We're asked to determine the product of the two binomials. We'll determine this
product using two methods. We'll use an area model as well as repeated distribution.
Using the area model we can determine this product by determining the area of
a rectangle.
Remember the area of a rectangle is equal to length times width. So for this
product we'll let x + 3 be equal to the length and x + 4 be equal to the width. Using
this rectangle here
let's label this length x plus 3. By labeling it like this
we're saying this blue length is equal to X units and this green length is
equal to three units. And we'll label length of this side x plus 4. So again
the blue length here is equal to X units and this green length is equal to four
units.
Notice how the area of the rectangle would be equal to the sum of these four
smaller areas and the sum of these four small areas is equal to the product of
the two binomials.
Now let's find the area of each of these smaller rectangles. Looking at the blue
area notice how this length is X and this length is x and therefore this is a
square. The area is equal to x times x which equals x squared.
Now let's look at this green rectangle. The dimensions are four units by X units and
therefore the area is equal to 4 times X or 4x. Looking at the second green area,
notice the dimensions are three units by X units and therefore the area is equal
to 3. And then finally this last rectangle has dimensions four units by three units
and therefore the area is 3 times 4 which equals 12. The sum of these four areas
equals the area of the rectangle and also the product of the binomials. Using
the area model, the quantity x + 3 times quantity x + 4 is equal to x squared
plus 4x
plus 3x plus 12.
Notice how this can be simplified though because 4x and 3x are like terms.
4x + 3 x is equal to 7x. And therefore the product is equal to x squared plus
7x plus 12.
Now let's also find this product using repeated distribution.
Looking back at the model, notice how to determine the product of two binomials
we ended up finding four areas or four products. Now let's find the same
product using repeated distribution when multiplying two binomials we multiply
both terms in the first binomial by both terms in the second binomial which means
we can think of this as a repeated distribution. Beginning with the first
term in the first binomial we distribute this X into the second binomial giving
us one two products. And then we do distribute the second term in the first
binomial into the second binomial giving us three and four products.
Let's go and find the four products distributing the x, x times x equals x
squared x times 4 is equal to 4x so we have plus 4x and now we distribute the 3
x times 3 is 3x so plus 3x and finally plus 3 times 4 which gives us plus 12. Notice the
some of these four terms is the same sum we used to determine the area of
the rectangle shown here.
So combining like terms notice how we also get x squared plus 7x plus 12.
Let's look at one more example and just use the repeated distribution technique.
Let's determine the product of these two binomials. We have the quantity X minus 5
times the quantity 2x plus 7.
We now know we'll have 4 products. We find the first two products by
distributing the X in the first binomial into the second binomial. W find the
other two products by distributing the second term in the first binomial into
the second binomial. Again notice how we have 4 products. So we have x times 2x
which equals two x squared plus X times seven that would be 7x so plus 7x. Now
because of the subtraction here, we view this term is negative 5 and we
distribute negative 5.
So negative five x 2x is negative 10x instead of writing plus negative 10x
we'll just write minus 10x . And then we have plus negative 5 times positive 7 that wouuld be plus
negative 35 or minus 35. And now we simplify by combining like terms
7x minus 10 x equals negative 3x and therefore the product simplifies to 2 x
squared minus 3x minus 35
I hope you found this helpful