Representing Area with Binomial Multiplication
In the image below, we see a floor plan of an office space. In the office, there is the main section with workspaces and cubicles, a kitchenette, a break room, and a bathroom:
Note that we know some measurements of the areas in the office, while other measurements are unknown. Adding the partial dimensions together, we can express the area of the entire office using binomial multiplication.
The area of the room can be expressed as: (x + 8)(x + 5)
We can expand binomial multiplication using a process known as FOIL. FOIL stands for First, Outside, Inside, Last, and is used as an aid in remembering which terms to multiply together to expand two binomials being multiplied together.
Using FOIL, we see that we can equivalently express the area as:
Notice that when we express the area using 4 terms, each term represents the area of the individual sections to the office:
We can, and should, further simplify the area by combining like terms: x2 + 13x + 40
Subtracting Area with Binomial Multiplication
At a different office space, workers are coming in to install carpeting on the floor. However, they need to leave a boarder of wooden floor around the office for certain equipment. The floor plan is illustrated below:
We can represent the area of the carpet using binomial multiplication as well. However, our binomials will include subtraction, because we need to take away from the dimension of the office space.
The area of the carpet can be expressed as: (12 – 2x)(10 – 2x)
We subtract 2x in this case, because a distance of x feet is being trimmed from both sides of the length and width of the room.
Using FOIL, we can equivalently express the area as a polynomial in expanded form. This is illustrated below: