This blog is a sequel to the Basics of Factorization of Polynomials. The basics talk about polynomials and the first 2 methods of factorization, while this blog will talk about 3 other factoring methods.
We can rewrite the expression x2 + 6x + 9 in the form a2 + 2ab + b2 as;
x2+ 6x + 9 ⟹ (x)2 + 2 (x) (3) + (3)2
Applying the formula of a2 + 2ab + b2 = (a + b)2 to the expression gives;
= (x + 3)2
= (x + 3) (x + 3)
Factoring by Difference of Squares Method
To factorize by the difference of squares, we need to follow the steps given below:
Check if the four terms have anything in common (GCF). If yes, factor out the GCF. This GCF would be included in the final answer.
Every difference of squares problem can be factored as follows:
a2 – b2 = (a + b) (a – b) or (a – b) (a + b).
So, we have to see what numbers squares will produce the desired results.
Check whether the remaining factors can be factored any further or not.
x2 - 36
Step 1: Notice whether the four terms have anything in common, the GCF. If yes, factor the GCF out. Remember the GCF while noting down the final answer. In this case, the GCF is 1, which is of no use.
= x2 - 36
Step 2: Determine which term forms x2 and which number squared equals 36 to express in the form of (a + b)(a - b). In this case it is (x)(x) = x2 and (6)(6) = 36.
= x2 - 62
Step 3: Check for any possibility of further factorization. In this polynomial, there is no such opportunity.
= (x - 6) (x + 6)
This blog covers in-depth 3 methods of Factoring Polynomials with details on each of them. It is a sequel to Factoring Polynomials - 1, which talks about the basics of polynomials and two methods to factorize them. This blog helps understand a stepwise breakdown of each method.
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What is Factoring by Grouping?
Regrouping refers to the rearrangement of the polynomial, by finding common terms. Therefore to factor a polynomial by grouping we group similar terms together, as shown above with examples.
What is GCF?
GCF or the Greatest Common Factor is the largest quantity or term that is a factor of all the integers or polynomials involved.
The steps involved in finding the GCF of a given list of integers or polynomial terms are:
Prime factorize all the numbers
Identify the common prime factors
Take the product of all common prime factors
If there are no common prime factors, then the GCF is 1
What are the different methods of factoring?
There are 5 different factoring methods:
Factorizing a Polynomial by Factoring out the HCF or GCF