Lesson 5: Factor Expressions

Property

Factoring is the process of using the distributive property in reverse.
To factor an expression, find the greatest common factor (GCF) of the terms, write it outside the parentheses, and then determine what remains inside the parentheses.

This can be modeled by arranging algebra tiles into a rectangle and finding the dimensions.

Examples

• To factor $6x + 18$, find the greatest common factor of $6x$ and $18$, which is $6$. Write $6$ outside parentheses. This gives $6(x + 3)$.
• To factor $8a + 12b$, the greatest common factor of $8a$ and $12b$ is $4$. The factored expression is $4(2a + 3b)$.

Explanation

Factoring is like being a math detective. You start with the final expression and work backward to find the original factors that were multiplied together. It's the opposite of distributing; you're 'un-distributing' the expression.

Property

Before you can factor an expression, you must find the Greatest Common Factor (GCF) of its terms. The GCF is the largest number (and/or variable) that divides evenly into every single term in the expression.

• First, find the GCF of the numerical coefficients.
• Second, check if there is a variable common to ALL terms.

Examples

• Numbers Only: Find the GCF of 40 and 56.
  • Break them into primes: 40 is $2 \cdot 2 \cdot 2 \cdot 5$, and 56 is $2 \cdot 2 \cdot 2 \cdot 7$.
  • They share three 2s. The GCF is $2 \cdot 2 \cdot 2 = 8$.
• Variables Included: Find the GCF of $9x$ and $15x^2$.
  • The GCF of 9 and 15 is 3.
  • They both share at least one $x$.
  • The GCF is $3x$.
• Mixed Terms: Find the GCF of $8x$ and 12.
  • The GCF of 8 and 12 is 4.
  • The second term does not have an $x$, so $x$ cannot be part of the GCF.
  • The GCF is simply 4.

Explanation

Think of finding the GCF like being a detective. You are looking for the biggest common "ingredient" that every term shares. If even one term is missing the variable, the variable gets kicked out of the GCF club!

Property

Distributive Property

If $a, b, c$ are real numbers, then

The form on the left is used to multiply. The form on the right is used to factor.