Beginning Algebra, 11th Edition

SECTION 5.6 Special Products^339

EXERCISES 61–70

FOR INDIVIDUAL OR GROUP WORK

Special products can be illustrated by using areas of rectangles.

Use the figure, and work Exercises 61– 66 in orderto justify

the special product

.

61.Express the area of the large square as the square of a

binomial.

62.Give the monomial that represents the area of the red

square.

63.Give the monomial that represents the sum of the areas of the blue rectangles.

64.Give the monomial that represents the area of the yellow square.

65.What is the sum of the monomials you obtained in Exercises 62 – 64?

66.Explain why the binomial square you found in Exercise 61 must equal the

polynomial you found in Exercise 65.

To understand how the special product can be applied to

a purely numerical problem, work Exercises 67–70 in order.

67.Evaluate , using either traditional paper-and-pencil methods or a calculator.

68.The number 35 can be written as. Therefore,. Use the

special product for squaring a binomial with and to write an

expression for. Do not simplify at this time.

69.Use the order of operations to simplify the expression you found in Exercise 68.

70.How do the answers in Exercises 67 and 69compare?

130 + 522

a= 30 b= 5

30 + 5 352 = 130 + 522

352

1 a+b 22 =a^2 + 2 ab+b^2

1 a+b 22 =a^2 + 2 ab+b^2

RELATING CONCEPTS

ab

b

a

The special product

can be used to perform some multiplication problems. Here are two examples.

Once these patterns are recognized, multiplications of this type can be done mentally. Use this

method to calculate each product mentally.

Determine a polynomial that represents the area of each figure. ( If necessary, refer to the for-

mulas on the inside covers.)

30

1

3

* 29

2

3

20

1

2

* 19

1

2

301 * 299

101 99 103 97 201 * 199

= 2499 = 9996

= 2500 - 1 =10,000- 4

= 502 - 12 = 1002 - 22

51 49 = 150 + 12150 - 12 102 98 = 1100 + 221100 - 22

1 x+y 21 x-y 2 =x^2 - y^2

m – 2n

m + 2n

6 p + q

6 p + q

3 a – 2

3 a + 2