Lesson 7-4: Factor Linear Expressions

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

Factoring a linear expression involves rewriting it as a product of its factors. This is done by finding the greatest common factor (GCF) of the terms and applying the distributive property in reverse:

Here, $a$ is the GCF of the terms $ab$ and $ac$.

Examples

• Factor $12x + 18$: The GCF of $12$ and $18$ is $6$. So, $12x + 18 = 6(2x) + 6(3) = 6(2x + 3)$.
• Factor $4y - 20$: The GCF of $4$ and $20$ is $4$. So, $4y - 20 = 4(y) - 4(5) = 4(y - 5)$.
• Factor $\frac{1}{2}x + \frac{3}{2}$: The GCF of $\frac{1}{2}$ and $\frac{3}{2}$ is $\frac{1}{2}$. So, $\frac{1}{2}x + \frac{3}{2} = \frac{1}{2}(x) + \frac{1}{2}(3) = \frac{1}{2}(x + 3)$.

Explanation

Factoring is the process of rewriting an expression as a product of its factors, which is the reverse of distribution. To factor a linear expression, first identify the greatest common factor (GCF) of the numerical coefficients and constants. Then, divide each term in the expression by the GCF to determine the remaining factor inside the parentheses. This method can be applied to expressions with integer or fractional coefficients.

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.