Factoring Polynomials

Factorisation is a process which is necessary to simplify algebraic expressions and is used to solve the higher degree equations. It is the inverse procedure of the multiplication of the polynomials. Once you factorize a polynomial and then divide the polynomial with the same factors, you would get zero as the remainder. Remember that an algebraic expression is said to be in a factored form only when the whole expression is an indicated product. In this article, we will learn about factoring polynomials, the factorisation of polynomial, how to factor polynomials, and some solved examples. Let us first consider the following examples:



In Factored Form

Not in Factored Form

Example 1

2x(x+y)

2x+3y+z

Example 2

(x+y)(3x-2y)

2(x+y)+z

Example 3

(x+4)(x2+3x+1)

(x+y)(2x-y)+5


For factorisation of polynomials, you must know how to factor out the common factors using the distributing property. The distributive property tells you that a(b+c) = ab + ac.


Consider this Example:


Find the product of 3x2 and 4x + 3. 


To find the product, you need to multiply each term in the binomial by a common factor 3x2

3x2(4x+3) = 3x2(4x)+3x2(3)

Since this is a distributive property, the reverse of it is true as well.

3x2(4x)+3x2 = 3x2(4x+3)


If you start with 3x2(4x) + 3x2(3) you can use this property to factor out 3x2 and get 3x2 (4x + 3).


This resulting expression that you got is in the factored form since it is written as the product of two different polynomials, while the original expression is the sum of two terms.

How to Factor Polynomials?

Factoring polynomials can be done with the help of six different methods which are:

  1. Greatest Common Factor (GCF)

  2. Grouping Method

  3. Sum or difference in two cubes

  4. Difference in two squares method

  5. General trinomials

  6. Trinomial method


Let us learn factoring polynomials by using some of these methods which are used for factoring polynomials frequently. 

Factoring Polynomials by Greatest Common Factor (GCF)

As you learned that for factoring polynomials, you need to first find the greatest common factor of the polynomial that is given. And this is the reverse process of the distributive law. Follow these steps for factoring polynomials by the greatest common factor.

  1. The first step is finding the GCF of all the terms in the given polynomial.

  2. Then express each term as a product of the GCF and the other factor.

  3. Lastly, use the distributive property for factoring out the GCF.


Consider the following example.


Factorize 2x3 – 6x2.


The first step is to find the GCF. Doing so, you get,

2x3 = 2 . x . x . x

6x2 = 2 . 3 . x . x

Hence, the GCF of the expression is

2x3 - 6x2 is 2 . x . x = 2x2

The next step is expressing each term as the product of 2x2 and the other factor. Doing so, you get,


2x3 = (2x2)(x)

6x2 = (2x2)(3)

You can then write the polynomial as:

2x3 - 6x2 = (2x2)(x) - (2x2)(3)

Finally, factorize using the GCF using the distributive property.

(2x2)(x) - (2x2)(3) = (2x2)(x-3)

Factoring Polynomials By Grouping

Factoring polynomials by grouping is usually done with polynomials having 4 terms. The idea of this method is pairing the like terms together and then apply the distributive property for factorising them properly. Consider the following example.


Factorise x3 − 3x2 − x + 3.


You have x3 − 3x2 − x + 3.


Taking out the common pairs and factorising them further gives you,


= x2(x - 3) - ( x - 3 )

=(x2 - 1)(x - 3)

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

Factoring Polynomials Using Identities

Factoring polynomials using identities is done by using the algebraic identities. When it comes to factorisation, the commonly used identities are as follows:

[ (a – b)2 = a2 – 2ab + b2 ], [ (a + b)2 = a2 + 2ab + b2 ] and [a2 – b2= (a + b) (a – b)]

Consider this example:

Factorise the term (x2 – 132)

Using the algebraic identity, you can write the above polynomial as

(x+13) (x-13)

Factoring Polynomials By Factor Theorem

Factoring polynomials by factor theorem is done for a polynomial p(x) having a degree greater than or equal to one. For example,

x - a is considered a factor of p(x), if p(a) = 0

Also, if p(a) = 0, then x - a is called a factor of p(x),

wherein a is a real number.

Factoring Polynomials Examples

  1. Factorise the Following:

(x + 1 )2 - 9(x - 2)2

Solution

You have 

(x + 1 )2 - 9(x - 2)2

Taking out the common terms and factorising it, you get,

= (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)

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

Simplifying this further, your answer is

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


  1. Factorize the Expression  x2  + 5x + 6.

Solution

For this polynomial expression, we will try factorisation by splitting middle term.

When you factorise by splitting the middle term, you find two terms a and b in a way that you get a + b =5 and ab = 6. When you solve this, you get the values of a and b as 3 and 2 respectively.

Hence, you can write this expression as

x + 3x + 2x + 6 

Factorising this you get x (x + 3) + 2 (x + 3) 

= (x + 3)(x + 2)

Therefore, you can say that ( x + 3) and (x + 2) are the two factors of the polynomial  x2  + 5x + 6.


FAQ (Frequently Asked Questions)

1. How to Find Factors of a Polynomial?

You have several options for factoring when you are solving polynomial equations.

  1. In a polynomial, irrespective of how many terms it has, you should always check for the greatest common factors (GCF) first. The greatest common factor is your biggest expression which would go into all of the terms. When you use the GCF it is similar to performing the distributive property backwards.

  2. If the equation is a trinomial, which means that it consists of three terms, you can use the FOIL method to multiply the binomials backwards.

  3. If the expression is a binomial then you must look for the difference of squares, the difference of cubes, or even the sum of cubes.

  4. Finally, once the polynomial is factored fully, you can then use the zero product property for solving the equation.

2. How to Factor Polynomials with 2 Terms?

To factor polynomials with 2 terms, you need to keep in mind that you cannot factor out the terms easily since there are only 2 terms. For example, consider the following expression:

x2 – 4 = 0

Since this expression is a difference of two squares, you can factorise it. Doing so, you get,

x2 – 4 = 0

(x + 2) (x – 2) = 0

x + 2 = 0, 

and x – 2 = 0

Therefore, x = –2, and x = 2

Hence, your answer is x = ± 2.