Distributing & Factoring Basics

TipLearning Objectives
  • Use the distributive property to expand expressions.
  • Factor common terms from expressions.
  • Recognize when distributing vs factoring is most efficient.

Key Ideas

Distributive Property

\[ a(b + c) = ab + ac \]

Examples:

  • \(4(x + 3) = 4x + 12\)
  • \(-2(3y - 5) = -6y + 10\)

Factoring (Reverse Distributing)

Factor out the greatest common factor (GCF):

\[ ab + ac = a(b+c) \]

Examples:

  • \(6x + 9 = 3(2x + 3)\)
  • \(14y - 21 = 7(2y - 3)\)

Factoring simplifies expressions, solves equations, and prepares for advanced algebra.

Strategies

  • Distribute to every term inside the parentheses.
  • Treat a negative sign outside parentheses like multiplying by \(-1\).
  • When factoring, look for the greatest common factor first.
  • Check your factoring by distributing back out.
  • If every term has a variable, include the lowest variable power in the GCF.

Worked Examples

Example 1 — Distribute

Expand: \[ -3(2x - 5) \]

\[ -6x + 15 \]

Example 2 — Factor Out the GCF

Factor: \[ 12x - 18 \]

GCF = 6:

\[ 6(2x - 3) \]

Example 3 — Factor Expressions With Variables

Factor: \[ 8x^2 + 4x \]

GCF = \(4x\):

\[ 4x(2x + 1) \]

Common Mistakes

WarningCommon Mistakes
  • Forgetting to distribute the negative sign.
  • Incorrect GCF factoring (leaving out variables).
  • Factoring only part of the expression.

Practice Problems

  1. Expand: \(5(3x - 4)\)
  2. Expand: \(-(7y + 2)\)
  3. Factor: \(15x - 20\)
  4. Factor: \(9x^2 + 6x\)
  5. Expand: \(4(2x - 3y + 5)\)

1. \(15x - 20\)
2. \(-7y - 2\)
3. \(5(3x - 4)\)
4. \(3x(3x + 2)\)
5. \(8x - 12y + 20\)

Summary

  • Distributing multiplies a factor across every term in parentheses.
  • Factoring reverses distributing by pulling out a common factor.
  • The GCF may include numbers, variables, or both.
  • Checking by redistributing catches most factoring mistakes.
  • Negative outside parentheses? Distribute the negative carefully.
  • Factoring should make the expression shorter or cleaner.
  • Always ask: what does every term have in common?