Factoring Ugly Trinomials

We have now come to the end of week 6 which was our first week as a cohort class. This week we learned a lot about factoring polynomials which we already learned a lit bit last week. The difference this time is we were learning how to factor “ugly” trinomials. Ugly trinomials are ones that have a leading coefficient greater than one. I thought that this was a very important thing to learn as first of all factoring is very useful in many questions, and so are polynomials in general. Although since these expressions have a leading coefficient greater than one they are more likely to run into.

The factor the following “ugly” trinomials we are going to use something called the box method.

Here is the first example we are going to be working with:

Step 1 – identifying the greatest common factor (GCF): When factoring ugly trinomials there is always the chance of there being a greatest common factor among the three terms. In this case however, there is no GCF so we can move on to the next step.

Step 2 – setting up the box: since this is called the box method we are going to first draw a 2×2 box. After the box is drawn we can put the first term “” in the upper right corner and the third term “-15” in the bottom right box.

Step 3 – multiplying two terms: next, in order to find the what is going to be in the next two boxes we need to multiply the first and third term together which in the case will equal to ““. Then you need to find two factors that multiply to get -60 and add to get 17. In order to do this write out all the factor pairs of -60 that you can think of until you find two that add up to 17. In this case these numbers are “20” and “-3”, these two numbers will both have a variable of x so that they can multiply together to get ““. Now these two remaining numbers will go in the open spots in the box.

Step 4 – finding the factors: now we need to actually factor the expression. So for the first column we will look at what the two terms have in common. In this case the only thing they have in common is “x”. So we will write x on top of the box on the left. Once this is determined you will be able to figure out what the rest of the numbers on the side have to be. It is just like an area diagram that we used when multiplying polynomials. The numbers on the outside come across and multiply together to get the numbers on the inside. Once this is done, the expression has been factored. Just write the two expressions on the top of the box and on the left in two separate pairs of brackets.

 

Example 2: now we are going to look at another example with a very similar process.

Here is the sample question we are going to be working with:

Step 1 – identifying the greatest common factor (GCF): As usual, when factoring ugly trinomials there is always the chance of there being a greatest common factor among the three terms. In this question unlike the first example there is a GCF. In this case the GCF is “2” so we are going to go ahead and divide each term by “2”. Once this is done put a 2 outside the brackets and all the new terms inside the brackets.

Step 2 – setting up the box:  Once again we are going to start off by drawing a 2×2 box. After the box is drawn we can put the first term “” in the upper right corner and the third term “-5” in the bottom right box.

Step 3 – multiplying two terms: next, in order to find the what is going to be in the next two boxes we need to multiply the first and third term together which in the case will equal to ““. Then you need to find two factors that multiply to get -60 and add to get -4. In order to do this write out all the factor pairs of -60 that you can think of until you find two that add up to -4. In this case these two numbers are “6” and -10″, these two numbers will both have a variable of x so that they can multiply together to get ““. Now these two remaining numbers will go in the open spots in the box.

Step 4 – finding the factors: now we need to actually factor the expression. So for the first column we will look at what the two terms have in common. In this case the two terms have “2x” in common. So we will write 2x on top of the box on the left. Once this is determined you will be able to figure out what the rest of the numbers on the side have to be. It is just like an area diagram that we used when multiplying polynomials. The numbers on the outside come across and multiply together to get the numbers on the inside. Once this is done, the expression has been factored. Just write the two expressions on the top of the box and on the left in two separate pairs of brackets. In addition to this, in front of the two pairs of brackets you’ll need to bring back that 2 from the beginning of the question when we divided all the terms by 2.

 

Summary:

We have now completed two examples of factoring “ugly” trinomials. Although these questions may seem very challenging and tedious but once the proper method is learned it isn’t all that difficult. Once again it is more common than not that the leading coefficient will be greater than one when you have to factor polynomials so this is a very useful skill to have.