## 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*}

Let’s say we know that the trinomial, \begin{align*}x^2+4x+16\end{align*}

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*}

We can also factor the prefect square trinomial, \begin{align*}x^2-10x+25\end{align*}

As you can see from the picture, we showed one \begin{align*}x^2\end{align*}

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*}

### 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!)