Holt Algebra 1

7-8 Special Products of Binomials

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

Holt Algebra 1

7-8 Special Products of Binomials

Students will be able to: Find special products of binomials.

Learning Target

Holt Algebra 1

7-8 Special Products 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.

Holt Algebra 1

7-8 Special Products 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

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.

Holt Algebra 1

7-8 Special Products of Binomials

Multiply.

A. (x +3)2 (a + b)2 = a2 + 2ab + b2

= x2 + 6x + 9

B. (4s + 3t)2

= 16s2 + 24st + 9t2

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

Holt Algebra 1

7-8 Special Products of Binomials

Multiply.

C. (5 + m2)2 (a + b)2 = a2 + 2ab + b2

= 25 + 10m2 + m4

Holt Algebra 1

7-8 Special Products of Binomials

Multiply.A. (x + 6)2 (a + b)2 = a2 + 2ab + b2

= x2 + 12x + 36

B. (5a + b)2 (a + b)2 = a2 + 2ab + b2

= 25a2 + 10ab + b2

Holt Algebra 1

7-8 Special Products of Binomials

Multiply.

(1 + c3)2 (a + b)2 = a2 + 2ab + b2

= 1 + 2c3 + c6

Holt Algebra 1

7-8 Special Products 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

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).

Holt Algebra 1

7-8 Special Products of Binomials

Multiply.

A. (x – 6)2 (a – b) = a2 – 2ab + b2

= x – 12x + 36

B. (4m – 10)2 (a – b) = a2 – 2ab + b2

= 16m2 – 80m + 100

Holt Algebra 1

7-8 Special Products of Binomials

Multiply.

C. (2x – 5y )2(a – b) = a2 – 2ab + b2

= 4x2 – 20xy +25y2

D. (7 – r3)2 (a – b) = a2 – 2ab + b2

= 49 – 14r3 + r6

Holt Algebra 1

7-8 Special Products of Binomials

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 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.

Holt Algebra 1

7-8 Special Products of Binomials

Multiply.A. (x + 4)(x – 4) (a + b)(a – b) = a2 – b2

= x2 – 16

B. (p2 + 8q)(p2 – 8q)

= p4 – 64q2

Holt Algebra 1

7-8 Special Products of Binomials

Multiply.C. (10 + b)(10 – b) (a + b)(a – b) = a2 – b2

= 100 – b2

Holt Algebra 1

7-8 Special Products of Binomials

Multiply.a. (x + 8)(x – 8)

= x2 – 64

b. (3 + 2y2)(3 – 2y2)

= 9 – 4y4

Holt Algebra 1

7-8 Special Products of Binomials

Multiply.

c. (9 + r)(9 – r)

= 81 – r2

Holt Algebra 1

7-8 Special Products of Binomials

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

(a + b)(a – b) = a2 – b2

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

= x2 – 4

= x2 + 10x + 25

x2 + 10x + 25 – (x2 – 4)

= 10x + 29

Holt Algebra 1

7-8 Special Products of Binomials

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

25 – x2 + x2

25

Holt Algebra 1

7-8 Special Products of Binomials

HW pp. 505-507/21-63 Odd,64,67-70,75-82