10.2: Factoring Special Cases
This activity is intended to supplement Algebra I, Chapter 9, Lesson 6.
In this activity, you will explore:
- factoring a perfect-square trinomial
- factoring a difference of squares
- using geometry to prove rules for factoring special quadratic expressions
Problem 1 - Factoring a Perfect-Square Trinomial
Any trinomial of the form \begin{align*}a^2 + 2ab + b^2\end{align*} is a perfect-square trinomial. If you recognize a perfect-square trinomial, you can factor it immediately as \begin{align*}(a + b)^2\end{align*}.
To see why \begin{align*}a^2 + 2ab + b^2 = (a + b)^2\end{align*} , start the CabriJr app by pressing the APPS button and choosing it from the menu.
Open the file FACTOR1 by pressing Y= to open then F1: File menu, choosing Open, and choosing it from the list.
This file shows \begin{align*}2\end{align*} squares and \begin{align*}2\end{align*} rectangles, with their dimensions labeled.
What is the area of each shape? On the screenshot at right, label each shape with its area.
- Arrange the shapes to form a square. To move a shape, move the cursor over it (so that the entire shape becomes a moving dashed line) and press ALPHA to grab it, then move it with the arrow keys. When the shape is positioned where you want it, press ENTER to let it go.
- The area of this square is equal to the sum of the areas of the shapes that make it up. What is the area of the square? Have you seen this trinomial before?
- How long is one side of the square?
- Using the formula \begin{align*}A = s^2\end{align*} for the area of a square with side length \begin{align*}s\end{align*} what is the area of this square?
You have shown that the area of this square is equal to \begin{align*}a^2 + 2ab + b^2\end{align*} and also equal to \begin{align*}(a + b)^2\end{align*}. Therefore \begin{align*}a^2 + 2ab + b^2 = (a + b)^2\end{align*} You have proved the rule for factoring a perfect-square trinomial!
Problem 2 - Factoring a Difference of Squares
Any trinomial of the form \begin{align*}m^2 - n^2\end{align*} is a difference of squares. If you recognize a difference of squares, you can factor it immediately as \begin{align*}(m + n)(m - n)\end{align*}.
To see why \begin{align*}m^2 - n^2 = (m + n)(m - n)\end{align*}, start the CabriJr app by pressing the \begin{align*}A\end{align*} button and choosing it from the menu.
Open the file FACTOR2 by pressing \begin{align*}Y=\end{align*} to open then F1: File menu, choosing Open, and choosing it from the list.
This file shows \begin{align*}2\end{align*} squares with their dimensions labeled \begin{align*}m\end{align*} and \begin{align*}n\end{align*}.
What is the area of each square? On the screenshot at right, label each square with its area.
How can you represent the area \begin{align*}m^2 - n^2\end{align*} with these squares? Move the \begin{align*}n^2\end{align*} square on top of the \begin{align*}m^2\end{align*} rectangle so that their corners align. If you imagine cutting the smaller square out of the larger square, the \begin{align*}L-\end{align*}shaped area that remains is equal to \begin{align*}m^2 - n^2\end{align*}.
We know that the area of the \begin{align*}L-\end{align*}shape is \begin{align*}m^2 - n^2\end{align*}, but there is also another way to find its area: by taking it apart and rearranging the pieces into a single long rectangle.
Open the CabriJr file FACTOR3, which shows the same shapes, but with the \begin{align*}L-\end{align*}shaped area \begin{align*}(m^2 - n^2)\end{align*} divided into two rectangles.
Rotate the smaller rectangle about point \begin{align*}P\end{align*} (at the bottom of the screen) clockwise \begin{align*}90^\circ\end{align*}. Press TRACE to open the F4: Transform menu and choose Rotation. Move the cursor over the rectangle to highlight it and press \begin{align*}e\end{align*} to choose it. Then move the cursor to the point you want to rotate around and press ENTER. Finally, mark the angle of rotation by choosing points \begin{align*}A, B\end{align*}, and \begin{align*}C\end{align*} in turn.
Hide the original small rectangle and the vertices of the rotated image. (Press GRAPH to open the F5: Appearance menu and choose Hide/Show \begin{align*}>\end{align*} Objects. Then choose the rectangle and vertices you want to hide.) Now there are two rectangles whose combined area is equal to the area of the original \begin{align*}L-\end{align*}shape.
Move the larger rectangle (the small rectangle cannot be moved) alongside the rotated image to form one long rectangle.
What are the dimensions of the long rectangle?
Using the formula \begin{align*}A = lw\end{align*} for the area of a rectangle and these dimensions, what is the area of this rectangle?
You have shown that the \begin{align*}L-\end{align*}shaped area is equal to \begin{align*}m^2 - n^2\end{align*} and also equal to \begin{align*}(m + n)(m - n)\end{align*}. Therefore \begin{align*}m^2 - n^2 = (m + n)(m - n)\end{align*}. You have proved the rule for factoring a difference of squares!