Factorization and Expansion

On this page, the following topics are covered:

What Is Factorization or Factoring?

Factoring or factorization is the breaking down of a mathematical expression in order to achieve a more basic form of it. Factorization is a very important “tool” in math because it is frequently used in solving more complex problems.

So basically, factorization is when we get the numbers (called “factors”) which, when multiplied, brings back the original expression.

Here’s an example. The number 20 is composite, therefore it has factors. Its factors are 5 and 4. When we multiply 5 and 4, the product will be 20, so those are its factors. However, we can delve deeper into this concept. Since 4 is also composite and it is a factor of 20, we can also break it down into 2 and 2. We can now say that the factors of 20 are: 5, 2, and 2. There is such thing called “prime factorization” — you will learn more about it below.

Important Reminder

Some might be confused in saying that the factors of twenty are just 5 and 2, even if 2 appears twice. It’s just the same number anyway! Well, there is a concept called multiplicity and it is not to be taken for granted.

When we say that 5, 2, and 2 are the factors of 20, we mean that the number 2 has a multiplicity of two. This means that it must appear twice in the list (or set) of the factors of 20.

Prime Factorization

Prime factorization is a factoring method wherein we find all the prime factors of all the factors of a number. This is done in order to ensure that the list of factors contain nothing but prime numbers.

If you do not know what a prime number is yet, then here is a concise definition: it is a number which is divisible only by 1 and itself.

Here is an example:

Find the prime factors of 32. First, let’s find all of its factors. 32 = 8*4 But wait! 8 and 4 are not yet prime numbers. So, we still need to find their prime factors as well.

8 = 2*2*2 and 4 = 2*2

So, 32 in its prime factorized form is 2*2*2*2*2, or simply 2^5.

Sample Problems

Find the prime factors of the following numbers.

  1. 39
  2. 45
  3. 4
  4. 21
  5. 11

Solutions:

  1. 39
    13*3
    Prime factors: 3, 13
  2. 45
    9*5
    3*3*5
    Prime factors: 3, 3, 5
  3. 4
    2*2
    Prime factors: 2, 2
  4. 21
    7*3
    Prime factors: 3, 7
  5. 11
    11*1
    Prime factors: Only itself. 11 is a prime number.

Expansion

Expansion is just the opposite of factorization. It is when we multiply terms together in order to get the final, or expanded, result.

For example, 2*3*5*7 = 210. The expanded result is 210.

Just a note, expansion is more commonly used when dealing with algebraic terms and expressions.