Factoring polynomials is a "must know" to understand algebra and score excellent grades in algebra. In this presentation we will explore the ultimate approach to factoring polynomials. Starting from factoring a monomial we will discuss every aspect of factoring polynomials.
To factor any kind of polynomial, knowledge of greatest common factor (GCF) is mandatory. If the students don't have the basic understanding of factoring numbers, they should review prime factorization of numbers first.
There are the following steps for factoring polynomials:
1. Find if there is any greatest common factor in the terms of given polynomial:
If the polynomial has the GCF, pull it out from each term of the polynomial by using the brackets. For example;
3a² + 6ab - 9a has "3a" as the GCF. Pull "3a" out as shown below to factor the polynomial;
3a (a + 2b - 3)
2. If the polynomial is a binomial (having two terms only), find if it is the difference of squares. Some times, by taking the GCF away, the binomial becomes the difference of squares. There is a special method for factoring difference of squares and I will discuss that in detail in my coming articles.
3. If the polynomial is a trinomial, again try to pull the greatest common factor away if you can. There is a special way to factor a trinomials and I am going to explain this topic alone in a separate article.
4. If the polynomial has four terms, then try to rearrange the terms and separate them into pairs of two's having greatest common factors. Take the greatest common factor out from each pair and see if you get two same brackets. For example; consider we have a polynomial, 4u² + 3a + 2u + 6au and we want to factor it.
There are four terms in the polynomial and there is no greatest common factor other than one. If we rearrange the terms and try to find the greatest common factor in the pairs of two terms, it might be possible to factor the polynomial that way.
So rewrite the polynomial by rearranging the terms as shown below:
4u² + 2u+ 3a + 6au
Look at the first pair "4u² + 2u" it has "2u" its GCF, pull it out as "2u (2u + 1)". Similarly factor the second pair as "3a (1+2u)". But, (1 + 2u) is same as (2u + 1), hence we can interchange them for simplicity.
= 2u (2u + 1) + 3a (2u+1)
Now, (2u + 1) is the common factor and pull it out as shown below:
= (2u + 1) (2u + 3a)
All the steps can be written together as follows;
4u² + 3a + 2u + 6au
= 4u² + 2u + 3a + 6au
= 2u (2u + 1) + 3a (2u+1)
= (2u + 1) (2u + 3a)
You can FOIL it to check your answer, if you get back the original polynomial.
