factoring

I haven’t really learned much starting this week, but I have learned quite interesting new techniques on factoring polynomials.

Let’s start off with this acronym I have learned to help with the process.

CDPEU – Can Divers Pee Easily Underwater?

Common

Difference of Squares

Pattern

Easy

Ugly

Let’s breakdown what each term means before getting into the factoring process.

Common means find what is common between the terms, the GCF (Greatest common factor).
Difference of Squares means if there are two binomials and they are subtracting.
Pattern means if they have three terms following the pattern: $x^2$ $x$ #
Easy means the easy ones to factor out (straightforward) : $x^2$
Ugly ones are the polynomials that not so straightforward and need more work. We’ll see what method we can use later on. : $a{x^2}$

Now let’s apply these steps!

To start, refer to the list above we just talked about.

The first step is find what is common. To do so, write all the possible factors of both coefficients and see what variables both terms have in common.

After we found all possible factors of 49 and 14, we can see that the common number is 7. Both terms also have at least one x variable. Therefore, you would factor out 7x :

Write out the rest of the values in the bracket next to 7x, making sure when you multiply the values in the bracket with 7x, it will give you the same original expression.

Do we stop here? Well let’s check the list.

Difference of squares is the next one. To tell if it’s a difference of squares, it must have two terms and also be able to form conjugates (subtraction of two terms).

In our example, (7x – 2) does not have an $x^2$ , so we cannot factor them out any further. Therefore, it is not a difference of squares.

Next step is pattern. Well this doesn’t work because it doesn’t have three terms, therefore, we cannot simplify it any further.

It is therefore : 7x(7x – 2)

Now let’s look at an example with three terms.

First step is looking for something in common. As we can see, there’s nothing in common so let’s go to the next step.

Second, is it a difference of squares? No! There is three terms.

So, thirdly, let’s check if there’s the pattern : $x^2$ $x$ #

Yes!

The first thing to do is draw two brackets and start with putting in x to get $x^2$ . Then factor out all the possibilities for 24.

To decide which two numbers we must use for 24, the two numbers that are multiplying must equal to 10 (the inner term) when added together. So the pair to choose would be 6×4. The sign depends on the inner term. It’s positive and so is 24 so therefore we use ++.

This is the final expression factored.

(x + 4)(x + 6)

To check if it’s correct, we can foil back in the values, meaning distribute all the numbers back by multiplying.

So far, these are both easy. But let’s get into the uglier ones.

There are no common factors here, and there are no differences of squares. There is the pattern though with the three terms, but this one seems a little more complicated than the other one we just did.

A method I learned for these ugly trinomials is using the box.

You split a box into four sections, writing in first the value with $x^2$ which in this example is $25{x^2}$ on the top left corner and the constant on the bottom right corner.