Factoring|Definition & Meaning


In mathematics, the word factoring or factorization means to split an expression into its constituents (called factors) such that the expression can be obtained by multiplying its factors.

Factoring is a very common technique used in the field of mathematics. Its a way of splitting complex mathematical expressions or algebraic equations into simpler parts. This process is also known as factorization. The following diagram illustrates this concept. 

Example of Factoring

Figure 1: Example of Factoring (Factorization)

The figure 1 clearly indicates a simple example. Here the given expression 14 x + 16 y is being split into its factors. Here, it can be split into two parts or factors. One of the parts is 2 and the other one is 7x + 8y. It can be shown that the original expression 14x + 16y can be obtained by simply multiplying its factors. That is, (2)(7x+8y) = 14x + 16y.

Following paragraphs further explain the process of factoring or factorization in great detail with the help of suitable examples and diagrams.

Explanation of Factoring

A polynomial or integer can be simply divided into components that, when multiplied together, produce the original or starting polynomial or integer. The factorization approach can be used to simplify any algebraic or quadratic problem by expressing the equations as the product of factors rather than by enlarging the brackets. Any equation may have a variable, an integer, or an algebraic expression as a factor.

Lets explain this process with the help of an example. Consider the expression given in the figure 2 below:

The Process of Factoring Factorization

Figure 2: The Process of Factoring (Factorization)

Here the given expression is a very well know algebraic identity i.e. a2 + b2. As shown in the figure it can be easily converted into its factors by using a neat trick of adding and subtracting a factor $ ab $ in this equation. Considering the steps given in the figure, we can split the given expression into its factors i.e. a + b and a – b. And clearly we can obtain the given expression by multiplying these two factors.

As explained earlier, the mathematical process of breaking or decomposing any mathematical entity (e.g., a number, a matrix, or a polynomial) into a product of any number of other entities, or factors, such that the result is the original entity, is known as factorization. Sometimes, we also call it factoring. It should be clear that the concept is very vast in applications and its not limited to the algebra or numbers only. However, for simplicity we will restrict ourselves to these areas on purpose.

Lets have another example of simple integers. Consider the case of the number 72. Now this number can  be converted to its factors through division method. This method is illustrated in the figure below:

The Process of Factoring Integers 1

Figure 3: The Process of Factoring Integers

Any number can be converted into its factors by the division method. We simply divide the number by a smallest divisible integer. We continue the process until the quotient becomes one. The continuous division is shown on the right side of the diagram above. Here it can be seen that the number 72 can be divided into its five factors that are 2 x 2 x 2 x 3 x 3. And the product of all these equals 72.

In simple words, an algebraic equation has been factorized when it is written as the product of its factors. These factors may include variables, components, or algebraic expressions depending upon the type of problem.

There are several methods for factoring numbers as well. There are numerous methods for determining a number’s factors. Finding the factors of an integer is simple; however, doing it for algebraic equations is more difficult. So let’s learn how to determine a quadratic polynomial’s factors.

For the quadratic case consider the example given in the figure below:

Factoring a Quadratic Expression 1

Figure 4: Factoring a Quadratic Expression

Here a the quadratic equation can be factorized by finding its roots. The figure also shows the formula for finding the roots of standard quadratic equation. Using this formula, we can find the roots which come out to be -3 and +4. Now we can split the given expression as product of (x + 3) and (x – 4). This ‘finding the roots method’ can be extended to any quadratic equation.

Lets summarize some of the methods used for factorization or factoring:

1. Factorization using algebraic manipulation for example the one explained in figure 2.

2. Factorization using common divisors for example the one explained in figure 3.

3. Factorization using algebraic identities for example the quadratic equation explained in figure 4.

Most of the factorization numerical problems can be solved by using any of the above methods. Lets delve into some such examples for more clarity.

Numerical Problems for Factoring

Find the factors of the following quadratic equations:

Part (a): x2 – x – 42

Part (b): x2 – 16

Solution of Part (a):

Comparing the given equation with the standard quadratic equation given in the figure 4, we can identify that:

a = 1, b = -1, and c = -42

Plugging these values into the quadratic equation roots formula:

\[ x = \dfrac{ -(-1) \pm \sqrt{ (-1)^2-4(1)(-42)}}{ 2(1) } \]

\[ x = \dfrac{ 1 \pm \sqrt{ 1 + 168 }}{ 2 } \]

\[ x = \dfrac{ 1 \pm \sqrt{ 169 }}{ 2 } \]

\[ x = \dfrac{ 1 \pm 13 }{ 2 } \]

We can obtain following roots:

x = -6 and x = 7

These roots can be converted into factors as follows:

x + 6 = 0 and x – 7 = 0

So the given equation can be written in the form of its factors as:

x2 – x – 42 = (x + 6) (x – 7)

The same can be solved using the following simpler method:

x2 – x – 42 = x2 – 7x + 6x – 42

x2 – x – 42 = x (x – 7) +6 (x – 7)

x2 – x – 42 = (x + 6) (x – 7)

Solution of Part (b):

Using the identity given in the figure 2, we can see that the given expression can be written in the following form:

x2 – 16 = (x)2 + (4)2

So comparing it with the standard factors given in the figure 2, we can see that if a = x and b = 4 then the above expression can be split into its factors as follows:

x2 – 16 = (x + 4) ( x – 4)

All images/mathematical drawings were created with GeoGebra.

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