Distributive property

Definition

The distributive property is a fundamental algebraic principle that states when you multiply a number by a sum, you can distribute the multiplication across each term within the parentheses. This means that for any numbers a, b, and c, the equation a(b + c) equals ab + ac. This property not only simplifies expressions but also plays a vital role in solving equations and understanding the relationships between numbers.

5 Must Know Facts For Your Next Test

1. The distributive property allows you to simplify expressions by breaking down complex calculations into more manageable parts.
2. It can be applied in both addition and subtraction, as long as multiplication is involved with a sum or difference.
3. In algebra, using the distributive property is essential when solving equations with parentheses.
4. The property is a key concept in expanding polynomials, as it helps to simplify expressions involving multiple variables.
5. Understanding the distributive property is crucial for higher-level mathematics, including calculus and linear algebra.

Review Questions

• How does the distributive property relate to simplifying algebraic expressions?
  • The distributive property is essential for simplifying algebraic expressions because it allows you to eliminate parentheses by distributing a multiplication across each term inside. For example, in an expression like 3(x + 4), you can apply the distributive property to rewrite it as 3x + 12. This technique makes it easier to combine like terms and solve equations.
• Illustrate how the distributive property is used when solving equations with parentheses.
  • When solving equations with parentheses, applying the distributive property is crucial for isolating variables. For instance, in the equation 2(x + 3) = 14, you would first distribute the 2 to get 2x + 6 = 14. Then, you can proceed to isolate x by subtracting 6 from both sides, leading to 2x = 8 and finally x = 4. This method streamlines the process and makes it more straightforward.
• Evaluate how the distributive property serves as a foundation for understanding more complex mathematical concepts, such as polynomial expansion.
  • The distributive property is foundational for understanding polynomial expansion because it allows you to multiply binomials and higher-degree polynomials systematically. For example, using the distributive property on (x + 2)(x + 3) involves distributing each term in the first binomial across each term in the second, resulting in x^2 + 5x + 6. Mastering this property enables students to tackle increasingly complex equations and functions in advanced mathematics.

Related terms

Associative Property: The associative property states that the way in which numbers are grouped when adding or multiplying does not change their sum or product.

Commutative Property: The commutative property indicates that the order in which two numbers are added or multiplied does not affect the sum or product.

Like Terms: Like terms are terms that have the same variable raised to the same power, making them able to be combined through addition or subtraction.