Factoring or factor decomposition is the process of **presenting a mathematical expression or a number in multiplication form** . Remember that the **factors** are the elements of multiplication and the result is known as the **product** .

## Types of factoring

In general terms, we can talk about two types of factorization: the factorization of integers and the factorization of algebraic expressions.

### Prime factorization

Every whole number can be decomposed into its **prime factors** . A **prime number** is one that is only divisible between 1 and itself. For example, the 2 can only be divided by 1 and 2.

We can decompose a given number X as the multiplication of its prime factors. For example, the number 525 is made up of the prime numbers 5, 3, and 7 as follows:

### Factoring algebraic expressions

The goal of factoring is to take a complicated polynomial and express it as the product of its simple polynomial factors.

Are called **factors** or **dividers** of an algebraic expression to algebraic expressions multiplied each given as the first expression product. For instance:

The factors are:

## How to factor

When we talk about factoring, we can follow the following recommendations:

- Observe if there is a common factor, that is, if there is a factor that is repeated in the different terms.
- Order the expression: sometimes when arranging the expression we realize the possibilities of factoring.
- Find out if the expression is factorizable: sometimes we are in the presence of expressions that cannot be decomposed into factors.
- Check if the factors found are in turn factorable.

## Steps to find the common factor of a polynomial

The common factor of a polynomial is the step prior to factoring a polynomial. We are going to explain step by step how to find the common factor of the following polynomial:

### Step 1

We get the greatest common factor of 24 and 16. The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24; the factors of 16 are 1, 2, 4, 8, and 16. The greatest common factor is 8.

### Step 2

We get the common factors of the variables, in this case the common variables with the highest common power. The common variables are **x** and **y** . The greatest common power of **x** is **x ^{6}** and the greatest common power of

**y**is

**y**.

^{3}### Step 3

We write the common factor of the polynomial as the product of steps 1 and 2 above:

## Factorization of polynomials

We already know the common factor of the polynomial, so we can go on to factor:

### Step 1

We determine the common factor of the polynomial:

### Step 2

We rewrite each term of the polynomial as a function of the common factor. For this, we first divide the term by the common factor to obtain a second factor:

Now we substitute each term for the common factor and the respective second factor:

**Note** : **8x ^{6} y ^{3} (3x ^{2} ) – 8x ^{6} y ^{3} (2y ^{4} z ^{3} )** is not the factored form because the factors are not separated yet.

### Step 3

We use the distributive property to get the common factor:

### Step 4

We review the steps taken:

## Four-term factoring

We can factor a polynomial of four terms by grouping them into pairs. Let’s look at the following example:

### Step 1

We rearrange the terms such that the first two have a common factor and the other two also have a common factor:

### Step 2

We factor the **x** of the first term and the **y** as a common factor of the second term:

### Step 3

We use the distributive property to factor the term (a + b) from the expression:

## Factoring a quadratic equation

When we have a three-term polynomial, this can be a quadratic trinomial of the form **ax ^{2} + bx + c** . This expression is obtained from the multiplication of two binomials:

When factoring a quadratic equation like **x ^{2} + 9x + 14** , we want to get the two binomials that originated it:

**(x + 7) (x + 2)**.

### Factor a quadratic equation by trial and error

For the expression 4x ^{2} -11x-3 we look for two binomial factors. 4x ^{2} is the first term, so the multiplication of the first numerical coefficients of the binomials must be 4. The last term is -3, so the last terms of the factors have different signs whose product is -3. We can try various combinations:

This option is incorrect.

This option is incorrect.

This is the correct option.

### Factor a quadratic equation by grouping

To factor by grouping, we identify the coefficients **a** , **b,** and **c** and look for two factors **ac** whose sum is **b** . For example, for the equation **4x ^{2} -11x-3** , the coefficients are

**a**= 4,

**b**= -11, and

**c**= -3.

The factors **ac** = (4) (- 3) = – 12. Two factors of -12 that together give -11 are -12 and 1.

Now we replace the middle term of **4x ^{2} -11x-3** with

**-12x + 1x**.

We group the terms in pairs and look for the common factor:

We apply the distributive property to the factor (x-3):

The factored form is then as:

### Factorization of perfect square trinomials

A perfect square trinomial is one where the absolute value of the coefficient b is equal to twice the product of the roots of **a** and **c** :

For example, in the equation **4x ^{2} -20x + 25** ,

**a**= 4,

**b**= -20,

**c**= 25, then:

This indicates that **4x ^{2} -20x + 25** can be factored as the square of a binomial:

The first term will be the square root of 4x ^{2} and the last term is the square root of **c** :

The sign in the binomial is the same as the middle term of the trinomial.

See also Quadratic Equations of the Second Degree .

## Factoring of binomials

The factorizable binomials are:

- the difference of two squares (x
^{2}-y^{2}), - the difference of two cubes (x
^{3}-y^{3}) and - the sum of two cubes (x
^{3}+ y^{3}).

The factorization of the difference of two squares (x ^{2} -y ^{2} ) is:

Example:

The factorization of the difference of two cubes (x ^{3} -y ^{3} ) is:

Example:

The factorization of the sum of two cubes (x ^{3} + y ^{3} ) is:

Example:

See also

- Notable products .
- Laws of exponents

## Solved factoring exercises

### 1. Factor the following expression:

#### Step 1

The common factor is (x-1).

#### Step 2

Apply the distributive property to the factor (x-1):

### 2. Factor the following expression:

#### Step 1

We take as a common factor 25x ^{2} and ^{2} z

#### Step 2

We factor the difference of two squares which is 4x ^{2} -1.

#### Step 3

The full factored form is:

### 3. Factor the following expression:

#### Step 1

This expression is a quadratic equation, so we look for binomial factors:

#### Step 2

We are looking for two numbers that multiplied to -30 and added to -7. We test with -10 and 3:

#### Step 3

The factored form is:

### 4. Factor the following expression:

#### Step 1

We note that this expression has four terms. We group them in pairs so that we can get a common factor:

#### Step 2

We factor the square binomial (x ^{2} -1):

#### Step 3

The final factored form is: