Distributive Property with Variables

The distributive property is a powerful mathematical rule that lets you multiply a single term by two or more terms inside a set of parentheses. When working with variables, this property helps us simplify, expand, and factor algebraic expressions.

The core rule looks like this: $a(b + c) = ab + ac$

Expanding Expressions

Expanding an expression means getting rid of the parentheses by multiplying the outside term by every term inside the parentheses.

Example: Expand $4(3x + 2)$

1. Multiply the outside number ($4$) by the first term inside ($3x$): $4 \times 3x = 12x$
2. Multiply the outside number ($4$) by the second term inside ($2$): $4 \times 2 = 8$
3. Add the results together: $12x + 8$

So, $4(3x + 2) = 12x + 8$.

Factoring Expressions

Factoring is the exact reverse of expanding. Instead of multiplying to remove parentheses, you find the Greatest Common Factor (GCF) of the terms and pull it out to create parentheses.

Example: Factor $12x + 8$ using the GCF

1. Find the GCF of the coefficients and constants. The numbers are $12$ and $8$. The largest number that divides evenly into both is $4$.
2. Pull the $4$ out to the front of a set of parentheses.
3. Divide each original term by the GCF to find what goes inside:
   • $12x \div 4 = 3x$
   • $8 \div 4 = 2$
4. Write the final factored expression: $4(3x + 2)$

Simplifying Complex Expressions

Sometimes, you need to use the distributive property multiple times in a single problem before combining like terms.

Example: Rewrite $5(x + 3) - 2(x + 1)$ in simplest form

1. Distribute the $5$ into the first set of parentheses: $5(x) + 5(3) = 5x + 15$

2. Distribute the $-2$ into the second set of parentheses (be careful with the negative sign!): $-2(x) - 2(1) = -2x - 2$

3. Write it all out together: $5x + 15 - 2x - 2$

4. Combine like terms (group the $x$ terms and the constant numbers together): $(5x - 2x) + (15 - 2) = 3x + 13$

The simplest form is $3x + 13$.