Contents

# Factoring|Definition & Meaning

## Definition

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.Â

**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:

**Figure 2: The Process of Factoring (Factorization)**

Here the given **expression** is a very well know **algebraic identity** i.e. a^{2} + b^{2}. 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 ofÂ **breaking**Â orÂ **decomposing**Â any mathematical entity (e.g., a number, a matrix, or a polynomial) into aÂ **product **of any number of **other entities,**Â orÂ **factors,**Â such that the result is the original entity, is known asÂ **factorization.**Â Sometimes, we also call itÂ **factoring.**Â It should be clear that the concept is very vast inÂ **applications**Â and its not limited to theÂ **algebra**Â orÂ **numbers**Â only. However, forÂ **simplicity**Â 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 canÂ be **converted to its factors** through **division method.** This method is illustrated in the **figure** below:

**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:

**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): x ^{2} – x – 42**

**Part (b): x ^{2} – 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:

x^{2} – x – 42 = (x + 6) (x – 7)

The same can be solved using the** following simpler method:**

x^{2} – x – 42 = x^{2} – 7x + 6x – 42

x^{2} – x – 42 = x (x – 7) +6 (x – 7)

x^{2} – 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:**

x^{2} – 16 = (x)^{2} + (4)^{2}

So **comparing** it with the **standard**Â **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:

x^{2} – 16 = (x + 4) ( x – 4)

*All images/mathematical drawings were created with GeoGebra.*