1. Special Products of Binomials
7-8
Special Products of Binomials
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 1

2. Warm Up
Simplify.
1. 42
3. (–2) (x)2
5. –(5y2) 16
2. 72 49 4 x2
–25y2
6. (m2)2 m4
7. 2(6xy)
12xy 8.
2(8x2)
16x2

3. Objective
Find special products of binomials.

4. Vocabulary
perfect-square trinomial
difference of two squares

5. Imagine a square with sides of length (a + b):
The area of this square is (a + b)(a + b) or (a + b)2. The area of this square can also be found by adding the areas of the smaller squares and the rectangles inside. The sum of the areas inside is a2 + ab + ab + b2.

6. This means that (a + b)2 = a2+ 2ab + b2.
You can use the FOIL method to verify this: F L
(a + b)2 = (a + b)(a + b) = a2 + ab + ab + b2 I
= a2 + 2ab + b2 O
A trinomial of the form a2 + 2ab + b2 is called a perfect-square trinomial. A perfect-square trinomial is a trinomial that is the result of squaring a binomial.

7. Example 1: Finding Products in the Form (a + b)2
Multiply.
A. (x +3)2
Use the rule for (a + b)2.
(a + b)2 = a2 + 2ab + b2
Identify a and b: a = x and b = 3.
(x + 3)2 = x2 + 2(x)(3) + 32
= x2 + 6x + 9
Simplify.
B. (4s + 3t)2
Use the rule for (a + b)2.
(a + b)2 = a2 + 2ab + b2
Identify a and b: a = 4s and b = 3t.
(4s + 3t)2 = (4s)2 + 2(4s)(3t) + (3t)2
= 16s2 + 24st + 9t2
Simplify.

8. Example 1C: Finding Products in the Form (a + b)2
Multiply.
C. (5 + m2)2
Use the rule for (a + b)2.
(a + b)2 = a2 + 2ab + b2
Identify a and b: a = 5 and b = m2.
(5 + m2)2 = (5)(m2) + (m2)2
= m2 + m4
Simplify.

9. Check It Out! Example 1
Multiply.
A. (x + 6)2
Use the rule for (a + b)2.
(a + b)2 = a2 + 2ab + b2
Identify a and b: a = x and b = 6.
(x + 6)2 = x2 + 2(x)(6) + 62
= x2 + 12x + 36
Simplify.
B. (5a + b)2
Use the rule for (a + b)2.
(a + b)2 = a2 + 2ab + b2
Identify a and b: a = 5a and b = b.
(5a + b)2 = (5a)2 + 2(5a)(b) + b2
= 25a2 + 10ab + b2
Simplify.

10. Check It Out! Example 1C
Multiply.
Use the rule for (a + b)2.
(1 + c3)2
(a + b)2 = a2 + 2ab + b2
Identify a and b: a = 1 and b = c3.
(1 + c3)2 = (1)(c3) + (c3)2
Simplify.
= 1 + 2c3 + c6

11. You can use the FOIL method to find products in the form of (a – b)2.
(a – b)2 = (a – b)(a – b) = a2 – ab – ab + b2 I O
= a2 – 2ab + b2
A trinomial of the form a2 – ab + b2 is also a perfect-square trinomial because it is the result of squaring the binomial (a – b).

12. Example 2: Finding Products in the Form (a – b)2
Multiply.
A. (x – 6)2
Use the rule for (a – b)2.
(a – b) = a2 – 2ab + b2
Identify a and b: a = x and b = 6.
(x – 6) = x2 – 2x(6) + (6)2
= x – 12x + 36
Simplify.
B. (4m – 10)2
Use the rule for (a – b)2.
Identify a and b: a = 4m and b = 10.
(a – b) = a2 – 2ab + b2
(4m – 10) = (4m)2 – 2(4m)(10) + 102
= 16m2 – 80m + 100
Simplify.

13. Example 2: Finding Products in the Form (a – b)2
Multiply.
C. (2x – 5y )2
Use the rule for (a – b)2.
(a – b) = a2 – 2ab + b2
Identify a and b: a = 2x and b = 5y.
(2x – 5y)2 = (2x)2 – 2(2x)(5y) + (5y)2
= 2x2 – 20xy +25y2
Simplify.
D. (7 – r3)2
Use the rule for (a – b)2.
(a – b) = a2 – 2ab + b2
Identify a and b: a = 7 and b = r3.
(7 – r3) = 72 – 2(7)(r3) + (r3)2
= 49 – 14r3 + r6
Simplify.

14. Check It Out! Example 2
Multiply.
a. (x – 7)2
Use the rule for (a – b)2.
(a – b) = a2 – 2ab + b2
Identify a and b: a = x and b = 7.
(x – 7)2 = x2 – 2(x)(7) + (7)2
= x2 – 14x + 49
Simplify.
b. (3b – 2c)2
Use the rule for (a – b)2.
(a – b) = a2 – 2ab + b2
Identify a and b: a = 3b and b = 2c.
(3b – 2c)2 = (3b)2 – 2(3b)(2c) + (2c)2
= 9b2 – 12bc + 4c2
Simplify.

15. Check It Out! Example 2c
Multiply.
(a2 – 4)2
Use the rule for (a – b)2.
(a – b) = a2 – 2ab + b2
Identify a and b: a = a2 and b = 4.
(a2 – 4)2 = a4 – 2(a2)(4) + (4)2
= a4 – 8a2 + 16
Simplify.

16. You can use an area model to see that (a + b) = a2 – b2.
Begin with a square with area a2. Remove a square with area b2. The area of the new figure is a2 – b2.
Then remove the smaller rectangle on the bottom. Turn it and slide it up next to the top rectangle.
The new arrange- ment is a rectangle with length a + b and width a – b. Its area is (a + b)(a – b).
So (a + b)(a – b) = a2 – b2. A binomial of the form a2 – b2 is called a difference of two squares.

17. Example 3: Finding Products in the Form (a + b)(a – b)
Multiply.
A. (x + 4)(x – 4)
Use the rule for (a + b)(a – b).
(a + b)(a – b) = a2 – b2
Identify a and b: a = x and b = 4.
(x + 4)(x – 4) = x2 – 42
= x2 – 16
Simplify.
B. (p2 + 8q)(p2 – 8q)
Use the rule for (a + b)(a – b).
(a + b)(a – b) = a2 – b2
Identify a and b: a = p2 and b = 8q.
(p2 + 8q)(p2 – 8q) = p4 – (8q)2
= p4 – 64q2
Simplify.

18. Example 3: Finding Products in the Form (a + b)(a – b)
Multiply.
C. (10 + b)(10 – b)
Use the rule for (a + b)(a – b).
(a + b)(a – b) = a2 – b2
Identify a and b: a = 10 and b = b.
(10 + b)(10 – b) = 102 – b2
= 100 – b2
Simplify.

19. Check It Out! Example 3
Multiply.
a. (x + 8)(x – 8)
Use the rule for (a + b)(a – b).
(a + b)(a – b) = a2 – b2
Identify a and b: a = x and b = 8.
(x + 8)(x – 8) = x2 – 82
= x2 – 64
Simplify.
b. (3 + 2y2)(3 – 2y2)
Use the rule for (a + b)(a – b).
(a + b)(a – b) = a2 – b2
Identify a and b: a = 3 and b = 2y2.
(3 + 2y2)(3 – 2y2) = 32 – (2y2)2
= 9 – 4y4
Simplify.

20. Check It Out! Example 3
Multiply.
c. (9 + r)(9 – r)
Use the rule for (a + b)(a – b).
(a + b)(a – b) = a2 – b2
Identify a and b: a = 9 and b = r.
(9 + r)(9 – r) = 92 – r2
= 81 – r2
Simplify.

21. Example 4: Problem-Solving Application
Write a polynomial that represents the area of the yard around the pool shown below.

22. Understand the Problem
Example 4 Continued 1
Understand the Problem
The answer will be an expression that shows the area of the yard less the area of the pool.
List important information:
The yard is a square with a side length of x + 5.
The pool has side lengths of x + 2 and x – 2.

23. Example 4 Continued 2
Make a Plan
The area of the yard is (x + 5)2 less the area of the pool. The area of the pool is (x + 2)(x – 2). You can subtract the area of the pool from the yard to find the area of the yard surrounding the pool.

24. Example 4 Continued
Solve 3
Step 1 Find the total area.
(x +5)2 = x2 + 2(x)(5) + 52
Use the rule for (a + b)2: a = x and b = 5.
x2 + 10x + 25 =
Step 2 Find the area of the pool.
(x + 2)(x – 2) = x2 – 2x + 2x – 4
Use the rule for (a + b)(a – b): a = x and b = 2.
x2 – 4 =

25. Example 4 Continued
Solve 3
Step 3
Find the area of the yard.
Area of yard =
total area –
area of pool a =
x2 + 10x + 25
(x2 – 4) –
Identify like terms.
= x2 + 10x + 25 – x2 + 4
= (x2 – x2) + 10x + ( )
Group like terms together
= 10x + 29
The area of the yard is 10x + 29.

26.  Example 4 Continued 4 Look Back
Suppose that x = 20. Then the total area in the back yard would be 252 or 625. The area of the pool would be 22  18 or 396. The area of the yard around the pool would be 625 – 396 = 229.
According to the solution, the area of the pool is 10x If x = 20, then 10x +29 = 10(20) + 29 = 229. 

27. To subtract a polynomial, add the opposite of each term.
Remember!

28. Check It Out! Example 4
Write an expression that represents the area of the swimming pool.

29. Understand the Problem
Check It Out! Example 4 Continued 1
Understand the Problem
The answer will be an expression that shows the area of the two rectangles combined.
List important information:
The upper rectangle has side lengths of 5 + x and 5 – x .
The lower rectangle is a square with side length of x.

30. Check It Out! Example 4 Continued2
Make a Plan
The area of the upper rectangle is (5 + x)(5 – x). The area of the lower square is x2. Added together they give the total area of the pool.

31. Check It Out! Example 4 Continued
Solve 3
Step 1 Find the area of the upper rectangle.
Use the rule for (a + b)
(a – b): a = 5 and b = x.
(5 + x)(5 – x) = 25 – 5x + 5x – x2
–x2 + 25 =
Step 2 Find the area lower square.
= x x  x2 =

32. Check It Out! Example 4 Continued
Solve 3
Step 3
Find the area of the pool.
Area of pool =
rectangle area +
square area a =
–x2 + 25 + x2
= –x x2
Identify like terms.
= (x2 – x2) + 25
Group like terms together
= 25
The area of the pool is 25.

33. Check It Out! Example 4 Continued
Look Back
Suppose that x = 2. Then the area of the upper rectangle would be 21. The area of the lower square would be 4. The area of the pool would be = 25.
According to the solution, the area of the pool is 25. 

35. Lesson Quiz: Part I
Multiply.
1. (x + 7)2
2. (x – 2)2
3. (5x + 2y)2
4. (2x – 9y)2
5. (4x + 5y)(4x – 5y)
6. (m2 + 2n)(m2 – 2n)
x2 + 14x + 49
x2 – 4x + 4
25x2 + 20xy + 4y2
4x2 – 36xy + 81y2
16x2 – 25y2
m4 – 4n2

36. Lesson Quiz: Part II
7. Write a polynomial that represents the shaded area of the figure below.
x + 6
x – 7
x – 6
x – 7
14x – 85