In mathematics, factorization is the process of breaking apart a polynomial into a product of other smaller polynomials. On this page, we will discuss the Factors of a Polynomial, types, how to find polynomial factors, some solved examples, and so on. Students of 10th Grade Math can answer these Factoring Polynomials Questions over here and apply the related knowledge to solve problems they come across in the future fastly.
Also, Read:
Factors of a Polynomial – Definition
Factoring Polynomials means decomposing the given polynomial into a product of two or more polynomials using prime factorization. It will help in simplifying the polynomials easily. The first step is to write each term of the larger expression as a product of its factors, and the second step is for the common factors across the terms to be taken out to create the required factors.
Methods of Factoring Polynomials
Factorizing the polynomials can be done with the help of six different types which are as such
-
- Greatest Common Factor (GCF)
- Grouping Method
- Sum or difference in two cubes
- The difference in the two squares method
- General trinomials`
- Trinomial method
How do find Factors of a Polynomial?
Let us learn how to find factors of a polynomial by using some of these methods which are used for factoring polynomials frequently.
Factoring Polynomials by Greatest Common Factor (GCF): As you learn that for factoring polynomials, you first need to find the greatest common factor of the polynomial that is given. This will be the reverse process of distributive law. The Following are the steps for factoring polynomials by the greatest common factor.
- Step 1: The first step is finding the GCF of all the terms in the given polynomial.
- Step 2: Then express each term as a product of the GCF and the other factor.
- Step 3: Finally, use the distributive property for factoring out the GCF.
Factoring Polynomials By Grouping: To factorize the polynomials by grouping is usually done with polynomials having 4 terms. The idea of this method is to pair the like terms together and then apply the distributive property for factorizing them properly. Consider the following example,
Example: Factorise x3 − 3x2 − x + 3.
Solution:
First, taking out the common pairs and factorizing them further gives you,
Then the value is, 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 factorization, the identities are commonly used as follows: [(a – b)2 = a2 – 2ab + b2], [(a + b)2 = a2 + 2ab + b2] and [a2 – b2= (a + b) (a – b)]. Consider the example,
Example: Factorise the term (x2 – 132)
Solution: 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. Consider an example,
x – a is considered as 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.
Factors of a Polynomial Examples
Problem 1: To Factorise the following is:
(x + 1 )2 – 9(x – 2)2
Solution:
As given in the question,
The equation is (x + 1 )2 – 9(x – 2)2
Now, we need to find the factorization.
First, taking out the common terms and factorizing it, you get,
(x + 1 )2 – 3(x – 2)2
Next, the equation is ((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)
Now, Simplify this further, we get
(-2x + 7)(4x – 5)
So, after factorization the factors of the equation is (-2x + 7)(4x – 5).
Problem 2: Factorize the Expression x2 + 5x + 6.
Solution:
In the given question,
The polynomial expression is x2 + 5x + 6.
For this polynomial expression, we will try factorization by splitting the middle term.
So, when you factorize 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.
Then, you can write this expression as,
x2 + 3x + 2x + 6
After factorising this you get,
x (x + 3) + 2 (x + 3)
The factors are (x + 3)(x + 2).
Therefore, you can say that ( x + 3) and (x + 2) are the two factors of the polynomial x2 + 5x + 6.
Problem 3: Using factor theorem, factorize the polynomial x3 – 6x2 + 11 x – 6.
Solution:
Given that,
Let f(x) = x3 – 6x2 + 11 x – 6.
The constant term in f(x) is equal to – 6 and the factors of – 6 are ±1, ± 2, ± 3, ± 6.
Now, Putting x = 1 in f(x), then we have
f(1) = 13 – 6 ×12 + 11× 1– 6
= 1 – 6 + 11– 6 = 0
So, (x– 1) is a factor of f(x).
Next, Similarly, x – 2 and x – 3 are the factors of f(x).
Since f(x) is a polynomial of degree 3. This means, it can not have more than three linear factors.
Let the f(x) is k (x–1) (x– 2) (x – 3).
Then, x3 – 6x2 + 11 x – 6 = k(x–1) (x– 2) (x– 3)
Now, Put x = 0 on both sides, then
we get – 6 = k (0 – 1) (0 – 2) (0 – 3)
i.e., – 6 = – 6 k ⇒ k = 1
Next, Putting k = 1 in f(x) = k (x– 1) (x– 2) (x–3), we get
f(x) = (x–1) (x– 2) (x – 3)
Hence, the polynimal of p(x) is x3 – 6x2 + 11 x – 6 factors are (x– 1) (x – 2) (x–3).
FAQ’s on Factors of a Polynomial
1. What is meant by a polynomial factor?
A factor of polynomial P(x) refers to any polynomial whose division takes place evenly into P(x). For example, x + 2 is a factor belonging to the polynomial x2 – 4. The polynomial factorization is represented as a product of its various factors. A good example can be the factorization of x2 – 4 is (x – 2)(x + 2).
2. How to factor a polynomial?
A polynomial can be factorized using different methods such as, finding the greatest common factor(GCF) of all the terms, splitting the polynomial into two parts, using algebraic identities, etc.
3. What are the four major types of factoring?
The four main types of factoring are the Greatest common factor (GCF), the Grouping method, the difference in two squares, and the sum or difference in cubes.
4. How to factor a polynomial with two terms?
To factorize the polynomial with two terms, first, find the GCF of the terms and take the common factor out. For example, x2 – x is the polynomial, x is the GCF of x2 and x, therefore x2 – x = x(x-1).
Thus, x and x-1 are the factors of x2 – x.
5. How to factorize polynomial with three terms? Give example.
To factorize the polynomial with three terms are,
Let x2 – 7x -18 be a three-term polynomial. Now we need to find two such numbers, whose the product will give -18 and the sum will give -7. Thus,
-9 x 2 = -18
-9 + 2 = -7
Then, we can write the given polynomial as:
x2– 9x + 2x – 18
= x(x – 9) + 2(x – 9)
= (x – 9)(x + 2)
Hence, the required factors of the given polynomial are (x-9)(x+2)