## Real World Applications – Algebra I

### Topic

Using Area Models to Factor “Special” Polynomials

### Student Exploration

The most common way to know how to factor special polynomials that are either a difference of squares or perfect square trinomials is to identify them by square roots. But how can we know how to factor them (or know which are the correct factors) once we’ve identified the trinomial as “special”?

Algebra tiles are a great tool to help students factor trinomials. First, let’s identify algebra tiles and understand what they mean.

The picture below shows the different algebra tiles, along with what they represent. Note that the “ones” tile represents 1 unit, and notice the \begin{align*}x-\end{align*}tile, the \begin{align*}x^2\end{align*} tile, the \begin{align*}y-\end{align*}tile, the \begin{align*}y^2\end{align*} tile, and the \begin{align*}xy\end{align*} tile. These tiles are identified by their areas \begin{align*}(length \times width)\end{align*}. (For the examples throughout, we will not be focusing on the \begin{align*}xy-\end{align*}tile.)

Let’s say we know that the trinomial, \begin{align*}x^2+4x+16\end{align*} is a perfect square trinomial, but you don’t know how to factor it. We can show this using an area model using the algebra tiles above. See the picture below. We have one \begin{align*}x^2\end{align*} tile, four \begin{align*}x-\end{align*}tiles, and 16 ones tiles.

We want to make sure that the structure we make with the algebra tiles in the polynomial is a square so it’s easy for us to factor.

I’m also going to label the sides of our rectangle. Each side of our area model is a factor in our expression!

Notice that in this picture above, I labeled everything. I labeled each tile, to make sure that there was one \begin{align*}x^2-\end{align*}tile, 8 \begin{align*}x-\end{align*}tiles, and 16 ones tiles. I also labeled each side of the big rectangle, to show that the whole top is length \begin{align*}x + 4\end{align*}, and the whole vertical side is also length \begin{align*}x + 4\end{align*}. We know that area of a rectangle is length \begin{align*}x\end{align*} width, so the area of the large square is \begin{align*}(x + 4)(x + 4)\end{align*}, or \begin{align*}(x + 4)^2\end{align*}. We just factored this perfect square trinomial!

We can also factor the prefect square trinomial, \begin{align*}x^2-10x+25\end{align*}. We know this is a perfect square trinomial because 25 is a perfect square, and \begin{align*}x^2\end{align*} is also a perfect square. Let’s make a diagram for this trinomial.

As you can see from the picture, we showed one \begin{align*}x^2\end{align*} tile, 10 of the \begin{align*}-x\end{align*} tiles (they’re shaded in a darker color), and 25 ones tiles. In the picture below, we’ve labeled each of the sides of the big square that this expression makes.

One step further, we simplify each of the sides of this square to find the factors of this perfect square trinomial.

So, we know that our factors are \begin{align*}(x-5)(x-5)\end{align*}, or \begin{align*}(x-5)^2\end{align*}.

### Extension Investigation

How do you think you can demonstrate a difference of squares using an area model? What other math concepts do you need to know in order to make this area model? (Hint: You have to know something about creating “zeros” from algebra tiles!)