Exercise 55 Page 30

$2x+6$

We have already made a rectangle in Part A to represent this area and concluded that the factored form of $2x+6$ is $2(x+3).$

$3x+3$

Looking at this expression, we can tell that we need $3$ $x\text{-}$tiles and $3$ positive $1\text{-}$tiles to form a rectangle to represent this area. Let's stack the $x\text{-}$tiles on top of each other.

Now we have to add the $1\text{-}$tiles to form a rectangle. Since there are $3$ rows and $3$ positive $1\text{-}$tiles, we can add one tile to each row.

Finally, we can tell that the area of this rectangle is $3(x+1),$ which is the factored form of $3x+3.$

$3x-12$

As in the previous example, we can see that we need $3$ $x\text{-}$tiles to form a rectangle with area $3x-12.$ Since we have $\text{-}12,$ we also need $12$ $\text{negative}$ $1\text{-}$tiles for our rectangle. Let's begin with the $x\text{-}$tiles!

Since we have $3$ rows to distribute the $\text{negative}$ $1\text{-}$tiles equally, we have to divide $12$ by $3.$ We need to add $4$ $\text{negative}$ $1\text{-}$tiles to each row.

The area of this rectangle is $3(x-4),$ which is also the factored form of $3x-12.$

$5x+10$

Again, analyzing this expression we can tell that we need $5$ $x\text{-}$tiles and $10$ positive $1\text{-}$tiles to form a rectangle with area $5x+10.$ Let's begin with the $x\text{-}$tiles!

We have $5$ rows and we want to place the $10$ remaining tiles equally to form a rectangle. Since $10÷5=2$ we can add $2$ positive $1\text{-}$tiles to each row.

The area of this rectangle is $5(x+2),$ which is also the factored form of $5x+10.$

Table

With the factored form of each expression we can complete the table.

Area	Factored Form
$2x+6$	$2(x+3)$
$3x+3$	$3(x+1)$
$3x-12$	$3(x-4)$
$5x+10$	$5(x+2)$

Let's illustrate this using the expression $4x+16.$

Each time we divided by the coefficient of $x.$ However, the constant might not be divisible by this number. Then we can find the greatest number which divides both terms and factor it out. Let's factor $6x+16$ using this method.