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Section 7.1 Organization Will Set You Free

Multiplication can be represented in several different ways. For this section, we're going to focus on the idea of multiplication as the area of a rectangle. The area of a rectangle with length and width is given by \(A = \ell w\text{.}\) This can also be seen by drawing out the grid and counting the squares, though it's a waste of time to do it in practice. So we will often represent this symbolically by simply using a box to represent the idea of the calculation.

One of the advantages of reducing the idea to just a representation is that it creates a space for us to work with abstract ideas. Once we stop focusing on specific numbers, we can start to use symbols to represent the idea of creating a grid. Consider the following examples:

We may not know the particular values of the variables and but we see that if we knew what those values were, then we would be able to draw out the grid to count out the number of squares, and that the number of squares would correspond with the expression inside the box.

We can push this even one step further and think about products that don't even have a proper physical representation:

Activity 7.1.1. Products of Monomials.

Before we can put this to work for polynomials, we first need to focus a bit on multiplying monomials. There were a few problems in the worksheets that hinted at how this works, but we'll formally practice some of those manipulations here.

When multiplying monomials, the result is actually just a single product involving a bunch of different factors. There is usually a number part (from the coefficients) and then there is usually some combination of variables. And all we need to do is properly account for them.

\begin{equation*} \begin{aligned} 5x^2y \cdot 2 x^2 y^2 \amp = (5 \cdot x \cdot x \cdot y) \cdot (2 \cdot x \cdot x \cdot y \cdot y) \amp \eqnspacer \amp \text{Definition of exponents} \\ \amp = 5 \cdot 2 \cdot x \cdot x \cdot x \cdot x \cdot y \cdot y \cdot y \amp \amp \text{Rearranging the factors} \\ \amp = 10 x^4 y^3 \amp \amp \text{Definition of exponents} \end{aligned} \end{equation*}

This can also be done using the properties of exponents:

\begin{equation*} \begin{aligned} 5x^2y \cdot 2 x^2 y^2 \amp = 5 \cdot 2 \cdot x^{2 + 2} \cdot y^{1 + 2} \amp \eqnspacer \amp \text{Properties of exponents} \\ \amp = 10 x^4 y^3 \amp \amp \text{Arithmetic} \end{aligned} \end{equation*}

But in the long run, it will be important to be able to do this as just one step.

\begin{equation*} \begin{aligned} 5x^2y \cdot 2 x^2 y^2 \amp = 10 x^4 y^3 \amp \eqnspacer \amp \text{Multiplying monomials} \end{aligned} \end{equation*}

Try it!

Compute \(-3a^3 b^2 \cdot 6 ab^3\text{.}\) Use a presentation that matches each of the three examples above.

Solution.
\begin{equation*} \begin{aligned} -3a^3 b^2 \cdot 6ab^3 \amp = (-3 \cdot a \cdot a \cdot a \cdot b \cdot b) \cdot (6 \cdot a \cdot b \cdot b \cdot b) \amp \amp \text{Definition of exponents} \\ \amp = -3 \cdot 6 \cdot a \cdot a \cdot a \cdot a \cdot b \cdot b \cdot b \cdot b \cdot b \amp \amp \text{Rearrange the factors} \\ \amp = -18a^4 b^5 \amp \amp \text{Definition of exponents} \end{aligned} \end{equation*}
\begin{equation*} \begin{aligned} -3a^3 b^2 \cdot 6ab^3 \amp = -3 \cdot 6 \cdot a^{3+1} \cdot b^{2+3} \amp \amp \text{Properties of exponents} \\ \amp = -18a^4 b^5 \amp \amp \text{Arithmetic} \end{aligned} \end{equation*}
\begin{equation*} \begin{aligned} -3a^3 b^2 \cdot 6ab^3 \amp = -18a^4 b^5 \amp \amp \text{Multiplying monomials} \end{aligned} \end{equation*}

Activity 7.1.2. Products of Monomials with Grid Representations.

We can use this framework to set up products using the grid representation that was introduced above.

Try it!

Complete the products in the following boxes.

Solution.

We will now start thinking about how to extend this idea to products of polynomials. Just as before, we will start with concrete values before looking at abstract ideas. Let's say that we wanted to calculate \(18 \cdot 6\text{.}\) We can draw out the picture, but there are a whole lot of squares to count.

So we might start to think about how we can organize this in a more sensible manner. With a little bit of thinking, you might realize that breaking up the into \(10 + 8\) might be helpful. Notice that this doesn't change the number of boxes. However, it does rearrange the information in a way that's more useful to us because we can work with more familiar multiplication calculations.

Activity 7.1.3. Product of Binomials Using a Grid.

We can use the same concept for polynomials. For example, we could write the product \((x + 2)(x - 4)\) using the following grid:

The grid itself is just a representation of the calculation. We would still need to use formal mathematical writing to present the calculation.

\begin{equation*} \begin{aligned} (x + 2)(x - 4) \amp = x^2 + 2x - 4x - 8 \amp \eqnspacer \amp \text{Distributive Property} \\ \amp = x^2 - 2x - 8 \amp \amp \text{Combining like terms} \end{aligned} \end{equation*}

The grid should be understood as scratch work. Scratch work is similar to an outline for an essay. It's important and helpful for keeping yourself organized, but it's not part of the final product. In the end, there must always be enough information in the final presentation so that other people reading your work can know what you did.

Try it!

Calculate \((2x - 3)(x + 4)\) using a grid. Write up your result using a complete presentation, being sure to simplify by combining like terms.

Solution.
\begin{equation*} \begin{aligned} (2x - 3)(x + 4) \amp = 2x^2 - 3x + 8x - 12 \amp \amp \text{Distributive property} \\ \amp = 2x^2 + 5x - 12 \amp \amp \text{Combining like terms} \end{aligned} \end{equation*}

There are times when it's helpful to make variables a particular length so that they're different from numbers. For example, we could physically represent the product \((x + 2)(x + 1)\) using the following diagram.

We will be using this representation to help us to think about factoring in the next section.

You are probably familiar with FOIL (First-Outer-Inner-Last) as a way to do this product. One of the downfalls to FOIL is that students end up confused when there are more than two terms in the parentheses. Many are so trapped by FOIL that they try to force it to happen even in problems that do not call for it. The basic problem is that students do not have an organized sense of what FOIL is supposed to accomplish, and so they treat it like a rule to be blindly followed.

However, this grid method extends very naturally regardless of the number of terms in the parentheses. And it's completely built around the basic idea of thinking of multiplication as an area. Having the right organizational scheme frees you from having to memorize more and more rules.

Activity 7.1.4. Larger Products Using a Grid.

Try it!

Calculate \((x^2 + 3x -4)(x^2 - 5x + 2)\) using a grid. Write up your result using a complete presentation, being sure to simplify by combining like terms.

Solution.
\begin{equation*} \begin{aligned} \amp (x^2 + 3x -4)(x^2 - 5x + 2) \\ \amp \qquad = x^4 + 3x^3 - 4x^2 - 5x^3 - 15x^2 + 20x + 2x^2 + 6x - 8 \amp \amp \text{Distributive property} \\ \amp \qquad = x^4 + 3x^3 - 5x^3 - 4x^2 - 15x^2 + 2x^2 + 20x + 6x - 8 \amp \amp \text{Rearrange the terms} \\ \amp \qquad = x^4 - 2x^3 - 17x^2 + 26x - 8 \end{aligned} \end{equation*}