THE CASE OF THE MISSING MIDDLE TERM
EXAMPLES
Consider the two squares that are shown.
b
1. What is the area of the smaller
square? Answer Îb2
a
b
2. What is the area of the larger square?
Answer Îa2
a
3. If I took a pair of scissors and cut square “b” from square “a,”
what “area” would be remaining?
Answer Î a2 – b2
b
b
b
b
a
a
a
a
a-b
b
b
Region 1
b
b
a
Region 2
a-b
a
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a-b
a
b
Region 1
a
b
Region 1
a-b
a-b
Region 2
b
a
Rearranging Region 1, these regions can be reassembled as shown below. You can see
that a2 – b2 is equal to the product of (a – b)(a + b).
a
(a + b)
b
Region 1
a-b
a-b
Region 2
b
(a + b)
a
4. Have students complete these products:
a. (a + 4)(a – 4) Îa2 – 16
b. (p – 5)(p + 5) Îp2 – 25
c. (4m – 1)(4m + 1) Î16m2 – 1
d. (3x + 2y)(3x – 2y) Î9x2 – 4y2
e. (a + b)(a – b) Îa2 – b2
(a + 4)(a – 4) Î a2 – 16
The first term is the square of the first
product, and the last term is the square of the
last product.
5. From the examples above, have the students make comments about the pattern
that is being developed.
6. Introduce students to the “Difference of Squares” equation Î
a2 – b2 = (a + b)(a – b). Students should memorize the form of a difference of two
squares.
7. Factor a2 – 64 using “Difference of Squares.”
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a2 – 64
(a)2 – (8)2
(a – 8)(a + 8)
Rev. 06.09.03
8.
Factor m2 – 4 using “Difference of Squares.”
m2 – 4
(m)2 – (2)2
(m – 2)(m + 2)
9x2 – 100y2
(3x)2 – (10y)2
(3x – 10y)(3x + 10y)
9. Factor 9x2 – 100y2 using “Difference of Squares.”
4x2 – 25m2
(2x)2 – (5m)2
(2x – 5m)(2x + 5m)
10. Factor 4x2 – 25m2 using “Difference of Squares.”
11. Factor
1 2 4 2
x – y using “Difference of Squares.”
4
9
12. Factor
1 2 9 2
x –
y using “Difference of Squares.”
16
25
1 2 4 2
x – y
4
9
1 2
2
( x) – ( y)2
2
3
1
2
1
2
( x – y)( x + y)
2
3
2
3
1 2 9 2
x –
y
16
25
1
3
( x)2 – ( y)2
4
5
1
3
1
3
( x – y)( x + y)
4
5
4
5
13. Review the FOIL method with students. For example, multiply (x + 2)(x + 5).
F
O
I
L
FIRST OUTER INNER LAST
(x + 2)(x + 5) = x2 + 5x + 2x + 10 = x2 + 7x + 10
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14. Ask the students to use the FOIL method on (7 – 12) and explain why there is
only two terms in the finished product when you are multiplying two terms times
two terms (two terms)(two terms) which should yield four terms.
ANSWER Î Middle terms cancel.
Sometimes the terms of a binomial have common factors. If so, the GCF (Greatest
Common Factor) should be factored out before factoring the “difference of squares.”
Occasionally, factoring the “difference of squares” needs to be used more than once in
a problem.
2
2
15. Factor 8x –18y using “Difference of Squares.”
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8x2 –18y2
2(4x2 – 9y2)
2(2x – 3y)(2x + 3y)
Rev. 06.09.03
Name:___________________
Date:_________
Class:_________________
THE CASE OF THE MISSING MIDDLE TERM
WORKSHEET
State whether each binomial can be factored using the “difference of squares.”
1. x2 – y2
6. 9x2 – 5
2. x2 + y2
7. 1.44x2 – 0.45y2
3. x2 – 25
8.
4 2 1
x −
9
4
9.
16 2
x +1
25
4. x2 – 4y4
5. 9x2 + y2
Factor each polynomial completely! Check Using FOIL method.
10. a2 – 9
17. 2x2 – 98
11. x2 – 49
18. 12a2 – 48
12. 4x2 – 9y2
19. 8x2 – 18
13. x2 – 36y2
20.
1 2
x − 25
16
14. 1 – 9y2
15. 49 – 4a2b2
16. 2a2 + 18a
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Rev. 06.09.03
THE CASE OF THE MISSING MIDDLE TERM
WORKSHEET KEY
State whether each binomial can be factored using the “difference of squares.”
1. x2 – y2 ÎYes
6. 9x2 – 5 ÎNo
2. x2 + y2 ÎNo
7. 1.44x2 – 0.45y2 ÎYes
3. x2 – 25 ÎYes
8.
4 2 1
x − ÎYes
9
4
9.
16 2
x + 1 ÎNo
25
4. x2 – 4y4 ÎYes
5. 9x2 + y2 ÎNo
Factor each polynomial completely!
10. a2 – 9 Î(x – 3)(x + 3)
16. 2a2 + 18a Î2a(a + 9)
11. x2 – 49 Î(x – 7)(x + 7)
17. 2x2 – 98 Î2(x – 7)(x + 7)
12. 4x2 – 9y2 Î(2x – 3y)(2x + 3y)
18. 12a2 – 48 Î12(a + 2)(a – 2)
13. x2 – 36y2 Î(x – 6y)(x + 6y)
19. 8x2 – 18 Î2(2x + 3)(2x – 3)
14. 1 – 9y2 Î(1 – 3y)(1 + 3y)
20.
1 2
1
1
x − 25 Î ( x − 5)( x + 5)
16
4
4
15. 49 – 4a2b2 Î(7 – 2ab)(7 + 2ab)
The Case of the Missing Middle Term©2003 www.beaconlearningcenter.com
Rev. 06.09.03
Student Name: __________________
Date: ______________
THE CASE OF THE MISSING MIDDLE TERM CHECKLIST
1. On questions 1 thru 9, did the student state whether each binomial could be
factored using the difference of squares correctly?
a.
b.
c.
d.
e.
f.
g.
h.
All nine (40 points)
Eight of the nine (35 points)
Seven of the nine (30 points)
Six of the nine (25 points)
Five of the nine (20 points)
Four of the nine (15 points)
Three of the nine (10 points)
Two of the nine (5 points)
2. On question 10, did the student factor the polynomial completely?
a. Yes (10 points)
b. No, but polynomial was partially factored (5 points)
3. On question 11, did the student factor the polynomial completely?
a. Yes (10 points)
b. No, but polynomial was partially factored (5 points)
4. On question 12, did the student factor the polynomial completely?
a. Yes (10 points)
b. No, but polynomial was partially factored (5 points)
5. On question 13, did the student factor the polynomial completely?
a. Yes (10 points)
b. No, but polynomial was partially factored (5 points)
6. On question 14, did the student factor the polynomial completely?
a. Yes (10 points)
b. No, but polynomial was partially factored (5 points)
7. On question 15, did the student factor the polynomial completely?
a. Yes (10 points)
b. No, but polynomial was partially factored (5 points)
The Case of the Missing Middle Term©2003 www.beaconlearningcenter.com
Rev. 06.09.03
8. On question 16, did the student factor the polynomial completely?
a. Yes (10 points)
b. No, but polynomial was partially factored (5 points)
9. On question 17, did the student factor the polynomial completely?
a. Yes (10 points)
b. No, but polynomial was partially factored (5 points)
10. On question 18, did the student factor the polynomial completely?
a. Yes (10 points)
b. No, but polynomial was partially factored (5 points)
11. On question 19, did the student factor the polynomial completely?
a. Yes (10 points)
b. No, but polynomial was partially factored (5 points)
12. On question 20, did the student factor the polynomial completely?
a. Yes (10 points)
b. No, but polynomial was partially factored (5 points)
Total Number of Points _________
A
135 points and above
B
120 points and above
C
105 points and above
D
90 points and above
F
89 points and below
Any score below C
needs
remediation!
The Case of the Missing Middle Term©2003 www.beaconlearningcenter.com
Rev. 06.09.03