a The area of a rectangle is obtained by multiplying its length times its width.
B
b Expand the products of binomials representing the areas and compare them directly.
C
c Can you factor any of the expressions for the areas even more?
A
a (x+2)(2x+2) and (2x+4)(x+1).
B
b Yes.
C
c See solution.
Practice makes perfect
a We are given the two area models shown below.
Notice that the side lengths of these models is indicated in the diagram. Furthermore, since both of them are rectangles is will be useful to recall the formula for the area of a rectangle.
A = Length * Width
Therefore, to find an expression for the area of each model it is enough to substitute the side lengths given in the diagram into the formula for the area of a rectangle.
b One way to compare the areas of both area models is by expanding the expressions found in Part A. After that, we can compare them directly. To expand the products we can use the FOIL Method. Let's give it a try.
Now let's do the same with the expression for the area of the other model.
As we can see, both expressions expand to the trinomial 2x^2+6x+4. Therefore, the area of both models is the same.
c To understand why both products expand to the same trinomial, notice that both expressions for the area of the models has a binomial with a common factor of two. Therefore, they are not completely factored yet. Let's factor them completely and compare the resulting expressions.
Expression for the area of model 1
Expression for the area of model 2
( 2x+ 2)(x+2)
(x+1)( 2x+ 4)
Factor out 2
2(x+1)(x+2)
2(x+1)(x+2)
As we can see, after factoring completely both expressions become the same — 2(x+1)(x+2). This is why they equal the same trinomial, because they are equivalent expressions.
