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# Factoring

## What is factoring?

Factoring is the process of decomposing or splitting any given polynomial into a product of two or more polynomials. We always do this with numbers. For example, here are some possible ways to factor 24.

The process of factoring or decomposing the factors of polynomials is called factorization. The factors of a polynomial should always be less than the degree or equal to the original polynomial.

## What are the different ways to factor polynomials?

In this section, we will learn the three different methods to factor any polynomials.

### Factoring by greatest common factor (GCF)

The first and simplest method that we are going to talk about is factoring out the greatest common factor (GCF) of the terms in the polynomial. In general, this should be the first thing that you should try because it often simplifies a complex polynomial.

Here are some of the strategies and techniques you can try in finding the greatest common factor of any polynomials.

1. Completely factor out each term.
2. Write a product using factors that is common to all the terms.

Remember that factoring out the GCF is like using the distributive law in reverse. The distributive law states that if

a(b + c) = ab + ac

However, we simply use this law in reverse when we factor out the greatest common factor. Hence, if

ab + ac = a(b + c)

Notice that each term in the polynomial ab + ac has an “a”, so we can factor it out.

Example #1

Factor out the greatest common factor (GCF) of the polynomial x4 – x3 – x2 + x.

Solution

Example #2

Factor out the greatest common factor (GCF) of the polynomial 3x4 + 6x3 – 18x2 + 9x.

Solution

Example #3

Factor out the greatest common factor (GCF) of the polynomial x5y4 – x2y3 + xy2.

Solution

Example #4

Factor out the greatest common factor of the polynomial (5x7)(3x +2) + (25x5)(3x +2).

Solution

### Factoring by groupings

Factoring by groupings is done when no common factor exists to all of the terms of a polynomial, but there are factors common to some of its terms. Hence, our main goal here is to find groups with common factors.

Example #1

Given the polynomial 5x2 + 9x – 10x – 18, factor out using the method of groupings.

Solution

Example #2

Using factorization by groupings, factor out the polynomial 6x2 + 14x + 9x + 21.

Solution

Example #3

Factor out the polynomial x7 + x5 – 10x4 – 10x2 using factorization by groupings.

Solution

### Factoring quadratic polynomials

Quadratic polynomial is a polynomial where the highest degree of a term is in degree two. When factoring quadratic polynomials, we are simply doing the FOIL method in reverse. Here are some things that will help you factor quadratic polynomials.

1. Arrange the given polynomial in a standard form, ax2 + bx + c.
2. List all the possible factors of c
3. Make sure that the sum of the factors of c is the same as b
4. If the last term is negative, it will have the form (x + __)(x – __).
5. If the second term is negative, it will have the form (x – __)(x – __).
6. If all terms of the polynomial is positive, it will be in the form (x + __)(x + __).
7. If all terms of the polynomial is negative, it will be in the form –(x + __)(x + __).

Example #1

Factor the quadratic polynomial x2 + 5x + 6.

Solution

Example #2

What is the factored form of the quadratic formula x2 + x – 20?

Solution

Example #3

Factor the quadratic polynomial x2 – 15x + 54.

Solution

### Factoring using algebraic identities

Using algebraic identities, the process of factorization can be easily done. Below is the table that shows some algebraic identities you can use in factoring.

Example #1

Factor the polynomial x2 + 14x + 49 using algebraic identities.

Solution

Example #2

Factor the polynomial 4x2 – 81.

Solution

Example #3

Factor the polynomial 27x3 + 8.

Solution

## What is the importance of factoring polynomials?

Factoring is a vital knowledge and fundamental step that helps us easily understand equations. Every time we rewrite complex polynomials into a simpler polynomials, we apply the concept of factoring – hence, giving us more information about the components of the equation or algebraic expressions.