Best method for factoring polynomials

Students in grade nine or grade ten need to know factoring polynomials as an important segment of their algebra study. At this page, I am giving an overall approach to learning how to factor polynomials. You should know the following points very well to make factoring polynomials easy:

1. Greatest Common Factor (gcf):

Students should have basic understanding of gcf of two numbers. For example; gcf of 3 and 6 is 3, the gcf of 6 and 9 is 3 as well and gcf of 15 and 20 is 5. (By the way gcf is the greatest number which can divide the both numbers)

That was grade five work, as we are talking grades nine and ten here, therefore you should know the following as well:

gcf of “4 and 8” is 4.

gcf of “4a and 8a” is “4a”.

gcf of “4a² and 8a” is again 4a.

2. Factoring the polynomial by pulling gcf out

Now in a given polynomial look in each term for the gcf, and if there is any common factor in the terms of the polynomial pull it out as shown in the following example:

6y + 3

This polynomial is a binomial as it has two terms 6y and 3.

As you already know that gcf of “6 and 3 is 3” and there is only one variable “y” with 6, so there is no common variable in this question. Now pull 3 out from both the terms as shown below:

= 3(2y + 1)

Keep in mind that once we pull the gcf 3 out, we divide both the terms with 3 to get the new terms in the brackets.

In the next example, I changed both the terms of the same polynomial a little bit to take it to the next level and let’s factor “6y² + 3y” now.

Again, the gcf of 6 and 3 is 3, but in this problem, we have the variable “y” common as well and the gcf of both the terms is “3y” in this case. Hence pull “3y” out from both the terms to factor the polynomial as shown below:

3y (2y + 1)

Alternative way:

If you didn’t get the above solution, there is another way to do this. Break your terms into prime factors as shown below:

6y² + 3y

Break “6y² into 2.3.y.y” and “3y into 3.y” because 6 have prime factors of “2 and 3” and you can break “y² into y.y”. Notice that I used dots for multiplication signs.

After breaking the terms of the polynomial, our problem can be written as shown below and I have highlighted the common factors in both the terms.

= 2.3.y.y + 3.y

Pull the common factors out and write the remaining factors inside the brackets to complete your solution.

= 3.y (2.y + 1)

= 3y (2y + 1) is the answer.

Point to note is that the whole of the second term “3y” is gcf and after pulling out the whole of the terms as gcf, we still have “1” inside (because some students write zero, which is wrong)

3. When a polynomial has four terms:

To understand this consider the following example:

Factor 4x² + 3y + 2x + 6xy

To factor these kind of polynomials rearrange the terms and try to find the greatest common factor in the pairs of two terms as shown below.

4x² + 2x+ 3y + 6xy

= 2x (2x + 1) + 3y (1 + 2x)

Look at the first two terms “4x² + 2x” it has “2x” as its gcf, pull it out to get “2x (2x + 1)”. Similarly factor the second pair “3y + 6xy” as “3y (1+2x)”. But, (1 + 2x) is same as (2x + 1), hence we can interchange them for simplicity.

=2x (2x + 1) + 3y (2x + 1)

Now, (2x + 1) is the common factor and pull it out as shown below:

= (2x + 1) (2x + 3y)

All the above steps can be written to show your work together as follows;

4x² + 3y + 2x + 6xy

= 4x² + 2x + 3y + 6xy

= 2x (2x + 1) + 3y (1 + 2x)

= 2x (2x + 1) + 3y (2x + 1) [Because (1 + 2x) = (2x+1)]

= (2x + 1) (2x + 3y)

Now if you multiply back (1 + 2x) with (2x + 3y) you will get the original polynomial back to justify your answer.