In this chapter, we have used geometric figures as applications of factoring. In this project, we will illustrate come of these applications. In particular, we will explore the special products. To illustrate why the special products factor as they do, we want to visualize the rectangular objects that are formed.

Part I

We know that a² + 2ab + b²  =  (a + b) ² . We can demonstrate this fact geometrically using the square in the illustration.

1. Determine the area of the shaded square with sides (a + b).

2. Determine the sum of the areas of the four numbered sections A₁, A₂, A₃, and A₄ .

                                            A₁ + A₂ + A₃ + A₄

3. The areas found in exercises 1 and 2 are equivalent. Write an equation to represent this finding.

4. Choose a small positive integer value for a and a different value for b. Show that these values are solutions
    of the equation in exercise 3.

5. On graph paper, draw the square in the illustration, using the values chosen for a and b, from exercise 4.
 

Part II

We know that  a² - 2ab + b²  =  (a - b) ² . We can demonstrate this fact geometrically using the square in the illustration.

1. Determine the area of the shaded square with sides (a - b).

2. Determine the difference of the entire area of the figure, A, and the sum of the areas of the three numbered sections.
 
                                                        A  - (A₁ + A₂ + A₃ )

3. The areas found in exercises 1 and 2 are equivalent. Write an equation to represent this finding.

4. Choose as small positive integer value for a and a different value for b. Show that these values are solutions of the
    equation in exercise 3.

5. On graph paper, draw the square in the illustration, using the values chosen for a and b, from exercise 4.

Part III

We know that a² - b²  =  (a + b)(a - b)  .  We can demonstrate this fact geometrically using the rectangle in the illustration.

1. Determine the area of the shaded rectangle with sides (a + b) and (a - b).

2. Determine the difference of the entire area of the figure, A, and the sum of the areas of the two numbered sections.
 
                                                        A  - (A₁ + A₂ )

3. The areas found in exercises 1 and 2 are equivalent. Write an equation to represent this finding.

4. Choose a small positive integer value for a and a different value for b. Show that these values are solutions of the equation in exercise 3.

5. On graph paper, draw the rectangle in the illustration, using the values chosen for a and b, from exercise 4.