Investigation: Multiplying Binomials
In this investigation, we will explore how to multiply two binomials together, using area of rectangles. After your comfortable with that method, we will use the distributive property and what is called the FOIL method to multiply binomials together. Finally, we will explore ‘special cases’ of binomials to see patterns that we can use to ‘quickly’ multiply these special cases. First, given the picture of a rectangle with length and width labeled, write the area formula for a rectangle here: A = _____________________. Now look at the area model below, where a binomial is written along the side of the boxes vertically and another binomial is written at the top of the box horizontally. X
X
+
+
4
2
In the above example, X represents the length and width of the first square. In the top right rectangle, X represents the length and 4 represents the width. In the bottom left rectangle 2 represents the length of the rectangle and X represents the width. In the bottom right rectangle, 2 represents the length and 4 represents the width. Using the concept of area (formula you wrote above), write the area of each rectangle inside each box. Now, take the area of each and write them in the blanks below and then combine like terms on the right hand side of the equation: __________ + __________ + __________ + __________ = _____________________________________ 1 You have just successfully multiplied two binomials together. Examine your result. What type of polynomial did you get as a result? ____________________________ Let’s try another one: 2X
+
1
X
+
2
Write the area of each rectangle inside each box. Then list the areas in the blanks below (left side of the equation) and then combine like terms on the right hand side of the equation. __________ + __________ + __________ + __________ = _____________________________________ Again, what type of polynomial did you end up with? ____________________________ In the above example, you multiplied X by 2X and X by 1. Then you multiplied 2 by 2X and 2X by 1. Did you notice that all terms from one binomial got multiplied by both terms from the other binomial? We now explore another method: Using the Distributive Property The example from above is written as (X + 2)(2X+1). Use the Distributive Property to multiply this out. In other words, distribute the X and the 2. (X + 2)(2X+1) = X(2X + 1) + 2(2X+1) Notice that the X and 2 are being distributed here. = __________ + __________ + __________ + __________ (Use Distributive Property = _____________________________________ (Check your answer from the model above…what do you notice?) 2 You should have found that your answer using areas of rectangles is the same as that of using distributive property. There is a method that we use that helps us remember this method. It is called FOIL. FOIL stands for: First Outer Inner Last This means that we multiply the First term in each binomial together, followed by the two ‘outside’ terms. Then we multiply the two ‘inside terms’, and then the last term in each binomial. Here is a diagram to help (using the previous example): First Outer Inner Last (x + 2)(2x+1) = (x)(2x) + (x)(1) + (2)(2x) + (2)(1) = __________ + __________ + __________ + __________ = _____________________________________ Again, you should be finding the same answer as on the previous page. Note: In mathematics we usually write things in descending order which means that the first term includes the highest power of x, the second term has the second highest power of x, etc. So, now that you know the FOIL method, try some: 1. (x + 4)(x + 3) 2. (x -­‐ 6)(x + 2) 3. (x -­‐ 5)(x -­‐ 4) 4. (y + 6)(y + 5) 5. (2x + 1)(x + 2) 6. (y + 6)(3y + 2) 3 7. (x + 5)(3x -­‐ 1) 8. (2x -­‐ 1)(x -­‐ 3) 9. (3x -­‐ y)(x + 2y) 10. (3x + y)(x + y) What type of polynomial did all of these problems result in? _____________________________ Now we are going to look at some ‘special cases’. First Case: (a + b)2. Question: What happens if the terms in each
binomial are the same?
Multiply these binomials. Simplify your answer.
1. (x+7)(x+7)
3. (2x + 1)(2x + 1)
2. (x + 3)(x + 3)
4. (4x + 1)(4x + 1)
Note: The above can be written as (x+7)2, (x+3)2, (2x+1)2, etc.
Now fill out the table below: (Remember descending order!)
Problem
Number
1.
2.
3.
4.
First Term of
binomial
First term of
answer
Last Term of
binomial
Last term of
answer
4 Do you notice a pattern between the first term of the binomial in the
problem and the first term of your answer? If so, what is it?
Do you notice a pattern between the last term of the binomial in the
problem and the last term of your answer? If so, what is it?
Problem
Number
1.
2.
3.
4.
Product of
‘Outside Terms’
Product of
‘Inside Terms’
Middle Term of
Trinomial (Answer)
Examining the table above, what do you notice about the product of the
outside terms and inside terms?
What is the relationship between the product of the outside terms and the
middle term of the trinomial final answer?
5 Using the area model below, express a general rule for multiplying two
binomials that are alike:
a
+
b
a
+
b
General rule: (a +
= (a + b)2 =
b)(a + b)
_______________________________
Now try these examples to come up with a general statement for
the Second Case: (a – b)2. You may use a table to help you (space
provided on the next page for you to create your own).
1. (x – 3)2
2. (x – 4)2
6 3. (2x – 1)2
4. (3x – 4)2
Generalizing (a – b)2. Use the space below to create a table or tables to
look for patterns, as well as explain your findings. At the bottom of the
page, write your general rule for finding (a – b)2.
7 General rule: (a - b)2 = _______________________________
Final Case: (a + b)(a – b)
Use FOIL to multiply these examples to their final answer. Note the type
of polynomial you find, and then use strategies to look for a pattern and
come up with a general statement for your findings.
1. (d + 11)(d – 11)
2. (x + 7)(x – 7)
3. (2x + 3)(2x – 3)
4. (5y – 2)(5y + 2)
What type of polynomial did you get in your answer?
__________________
Now use strategies to look for a pattern and come up with a general
statement for your findings.
8 General Statement:
________________________________________________________
Conclusion:
You learned how to multiply binomials using an area model and FOIL.
Then you explored special cases, including (a + b)2, (a – b)2, and (a+b)(a
– b). Summarize your findings below:
FOIL represents:
F
O
I
L
(a + b)2 = _______________________________
(a - b)2 = _______________________________
(a + b)(a – b) = _______________________________
9