7-9 Special Products of Binomials Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview

7-9 Special Products of Binomials

Warm UpWarm Up

Lesson Presentation

California Standards

PreviewPreview


Warm UpSimplify.

1. 42

3. (–2)2 4. (x)2

5. –(5y2)

16 49

4 x2

2. 72

6. (m2)2 m4

7. 2(6xy) 2(8x2)8. 16x2

–5y2

12xy


California Standards

10.0 Students add, subtract, multiply, and divide monomials and polynomials. Student solve multistep problems, including word problems, by using these techniques.


Vocabularyperfect-square trinomialdifference of two squares


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.


This means that (a + b)2 = a2+ 2ab + b2.You can use the FOIL method to verify this:

(a + b)2 = (a + b)(a + b) = a2 + ab + ab + b2

F L

I

O = a2 + 2ab + b2

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.


Multiply.

Additional Example 1: Finding Products in the Form (a + b)2

A. (x + 3)2

(a + b)2 = a2 + 2ab + b2

Use the rule for (a + b)2.

(x + 3)2 = x2 + 2(x)(3) + 32

= x2 + 6x + 9

Identify a and b: a = x and b = 3.

Simplify.

B. (4s + 3t)2

(a + b)2 = a2 + 2ab + b2

(4s + 3t)2 = (4s)2 + 2(4s)(3t) + (3t)2


= 16s2 + 24st + 9t2

Identify a and b: a = 4s and b = 3t.

Simplify.


Multiply.

Additional Example 1: Finding Products in the Form (a + b)2

C. (5 + m2)2

(a + b)2 = a2 + 2ab + b2

(5 + m2)2 = 52 + 2(5)(m2) + (m2)2

= 25 + 10m2 + m4


Identify a and b: a = 5 and b = m2.

Simplify.

7-9 Special Products of BinomialsCheck It Out! Example 1

Multiply.A. (x + 6)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

(a + b)2 = a2 + 2ab + b2


(5a + b)2 = (5a)2 + 2(5a)(b) + b2

Identify a and b: a = 5a and b = b.

= 25a2 + 10ab + b2 Simplify.


Check It Out! Example 1

Multiply.

c. (1 + c3)2 Use the rule for (a + b)2.

(a + b)2 = a2 + 2ab + b2 Identify a and b: a = 1 and b = c3.

(1 + c3)2 = 12 + 2(1)(c3) + (c3)2

= 1 + 2c3 + c6 Simplify.


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

F L

IO = 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).


Multiply.

Additional Example 2: Finding Products in the Form (a – b)2

A. (x – 6)2

(a – b)2 = a2 – 2ab + b2

(x – 6)2 = x2 – 2x(6) + (6)2

= x2 – 12x + 36

Use the rule for (a – b)2.


Simplify.

B. (4m – 10)2

(a – b)2 = a2 – 2ab + b2

(4m – 10)2 = (4m)2 – 2(4m)(10) + 102

= 16m2 – 80m + 100


Identify a and b: a = 4m and b = 10.

Simplify.


Multiply.

Additional Example 2: Finding Products in the Form (a – b)2

C. (2x – 5y )2

(a – b)2 = a2 – 2ab + b2

(2x – 5y)2 = (2x)2 – 2(2x)(5y) + (5y)2

= 4x2 – 20xy +25y2


Identify a and b: a = 2x and b = 5y.

Simplify.

D. (7 – r3)2

(a – b)2 = a2 – 2ab + b2

(7 – r3)2 = 72 – 2(7)(r3) + (r3)2

= 49 – 14r3 + r6


Identify a and b: a = 7 and b = r3.

Simplify.


Multiply.

a. (x – 7)2

(a – b)2 = a2 – 2ab + b2

(x – 7)2 = x2 – 2(x)(7) + (7)2

= x2 – 14x + 49



Simplify.

b. (3b – 2c)2

(a – b)2 = a2 – 2ab + b2

(3b – 2c)2 = (3b)2 – 2(3b)(2c) + (2c)2

= 9b2 – 12bc + 4c2


Identify a and b: a = 3b and b = 2c.

Simplify.


Multiply.

c. (a2 – 4)2

(a – b)2 = a2 – 2ab + b2

(a2 – 4)2 = a4 – 2(a2)(4) + (4)2

= a4 – 8a2 + 16


Identify a and b: a = a2 and b = 4.

Simplify.


You can use an area model to see that (a + b)(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 90 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.


Multiply.

Additional Example 3: Finding Products in the Form (a + b)(a – b)

A. (x + 4)(x – 4)

(a + b)(a – b) = a2 – b2

(x + 4)(x – 4) = x2 – 42

= x2 – 16

Use the rule for (a + b)(a – b).


Simplify.

B. (p2 + 8q)(p2 – 8q)

(a + b)(a – b) = a2 – b2

(p2 + 8q)(p2 – 8q) = p4 – (8q)2

= p4 – 64q2


Identify a and b: a = p2 and b = 8q.

Simplify.


Multiply.

Additional Example 3: Finding Products in the Form (a + b)(a – b)

C. (10 + b)(10 – b)

(a + b)(a – b) = a2 – b2

(10 + b)(10 – b) = 102 – b2

= 100 – b2


Identify a and b: a = 10 and b = b.

Simplify.


Multiply.a. (x + 8)(x – 8)

(a + b)(a – b) = a2 – b2

(x + 8)(x – 8) = x2 – 82

= x2 – 64



Simplify.

b. (3 + 2y2)(3 – 2y2)

(a + b)(a – b) = a2 – b2

(3 + 2y2)(3 – 2y2) = 32 – (2y2)2

= 9 – 4y4


Identify a and b: a = 3 and b = 2y2.

Simplify.


Multiply.

c. (9 + r)(9 – r)

(a + b)(a – b) = a2 – b2

(9 + r)(9 – r) = 92 – r2

= 81 – r2


Identify a and b: a = 9 and b = r.

Simplify.


Write a polynomial that represents the area of the yard around the pool shown below.

Additional Example 4: Problem-Solving Application


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.

11 Understand the Problem

The answer will be an expression that shows the area of the yard less the area of the pool.

Additional Example 4 Continued


22 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.



Solve33

Step 1 Find the total area.

(x + 5)2 = x2 + 2(x)(5) + 52Use the rule for

(a + b)2: a = x and b = 5.

Step 2 Find the area of the pool.

(x + 2)(x – 2) Use the rule for (a + b)(a – b): a = x and b = 2.

x2 + 10x + 25=

x2 – 4 =



Step 3 Find the area of the yard.

Area of yard = total area – area of pool

a = x2 + 10x + 25 (x2 – 4) –

= x2 + 10x + 25 – x2 + 4

= (x2 – x2) + 10x + ( 25 + 4)

= 10x + 29

Identify like terms.

Group like terms together

The area of the yard is 10x + 29.


Solve33


Look Back44

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 + 29. If x = 20, then 10x +29 = 10(20) + 29 = 229.



Write an expression that represents the area of the swimming pool.

7-9 Special Products of BinomialsCheck It Out! Example 4 Continued

11 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.


22 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.

Check It Out! Example 4 Continued


Solve33

Step 1 Find the area of the upper rectangle.

Step 2 Find the area lower square.

(5 + x)(5 – x) = 25 – 5x + 5x – x2 Use the rule for (a + b)

(a – b): a = 5 and b = x.–x2 + 25=

x2 =

= x x



Step 3 Find the area of the pool.

Area of pool = rectangle area + square area

a x2 +

= –x2 + 25 + x2

= (x2 – x2) + 25

= 25

–x2 + 25=

Identify like terms.

Group like terms together

The area of the pool is 25.

Solve33



Look Back44

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 21 + 4 = 25.

According to the solution, the area of the pool is 25.



7-9 Special Products of BinomialsLesson 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 – 4x + 4

x2 + 14x + 49

25x2 + 20xy + 4y2

4x2 – 36xy + 81y2

16x2 – 25y2

m4 – 4n2

7-9 Special Products of BinomialsLesson Quiz: Part II

7. Write a polynomial that represents the shaded area of the figure below.

14x – 85

x + 6

x – 6x – 7

x – 7

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7-9 Special Products of Binomials Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview