Understanding the Distributive Property

Think about multiplying 3 by 12, we could instead think of 3 × (10 + 2) and we can model it with the base ten blocks below.

The model shows us that this is clearly three sets of ten and three sets of two. We could also consider an area model that is three rows high and 12 units long as in the diagram below.

There are three rows of ten (the long rectangle) and three rows of two (the small unit square), giving the solution 36.

Similarly, we can use the same type of model for 3 (x + 2). In this case the length of the rectangle is x, an unknown value, and the unit square still represents 1. The area of the rectangle is x units since its dimensions are x by 1. Thus the area of the figure below is 3x + 6.

3 (x + 2) = 3 ∙ x + 3 ∙ 2 = 3x + 6

We can use the same method to multiply an algebraic term by a binomial such as 4x(x–4) as in the diagram below.

In this diagram we see large squares with dimensions x by x and area x2. The rectangles have area –x because they are -1 by x. The height of the rectangle is 4x units and the width is x – 4 units. This gives the area 4x2 – 16x.

4x (x – 4) = 4x ∙ x + 4x ∙ -4 = 4x2 – 16x

Use the examples above to expand the following:

  1. 5 (x – 2)
  1. 3x (x + 4)
  1. 2x(3x + 5)
  1. 6x(2x – 3)

Binomial Expansion

We can use the method of area model to multiply two binomials as well. Consider the example (x + 3)(x + 2). We create a rectangular area with height (x + 3) units and width (x + 2) units. The solution is the area of the rectangle.

The large square has dimensions x by x so its area is x2. The long rectangles have dimensions 1 by x, thus area x units. The small squares have dimension 1 by 1 and area 1 unit. The area of the entire rectangle is x2 + 5x + 6.

Here is another example: (2x – 1)(x + 5)

Here we get an area that is 2x2 + 10x – x – 5 = 2x2 + 9x – 5. Note that it is very important to pay attention to the sign (+ or -) of a term in order to ensure the proper solution.

Use area models to expand the following.

  1. (x + 4)(x + 2)
  1. (x – 3)(x + 5)
  1. (x – 2)(x + 2)
  1. (3x + 2)(2x – 1)
  1. (2x + 3)(x + 4)

Do you notice a pattern? What generalizations might you make about a process for expanding binomials?


Factoring

If you have 12 squares, you can arrange them into a rectangle to find factors of 12. Two possible arrangements are shown below. One shows that 4 x 3 = 12, the other shows that 2 x 6 = 12.

We can also do this with polynomial terms to find the factors. Consider 2x2 + 6x.

When we arrange them like this we see that the factors are 2x and x + 3.

2x2 + 6x = 2x(x + 3)

Consider another example: 3x – 12

We see that the rectangle is 3 units high and x – 4 units wide.

Note that it is very important to pay attention to the signs.

3x – 12 = 3(x – 4)

Factor each of the following using area models.

  1. 4x + 12
  1. 3x2 - 9x
  1. 12x2 + 8x
  1. 2x2 - 10x
  1. 10x2 + 5x


We can also factor trinomials in the same way. Consider a model for x2 + 5x + 6. Can we arrange the tiles into a rectangle?

The height of the rectangle is (x + 2) and the width is (x + 3). Thus we see that (x + 2)(x + 3) = x2 + 5x + 6. So (x + 2) and (x + 3) are the factors of the trinomial x2 + 5x + 6.

When there are negatives the factoring becomes more challenging. Consider . Can this be arranged into a rectangle?

The height is (x – 1) and the width is (x – 4), so these are the factors of , i.e. (x – 1)(x – 4) = .

Sometimes you may need to use the zero principle in order to create the rectangle. Take the example of . With only these tiles you cannot form a proper rectangle but if you add a +x and a –x you can complete the rectangle.

We now see the factors are (x – 1) and (x + 3).

Use your tiles to find the factors of each of the following.

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