The Distributive Property in Algebra

The Distributive Property

Definition and Application

The distributive property is a fundamental property of real numbers used to simplify expressions and solve equations. It states that multiplying a number by a sum is the same as multiplying the number by each addend and then adding the results.

• General Form: For any real numbers a, b, and c:

$a(b + c) = ab + ac$

• This property is used to remove parentheses in algebraic expressions.

• It applies to both addition and subtraction inside the parentheses.

Example Problem

Question: Multiply $x(9x^2 + 5)$ using the distributive property.

• Step 1: Distribute x to each term inside the parentheses:

$x(9x^2 + 5) = x imes 9x^2 + x imes 5$

• Step 2: Simplify each term:

$x imes 9x^2 = 9x^3$ $x imes 5 = 5x$

• Final Answer:

$x(9x^2 + 5) = 9x^3 + 5x$

Additional Examples

• Example 1: $2(a + 3) = 2a + 6$

• Example 2: $y(4y - 7) = 4y^2 - 7y$

Key Points

• The distributive property helps simplify expressions and solve equations.

• Always multiply the term outside the parentheses by each term inside.

• Combine like terms if possible after distributing.