Here are the steps required for factoring a sum of cubes:

Step 1: Decide if the two terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer.
Step 2 : Rewrite the original problem as a difference of two perfect cubes.
Step 3 : Use the following sayings to help write the answer.
  a) “Write What You See”
  b) “Square-Multiply-Square”
  c) “Same, Different, End on a Positive”
Step 4 : Use these three pieces to write the final answer.

Example 1 – Factor: Example 1

Step 1: Decide if the two terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer. In this case, the two terms only have a 1 in common which is of no help.
Step 1
Step 2: Rewrite the original problem as a difference of two perfect cubes.
Step 2
Step 3a: “Write What You See” If you disregard the parenthesis and the cubes in step 2, you should see:
Step 3a
Step 3b: “Square-Multiply-Square” If you square the first term, x, you get x2. If you multiply the two terms, x and 4, you get 4x. Finally, if you square the second term, 4, you get 16.
Step 3b
Step 3c: “Same, Different. End on a Positive” This will determine the signs of the problem. The first sign should be the same as the original question, the next sign should be different then the first, and the last sign should always be positive.
Step 3c
Step 4: Write the final answer.
Step 4

Example 2 – Factor: Example 2

Step 1: Decide if the two terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer. In this case, the two terms only have a 1 in common which is of no help.
Step 1
Step 2: Rewrite the original problem as a difference of two perfect cubes.
Step 2
Step 3a: “Write What You See” If you disregard the parenthesis, the cubes, and the 2 in step 2, you should see:
Step 3a
Step 3b: “Square-Multiply-Square” If you square the first term, 2x, you get 4x2. If you multiply the two terms, 2x and 5, you get 10x. Finally, if you square the second term, 5, you get 25.
Step 3b
Step 3c: “Same, Different. End on a Positive” This will determine the signs of the problem. The first sign should be the same as the original question, the next sign should be different then the first, and the last sign should always be positive.
Step 3c
Step 4: Write the final answer.
Step 4

Click Here for Practice Problems

Example 3 – Solve: Example 3

Step 1: Decide if the two terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer. In this case, the two terms only have a 1 in common which is of no help.
Step 1
Step 2: Rewrite the original problem as a difference of two perfect cubes.
Step 2
Step 3a: “Write What You See” If you disregard the parenthesis, the cubes, and the 2 in step 2, you should see:
Step 3a
Step 3b: “Square-Multiply-Square” If you square the first term, 5x, you get 25x2. If you multiply the two terms, 5x and 6y, you get 30xy. Finally, if you square the second term, 6y, you get 36y2.
Step 3b
Step 3c: “Same, Different. End on a Positive” This will determine the signs of the problem. The first sign should be the same as the original question, the next sign should be different then the first, and the last sign should always be positive.
Step 3c
Step 4: Write the final answer.
Step 4

Click Here for Practice Problems

Example 4 – Solve: Example 4

Step 1: Decide if the two terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer. In this case, the two terms have a 3 in common, which leaves:
Step 1
Step 2: Rewrite the original problem as a difference of two perfect cubes.
Step 2
Step 3a: “Write What You See” If you disregard the parenthesis, the cubes, and the 3 in step 2, you should see:
Step 3a
Step 3b: “Square-Multiply-Square” If you square the first term, 3x, you get 9x2. If you multiply the two terms, 3x and 4y, you get 12xy. Finally, if you square the second term, 4y, you get 16y2.
Step 3b
Step 3c: “Same, Different. End on a Positive” This will determine the signs of the problem. The first sign should be the same as the original question, the next sign should be different then the first, and the last sign should always be positive.
Step 3c
Step 4: Write the final answer, do not forget the 3 that was factored out in the first step.
Step 4

Click Here for Practice Problems