CHAPTER 6 6-6 Special products of binomials

OBJECTIVES

Find special products of binomials.

SPECIAL PRODUCT OF BINOMIALS

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.

SPECIAL PRODUCT OF BINOMIALS

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= 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.

EXAMPLE 1: FINDING PRODUCTS IN THE FORM (A + B)2

Multiply.

Solution:

A. (x +3)2

Use the rule for (a + b)2.

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

(x + 3)2 = x2 + 2(x)(3) + 32Identify a and b: a = x and

b = 3.

= x2 + 6x + 9

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.

CHECK IT OUT! EXAMPLE 1

Multiply.

Sol:

A. (x + 6)2

= x2 + 12x + 36

B. (5a + b)2

Sol= 25a2 + 10ab + b2

SPECIAL PRODUCT OF BINOMIALS

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

I= 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).

EXAMPLE 2: FINDING PRODUCTS IN THE FORM (A – B)2

Multiply.

Sol: Identify a and b: a = x and b = 6.

(a – b)2 = a2 – 2ab + b2

Identify a and b: a = 4m and b = 10.4m – 10)2 = (4m)2 – 2(4m)(10) + (10)2

A. (x – 6)2

(a – b)2 = a2 – 2ab + b2

(x – 6)2 = x2 – 2x(6) + (6)2

= x2 – 12x + 36

B. (4m – 10)2

= 16m2 – 80m + 100

CHECK IT OUT! EXAMPLE 2

Multiply a. (x – 7)2

Sol:

= x2 – 14x + 49 b. (3b – 2c)2

Sol:= 9b2 – 12bc + 4c2

DIFFERENCE OF SQUARES

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.

Remove the 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).

DIFFERENCE OF SQUARES

So (a + b)(a – b) = a2 – b2. A binomial of the form a2 – b2 is called a difference of two squares.

EXAMPLE 3: FINDING PRODUCTS IN THE FORM (A + B)(A – B)

Multiply. A. (x + 4)(x – 4) Sol: (a + b)(a – b) = a2 – b2Use the rule for (a +

b)(a – b). (x + 4)(x – 4) = x2 – 42 Identify a and b: a = x

and b = 4.

= x2 – 16

EXAMPLE

B. (p2 + 8q)(p2 – 8q) Sol: (p2 + 8q)(p2 – 8q) = (p2)2 – (8q)2

= p4 – 64q2

CHECK IT OUT !!

Multiply a. (x + 8)(x – 8) Sol: x2 – 64

b. (3 + 2y2)(3 – 2y2) Sol:= 9 – 4y4

EXAMPLE 4: PROBLEM-SOLVING APPLICATION

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

SOLUTION

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

The area of the yard around the pool is 10x + 29.

CHECK IT OUT! EXAMPLE 4

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

The area of the pool is 25.

SUMMARY

STUDENT GUIDED PRACTICE

Do problems 2,5,6,7,9,10,19 and 20 in your book page 437

HOMEWORK

Do even problems from 21-38 in your book page 437