Question

How do you factor `12y^2 + 7y + 1`?

Answer 1

Hi there. The polynomial you have here is called a " complex trinomial" where a (the coefficient with `x^2`) > 1. For these polynomials, there are several ways to factor it, but the most straight forward method is called "decomposition".

Explanation:

Let's start off by recalling the general/standard form of a quadratic expression:

`f(x) = ax^2 + bx + c`

Now, to factor a complex trinomial using decomposition follow the following method:

First, multiply your first and last numbers (a and c) together and write this off to the side. I'll refer to this as the "ac" term. In your case, this is 12 (since 12 x 1 = 12).

Now, you want to determine factors of ac that ADD to give the "b" term (7). Your options include:

1 and 12
2 and 6
3 and 4

So, which of these pairs add to give 7? Obviously, 3 and 4!

Now, we call this method decomposition because we are going to break down the b term into the 3 and 4 that we determined in the last step.

The new polynomial will look something like this:

`12x^2 +`( blank )`x +` ( blank )`x + 1`

Now, this part requires a bit of strategy. With the numbers you determined earlier (3 and 4), you want to put them in a place so that if you were to group the first two terms together, and the last two terms together, then common factor something out, you would be left with the exact same thing left over. Lets see this in action:

Lets try it by writing the 4 first, then the 3 like so...

`12x^2 + 4x + 3x + 1`

Now group the first two and last two terms together like so:

`(12x^2 + 4x) + (3x + 1)`

Now, if possible, you'll want to pull out a common factor from these brackets. From the first bracket, you can pull out a 4x. From the second bracket, you can pull out a 1 (this does't change anything). Doing this should look like so:

`4x(3x+1) + 1(3x+1)`

Notice how the two resulting brackets are the same? This means you've factored correctly! To simplify this, take one copy of the (3x+1) and collect what is left over in another bracket like so:

`(4x+1)(3x+1)`

And that's it! You're fully factored! Just for your understanding, lets take a look at what the polynomial would have looked like if you were to put the 3 before the 4 when breaking down the "b" term:

`12x^2 + 3x + 4x +1`

`(12x^2 + 3x)+(4x+1)`

`3x(4x+1)+1(4x+1)`

You end up getting the same thing! It's important to note that this will not always happen. Always make sure that you get two sets of brackets that contain the same thing!

Hopefully everything was clear! If you have any questions, feel free to ask! :)