Factoring Polynomials: Definition, Types & Technique

Factoring polynomials is the opposite process for multiplying polynomial factors. Polynomials are algebraic expressions that consist of variables with exponents, coefficients, and constants that are combined via elementary mathematical operations like addition, subtraction, and multiplication. The word “Polynomial” is made up of two Greek terms – “poly” meaning “many” and “nomial” meaning “terms”. Put together it means many terms. Although a polynomial can have any number of terms, it cannot be infinite.

Terms in a Polynomial

The parts in the polynomial separated by a plus sign “+” are called terms. Therefore every part of a polynomial can be said to be a term. 

Polynomial	Terms
x3– 2x2+ 3x+4	x2,-2x2,3x, and 4
3x+1	3x and 1

Like and Unlike Terms

The terms having the same variable with the same exponential power are called like terms. Whereas, the polynomial terms having different variables or the same variable but with different exponential powers are called, unlike terms. Coefficients are not of significance here.

Terms	Type
2x, 3x, 5x and x	Like Terms
-5y2 and y2	Like Terms
2x3 and 3x5	Unlike Terms
x3 and y3	Unlike Terms

Degree of polynomial

The highest exponential power of any variable present in any term of the algebraic expression is called the degree of the polynomial.

Types of Polynomials

Depending on the degree of the polynomial, they are divided into the following categories:

• Constant or Zero Polynomial: The polynomials with degree zero are called constant or zero polynomials. When the power of the variable becomes zero, then its value ultimately becomes 1. Such polynomials have only constant terms.

Example: 10, 94, 0, -6 etc.

• Linear Polynomials: The polynomials with a degree of one are called linear polynomials. 

Example: x – 1, y + 1, a + 4 etc.

• Quadratic Polynomial: The polynomials with the degree of two are called quadratic polynomials.

Example: x²+ x, y²+ 1, a²+ 8 etc.

• Cubic Polynomial: The polynomials with the degree of three are called cubic polynomials.

Example: y³+ 8,  x³– 27,  5+a³, x³+x²– 2 etc.

• Quartic Polynomial: The polynomials with the degree of four are called quartic polynomials.

Example: x⁴+x³-x²+ x+1, y⁴-y²+1, etc.

Polynomial	Example	Degree
Constant or Zero Polynomial	10	0
Linear Polynomial	y-3x+2	1
Quadratic Polynomial	x2+ y2 + 2	2
Cubic Polynomial	x3– 2x2+ 3x+4	3
Quartic Polynomial	y4+3y3+3y2+6y+1	4

Classification of Polynomial

Depending on the terms of the polynomial, they are divided into the following categories:

• Monomial: The polynomials that consist of only one term.

Example: 4x, 3y, x²,y³,3a⁴ etc.

• Binomial: The polynomials that consist of two terms.

Example: x+1, x²– 1, y³+ 4, a+3, x²+ x, etc.

• Trinomial: The polynomials that consist of three terms.

Example: x²+ x+1, x²+ y²+2, y-3x+2, etc.

Factoring Polynomials

The process of obtaining the factors of a given value or an algebraic expression is called factorization. The process of factoring a polynomial basically involves turning the polynomial into the sum of its factors. In this way, a polynomial can be expressed as the sum of two or more simpler polynomials. In other words, it helps you make your polynomial simpler. Before we understand the methods of factoring polynomials, let us first see what factors are. Factors are simply numbers multiplied together to obtain the original required number. Similarly, for the case of polynomials, the factors are also polynomials themselves multiplied together to generate the original polynomial.

Let us now have a look at the various techniques of factoring polynomials and most common factoring polynomials formulas.

 Factoring Polynomials Techniques 

The subsequent steps help in the polynomial factoring process. The steps to factorizing a polynomial are listed below. 

• Check to see if the polynomial’s terms have a common factor. 
• Choose the best technique for factoring polynomials.  
• You can determine the polynomial’s factors by regrouping or by using algebraic identities. 
• Polynomial should be written as the product of its factors. 

How to factor a polynomial, and what are the different factoring polynomials formulas?

Before we answer how to factoring polynomials, let us first list down the various factoring polynomials formulas that we employ. There are five different factoring polynomials formulas as follows:

• Greatest Common Factor (GCF)
• Substitution Method
• Grouping Method
• Sum or difference in two cubes
• The difference in two squares method

Greatest Common Factor

In this method, we aim to find out the greatest common factor of the given polynomial to factorize it. It is simply a reverse procedure of the distributive law.

In the case of distributive law, we get:

p(q+r) = pq + pr

Whereas in the case of factorization, we invert the process

pq + pr = p(q+r)

Here p is the greatest common factor.

Grouping Method

Also known as factoring by pair, the polynomial is distributed in pairs or grouped in pairs to find the zeros. The basic idea is to pair like terms together. So we can conveniently apply the distributive property to factorize it nicely.

Example: Factorize x²– 15x+50

Firstly find the two numbers that on being added give -15 as their sum and on multiplication give 50 as their product. The two numbers are -5 and -10 respectively as,

(-5) + (-10) = -15

(-5) x (-10) = 50

Therefore, we can rewrite the given polynomial as;

x²-5x-10x+50

x(x-5)-10(x-5)

Taking x – 5 as common factor we get;

(x-5)(x-10)

Hence, the factors are (x – 5) and (x – 10).

Substitution Method

If the polynomial given is too complex, we can try substituting the complicated terms with a simpler term to solve. Thereby; making it much easier to factor out.

Example: Factorize (x – y)(x – y – 1) – 20

Let S = x-y. Now substitute S for x-y in the given expression.

(x – y)(x – y – 1) – 20 = (S)(S – 1) – 20

S²– S-20

(S – 5)(S + 4)

(x-y-5)(x-y+4)

Difference of Two Squares Identity

This technique applies to factorize the binomial expressions in the form of 

x²– y²= (x – y)(x + y)

Example: Factorize (x+1)²– 9(x-2)²

Solution:

(x+1)²– 9(x-2)²= (x+1)²– (3(x-2))²

=((x+1-3(x-2)) ((x+1)+3(x-2))

=(x+1-3x+6)(x+1+3x-6)

= (-2x+7)(4x-5)

How to factor cubic polynomials?

Students will often have to face the problem of how to factor cubic polynomials. The cubic polynomials are one of the most prominent polynomial forms asked in the questions to the students. The trick of factoring cubic polynomials lies in the use of identities. The sum and difference of the cube’s identity greatly help in factoring cubic polynomials. Let us now answer how to factor cubic polynomials in detail below.

Factoring Cubic Polynomials using the Sum and Difference of Cubes Identity

This technique applies to factorize the binomial expressions in the form of 

x³+ y³= (x+y) (x²– xy+y²⁾

And, x³-y³=(x-y) (x²+xy+y²)

Example: Factorize 27x³ – y³

Solution: 

27x³– y³= (3x)³– y³

= (3x-y) ((3x)²+3xy+y²)

=(3x-y) (9x²+3xy+y²)

Example: Factorize 64x³+ 27y³

Solution:

64x³+ 27y³= (4x)³+(3y)³

= (4x+3y) ((4x)²-12xy+(3y)²)

= (4x+3y) (16x²-12xy+9y²)

How to factor polynomials with four terms?

The next interesting question in our study is how to factor polynomials with four terms. The answer to how to factoring polynomials with four terms is a simple one. We’ll have to follow the same method, as we have applied in the previous questions. Let us understand it with the help of an example.

Example: Factorize x³+x²-x-1

Solution:

Break the polynomial into two parts as follows:

(x³+x²)+(-x-1)

Now determine the highest common factor from both the parts and take it out of the brackets. As we can see from the first part, x² is the greatest common factor. From the second part, we can take out -1 or the minus sign as common.

x²(x+1)-1(x+1)

Again, regrouping the terms as the factors we’ll finally arrive at

(x²-1)(x+1)

Now that we’ve completely understood the various factoring polynomial formulas let us solve a few problems to gain more practice.

Question: Check if x+3 is a factor of x³+3x²+5x+15

Start by breaking the polynomial into two parts as follows:

(x³+3x²)+(5x+15)

Take out the greatest common factors from both the parts. For the first part x² and the second part 5 is the greatest common factor.

x²(x+3)+5(x+3)

Finally grouping the common terms we’ll arrive at the final solution.

(x+3)(x²+5)

Hence the answer is yes, (x+3) is a factor of x³+3x²+5x+15.

Question: Factorize x²+5x+6

Start by splitting the middle term into two numbers whose sum is 5 and product is 6. The two numbers are 2 and 3.

x²+5x+6=x²+2x+3x+6

This is simply a polynomial with four terms. Break it into two parts.

(x²+2x)+(3x+6)

Just like in the previous examples, take out the greatest common factors from the two parts. The common factors will be x and 3 for the first and second parts respectively.

x(x+2)+3(x+2)

All that’s left now is to group the common terms and reach the final solution.

(x+2) (x+3)

Frequently Asked Questions

1. What is Factoring Polynomials?

Ans. Factoring polynomials is a way to write a polynomial as the product of two or simpler polynomials. In other words, it’s a way to simplify your polynomial.

2. How Do you Find the Factors of a Polynomial?

Ans. To find the factors of a polynomial, you need to use a method called ‘divisibility rules. Divisibility rules are a set of tests that tell you whether or not a number is divisible by another number. For example, if you want to know if 12 is divisible by 6, you could just divide 12 by 6 and see if it worked out 12/6 = 2Rounded up, which means that 12 is divisible by 6.

3. How to Factorize Polynomials in Two Variables?

Ans. There are several methods that can be used to find the factors of polynomials. One way is to look at each coefficient and see if it divides evenly into the constant term. Another method is to divide both sides of the equation by a variable, such as x or y. The variable that divides evenly into both sides is a factor of the polynomial.

4. What Are the Four Methods of Factoring Polynomials?

Ans. The four methods of factoring polynomials are:

-Factor by grouping

-Factor by using the difference of squares method

-Factor using the difference of cubes method

-Factor by using the sum and difference of cubes method

5. How to Factorize Polynomials in 3 Degree?

Ans. Factorizing a polynomial in 3 degrees is quite straightforward. First, identify the term with the highest exponent and drop it. Then, write the remaining terms in order of descending exponents to get your answer.