b How can we factor 2xy so that 2x can be created as well?
C
c -8xy and -21 cannot be in adjacent rectangles.
D
d Which terms cannot be in adjacent rectangles?
A
a (x+3)(y-6)
B
b (2x-3)(y+1)
C
c (3-2x)(4y-7)
D
d See solution
Practice makes perfect
a The given expression consists of four terms, which means we need a 2* 2 area model. Let's create such a model and include the first term in the lower left rectangle, which can only be factored as x* y.
Next, we will factor 3y in the upper left rectangle and - 6x in the lower right rectangle. They belong in these spots because of where we placed x and y. this leaves the upper right rectangle left for -18.
Finally, we will calculate the products and label the length and width of the area model.
Now we can write the factored expression.
(x+3)(y-6)
b Like in Part A we have four terms, which means we need a 2* 2 area model. Examining the expression, we should pay attention to the first and third term
2xy-3y+ 2x-3
Since both terms contain 2x, we know that 2xy must be factored as 2x* y in order to create 2x in an adjacent rectangle. With this information, we can determine the contents of two adjacent rectangles of the model.
Now we have enough information to determine the upper rectangles as well. Since -3y contains y as a factor it must go in the upper left rectangle. This leaves -3 for the upper right rectangle.
Finally, we will calculate the products and label the length and width of the area model.
Now we can write the factored expression.
(2x-3)(y+1)
c Like in previous parts we have four terms, which means we need a 2* 2 area model. Notice that -8xy contains both x and y, unlike the last term, 21, which is a constant. Therefore, these terms cannot be placed in adjacent rectangles.
Let's place the two remaining terms in the area model as well. Note that it does not matter which of the remaining rectangles they occupy.
Next, we will find common factors. Let's start with the top row and right column. If we factor -21 as 3*(-7) then 12y would be factored as 3* 4y, and 14x factors to - 2x*(-7).
With this information, we can also identify the factors that create -8xy.
Finally, we will calculate the products and label the length and width of the area model.
Now we can write the factored expression.
(-2x+3)(4y-7)
⇓
(3-2x)(4y-7)
d To determine where to place the terms, we should first identify which two terms should not be in adjacent rectangles of the model. For example, the constant of an expression cannot be in the adjacent rectangle of a term that contains two variables. Otherwise, the constant would not be a constant anymore.
