Factoring Polynomials
Factoring Polynomials means splitting the given algebraic expression into its factors. The first step is to write each term of the larger expression as a product of its factors. As a second step, the common factors across the terms are taken common to create the required factors. Factoring polynomials help in finding the values of the variables of the given expression or to find the zeros of the polynomial expression.
Here we aim to learn the process of factoring polynomials, methods of factoring, and some of the important concepts such as remainder theorem, factor theorem, GCF, long division.
What Is Factoring of Polynomials?
The process of factoring polynomials involves writing the factors of the individual terms and taking common factors, to form the factors of the polynomial expression. A polynomial is of the form ax^{n} + bx^{n  1} + cx^{n  2}+ .........px + q, which can be factorized using numerous methods. Here in this polynomial, the exponent of x is n and it has n factors. The number of factors is equal to the degree of the variable in the polynomial expression.
The factorization can be understood with the help of a simple example. The quadratic expression x^{2} + x(a + b) + ab can be factorized as (x + a)(x + b). Higher degree polynomials are reduced to a simpler lower degree, linear or quadratic expressions to obtain the required factors.
Process of Factoring Polynomials
The following steps help for the process of factoring polynomials. Follow the below sequence of steps to factorize a polynomial.
 Factor out if there is a factor common to all the terms of the polynomial.
 Identify the appropriate method to factorize the polynomial. You can use regrouping or algebraic identities to find the factors of the polynomial.
 Write polynomial as the product of its factors.
Let us solve an example problem to more clearly understand the process of factoring polynomials. Consider a polynomial: 8ab+8b+28a+28.
Notice that 4 is a single factor common to all the terms of this polynomial.
So, we can write
8ab+8b+28a+28 =4(2ab+2b+7a+7)
Here, we will do factoring polynomial by grouping. Let us write 2ab+2b and 7a+7 in the factor form separately.
2ab + 2b = 2b(a + 1), and 7a + 7 = 7(a + 1)
Now we have 8ab+8b+28a+28 =4(2ab+2b+7a+7)
= 4 (2b(a + 1) + 7(a + 1))
= 4(2b + 1)(a + 1)
Methods of Factoring Polynomials
There are numerous methods of factoring polynomials, based on the expression. The method of factorization depends on the degree of the polynomial and the number of variables included in the expression. The four important methods of factoring polynomials is as follows.
 Method of Common Factors
 Grouping Method
 Factoring
 Factoring Using Algebraic Identities
Let us check the details of each of the methods of factoring polynomials, in detail.
Method of Common Factors
This is the simplest method of factoring an algebraic expression by taking common factors of each of the terms of the given expression. As a first step, the factors of each of the terms of the algebraic expression are written. Further, the common factors across the terms are taken to obtain the possible factors. Let us understand this better with the help of an example.
Consider a simple example: 3x+9
By factoring each term we get, 3 x + 3 . 3
By distributive law, 3x+9= 3.x + 3.3=3(x+3)
Factoring Polynomials By Grouping
The method of grouping for factoring polynomials is a further step to the method of finding common factors. Here we aim at finding groups from the common factors, to obtain the factors of the given polynomial expression. The number of terms of the polynomial expression is reduced to a lesser number of groups. First, we split each term of the given expression into its factors and further aim at taking common terms to find the group of factors. Let us try to understand grouping for factorizing with the help of the following example.
The polynomial is factorized as
8ab+8b+28a+28=8.a.b + 8.b + 28.a + 28
= 8b(a + 1) + 28(a + 1)
= (8b + 28)(a + 1)
= (4.2.b + 4.7)(a + 1)
= 4(2b + 7)(a + 1)
Hence the given expression has been factorized by grouping. 8ab+8b+28a+28 = 4(2b + 7)(a + 1)
Factoring
The process of factoring is often used for quadratic equations. Also, many times across factoring polynomials we often reduce the higher degree polynomial into a quadratic expression. Further, the quadratic equation has to be factorized to obtain the factors needed for the higher degree polynomial. The general form of a quadratic equation is x^{2} + x(a + b) + ab = 0, which can be split into two factors (x + a)(x + b) = 0. The sequence of steps for factoring the quadratic equation is as follows.
x^{2} + x(a + b) + ab = 0
x.x + ax + bx + ab = 0
x(x + a) + b(x + a) = 0
(x + a)(x + b) = 0
Here in the above expression, the middle term is split as the sum of two factors, and the constant term is presented as the product of these two factors. Thus the given quadratic equation is now presented as the product of two expressions. Let us understand this better, by factoring a quadratic equation x^{2} + 7x + 12 = 0.
x^{2} + 7x + 12 = 0
x.x + 3x + 4x + 3.4 = 0
x(x + 3) + 4(x + 3) = 0
(x + 3)(x + 4) = 0
Factoring Using Algebraic Identities
The process of factoring polynomials can be easily performed using algebraic identities. The given polynomial expressions represent one of the algebraic identities. Also sometimes the given expression has to be modified so as to match with the expression of the algebraic identities. Please find listed below some of the algebraic identities which are helpful in factoring polynomials.
 a^{2}  b^{2} = (a + b)(a  b)
 a^{3}  b^{3} = (a  b)(a^{2} + ab + b^{2})
 a^{3} + b^{3} = (a + b)(a^{2}  ab + b^{2})
 a^{4}  b^{4} = (a^{2} + b^{2})(a + b)(a  b)
Let's factorize the polynomial 4z^{2}12z+9
Observe that 4z^{2}=(2z)^{2}, 12z=2 × 3 × 2z, and 9 = 3^{2}
So, we can write 4z^{2}12z+9 = (2x)^{2} + 2(2x)(3) + 3^{2}
= (2z  3)^{2}
Concepts Relating to Factoring Polynomials
The following concepts are helpful in factoring polynomials. Also
Remainder Theorem
The remainder theorem is helpful to find the remainder on dividing an algebraic expression with another expression, without actually performing the division. The remainder obtained when the algebraic expression f(x) is divided by ( x  a) is f(a). If f(a) = 0, then (x  a) is a factor of f(x). For a polynomial expression f(x) = 12x^{3}  9x^{2 } + 5x + 17, the remainder obtained on dividing it with (x  2) is f(2) = 12(2)^{3}  9(2)^{2} + 5(2) + 17 = 12(8) 9(4) + 10 + 17 = 96  36 + 27 = 87.
Factor Theorem
The factor theorem helps in connecting the factors and zeros of polynomials. If f(x) is a polynomial of degree n , a is a real number such that (x  a) is a factor of f(x), then if f(a) = 0. Also if f(a) = 0 then (x  a) is a factor of f(x). The factor theorem is helpful to find if a given expression is a factor of a higher degree polynomial expression without actually performing the division.
Greatest Common Factors
The process of obtaining the greatest common factor for two or more terms includes two simple steps. First, split each of the terms into its prime factors, and then take as many common factors as possible from the given terms. Let us understand this by taking a simple expression of two terms 12x^{2} + 9x. Here we split the terms into its prime factors 12x^{2} + 9x = 2.2.3.x.x + 3.3.x. Among these two terms, we can take the maximum common terms to obtain the greatest common factors. Here we have the maximum common factor as 3x, and hence 12x^{2} + 9x = 2.2.3.x.x + 2.3.x = 3x(4x + 3).
Long Division
The process of long division involving polynomials is similar to the process of long division of natural numbers. Long division of polynomials is greatly helpful to find the factors of the given algebraic expression. The division resulting in a remainder of zero has the divisor as a factor of the polynomial expression. Divisions resulting in a remainder of zero can be written as Dividend = Divisor × Quotient. Thus the given polynomial expression has been divided into two factors. Further, the division of the below polynomial expression can be written as 4x^{2}  5x  21 = (x  3)(4x + 7).
Related Topics
Solved Examples on Factoring Polynomials

Example 1: Carlos will begin attending classes for the sixth grade from next week. He suddenly realized that he doesn't have either a notebook or a pen. He goes to a shop to purchase them and realizes that the cost of a notebook is twice more than $4 for a pen. Represent this above information using a polynomial. Can you find the factors of the polynomial?
Solution:
Let's assume the following.
The cost of a pen = $x
According to the given information, the cost of the notebook can be expressed as (2x + 4)
2 is a common factor in the polynomial (2x + 4)
Answer: Therefore on factoring polynomials, the factors of (2x + 4) are 2 and (x + 2)

Example 2: Emma asked Jolly to fact polynomial 6xy4y+69x. Jolly wants to factorize it using the method of regrouping. Can you help them?
Solution:
For factoring polynomials we observe that we have no common factor among all the terms in the expression 6xy4y+69x.
Let's deal with (6xy4y) and (69x) separately.
(6xy  4y) + (6  9x) = 2y(3x  2)+ 3(3x  2)
= (3x  2)(2y + 3)
Answer: Therefore the factors of 6xy  4y + 6  9x, are (2y  3) and (3x  2).

Example 3: Jake is stuck with one question in his maths assignment. The question is on factoring polynomials for x^{3} +5x^{2} +6x. Can you help him?
Solution:
Notice that x is a common factor in x^{3} +5x^{2} +6x.
So, x^{3} + 5x^{2} +6x = x(x^{2} +5x+6)
We can now factorize x^{2} +5x+6 in two binomials.
x^{2} +5x+6 = x^{2} + 3x + 2x + 3×2
= x(x + 3) + 2(x + 3)
= (x + 3)(x + 2)
Thus, the cubic polynomial can be factored as follows.
x^{3} + 5x^{2} +6x = x(x+2)(x+3).
Answer: Therefore x^{3} + 5x^{2} + 6x = x(x+2)(x+3).
FAQs on Factoring Polynomials
What Is Factoring of Polynomials?
The process of factoring polynomials is to split the given expression and write it as a product of these expressions. For example, to factorize x^{2} + 2x, we split it into two factors x and (x + 2), and write it as a product of these two factors x^{2} + 2x = x(x + 2). Here the process of factoring polynomials involves polynomials of higher degrees and involves concepts of greatest common factor, factor theorem, long division.
How Do you Find the Factors of a Polynomial?
To write a polynomial in factored form, it must be expressed as a product of terms in their simplest form. The terms could be constant or linear equation or any polynomial expression, and which cannot be further factorized.
How to Factorize Polynomials in Two Variables?
For factoring polynomials in two variables we factorize using a factoring method or by using a formula. A polynomial in two variables is of the form x^{2} + (x(a + b) + ab = 0, and can be factorized as x^{2} + (x(a + b) + ab = (x + a)(x + b) . Also, the factoring polynomials in two variables is needed for further factoring polynomials of high degree.
How Do you Do the Prime Factorization of Polynomials?
The following methods mentioned below can be used for factoring polynomials into their prime factors:
 Method of Common Factors
 Method of Grouping
 Method Using Algebraic Identities
How to Factorize Polynomials in 3 Degree?
The process of factorization of polynomials of 3 degrees involves three simple steps. First for the given n degree polynomial f(x), substitute a value 'a' such that f(a) = 0, and (x  a) is a factor. As a second step divide f(x) by (x  a) to obtain a quadratic equation. Finally, factorize the quadratic equation to obtain its two factors and hence we can obtain all the three factors of the 3degree polynomial.
How Is Factor Theorem Useful in Factoring Polynomials?
The factor theorem is used to find the factors of a ndegree polynomial without actual division. If a value x = a satisfies a ndegree polynomial f(x), and f(a) =0, then (x  a) is a factor of the polynomial expression. Further, we can find a few factors using the factor theorem and the remaining can be found using the factorization of a quadratic equation.
What is the Meaning of Factoring Polynomials by Grouping?
Factoring polynomials by grouping means factoring the polynomial by the method of grouping that allows us to rearrange the terms of the expression, to easily identify and find factors of the polynomial expression..
How Do you Find the Factors a Polynomial With 5 Terms?
The process of factoring polynomials with 5 terms is as follows.
 Write the polynomial in the standard form.
 Take the greatest common factor out if it exists.
 Try to find at least 3 roots of the polynomial. If \(\alpha\) is a root of the polynomial, then \(x\alpha\) is a factor of the polynomial.
 After finding the 3 linear factors, we are left with a quadratic polynomial. Find the product of the leading coefficient and the constant term.
 Determine the factors of the product found in step 3 and check which factor pair will result in the coefficient of x..
 After choosing the appropriate factor pair, keep the sign in each number such that while operating them we get the result as the coefficient of \(x\), and on finding their product the number is equal to the number found in step 3.
 Now, you have 4 terms in the expression and so we use the method of regrouping to factorize.
What Are the Four Methods of Factoring Polynomials?
The four methods of factoring polynomials are:
 Method of Common Factors
 Method of Grouping
 Method Using Algebraic Identities
 Method of Finding Roots