Table of Contents

  1. Concept of Common Factors
  2. Identifying the GCF
  3. Factoring Out the GCF
  4. Examples
    1. Example 1: $ 6x^2 + 12x $
    2. Example 2: $ 9y^3 - 3y^2 + 6y $
  5. R Code Example
  6. Practice Problems

Factoring common factors involves identifying and extracting the greatest common factor (GCF) from a polynomial. This technique simplifies expressions and is fundamental in solving algebraic equations. The GCF of a set of terms is the largest expression that divides all of them without a remainder.

Concept of Common Factors

The common factor in a set of terms is a term that divides each of them evenly. The greatest common factor is the largest of these common factors. For example, in the expression \( 4x^3 + 8x^2 \), both terms are divisible by \( 4x^2 \), making it their GCF.

Identifying the GCF

To find the GCF of an algebraic expression:

  1. List the factors of each term.
  2. Identify the common factors.
  3. Choose the largest factor common to all terms.

Factoring Out the GCF

Once the GCF is identified, the expression can be rewritten as the product of the GCF and another expression. For instance, factoring \( 4x^3 + 8x^2 $\)by its GCF \( 4x^2 $\)gives \( 4x^2(x + 2) \).

Examples

Here are some examples of factoring common factors in algebraic expressions:

Example 1: \( 6x^2 + 12x \)

  • Factors of \( 6x^2 \): \( 1, 2, 3, 6, x, x^2, 2x, 3x, 6x, 2x^2, 3x^2, 6x^2 \)
  • Factors of \( 12x \): \( 1, 2, 3, 4, 6, 12, x, 2x, 3x, 4x, 6x, 12x \)
  • GCF: \( 6x \)
  • Factored form: \( 6x(x + 2) \)

Example 2: \( 9y^3 - 3y^2 + 6y \)

  • Factors of \( 9y^3 \): \( 1, 3, 9, y, y^2, y^3, 3y, 9y, 3y^2, 9y^2, 3y^3, 9y^3 \)
  • Factors of \( 3y^2 \): \( 1, 3, y, y^2, 3y, 3y^2 \)
  • Factors of \( 6y \): \( 1, 2, 3, 6, y, 2y, 3y, 6y \)
  • GCF: \( 3y \)
  • Factored form: \( 3y(3y^2 - y + 2) \)

R Code Example

We can use R to visualize the process of finding GCFs. Let's plot a simple bar graph showing the factors of two numbers and highlight their GCF.

x <- c(1, 2, 4, 8)
y <- c(1, 2, 3, 6, 9, 18)
common_factors <- intersect(x, y)

barplot(c(table(x), table(y)), beside=TRUE, col=c(rep("blue", length(x)), rep("red", length(y))),
        legend.text=c("Factors of 8", "Factors of 18"))
abline(h=common_factors, col="green", lwd=2)
title("Common Factors of 8 and 18")

Practice Problems

  • Factor out the GCF from the following expressions:
    1. \( 15x^4 - 25x^3 \)
    2. \( 7y^2 + 14y - 21 \)

Understanding and applying the concept of factoring common factors is crucial in algebra, especially when simplifying expressions and solving equations.