, solving equations, and evaluating numerical expressions involving multiplication.
3. The distributive property can be used with both whole numbers and variables, and it applies to both addition and subtraction.
4. Understanding the distributive property is crucial for performing operations with integers, fractions, and polynomials.
5. The distributive property is one of the fundamental properties of operations that form the foundation for more advanced algebraic concepts and problem-solving techniques.

Review Questions

• Explain how the distributive property can be used to simplify the expression $3(2x + 5)$.
  • To simplify the expression $3(2x + 5)$ using the distributive property, we multiply each term inside the parentheses by the factor outside the parentheses. This gives us $3(2x) + 3(5)$, which can then be simplified to $6x + 15$. The distributive property allows us to distribute the 3 across the two terms inside the parentheses, resulting in the simplified expression.
• Describe how the distributive property can be used to solve the equation $4(x - 3) = 16$.
  • To solve the equation $4(x - 3) = 16$ using the distributive property, we first distribute the 4 to the terms inside the parentheses: $4x - 12 = 16$. Then, we can isolate the variable $x$ by adding 12 to both sides: $4x = 28$. Finally, we divide both sides by 4 to find the solution: $x = 7$. The distributive property allows us to simplify the equation by breaking down the expression on the left-hand side, making it easier to solve for the unknown variable.
• Analyze how the distributive property can be used to multiply binomials, such as $(2x + 3)(4x - 5)$, and explain the significance of this application.
  • To multiply the binomials $(2x + 3)(4x - 5)$ using the distributive property, we distribute each term in the first binomial to the terms in the second binomial: $(2x)(4x) + (2x)(-5) + (3)(4x) + (3)(-5)$. This results in the expanded expression $8x^2 - 10x + 12x - 15$, which can then be simplified to $8x^2 + 2x - 15$. The ability to use the distributive property to multiply binomials is crucial for working with polynomial expressions, as it allows for the expansion and simplification of more complex algebraic terms. This skill is foundational for understanding and manipulating higher-order polynomials.

Related terms

Commutative Property: The commutative property states that the order of factors in a multiplication or addition operation does not change the result.

Associative Property: The associative property states that the grouping of factors in a multiplication or addition operation does not change the result.

Identity Property: The identity property states that adding or multiplying by the identity elements (0 for addition, 1 for multiplication) does not change the value of the expression.