Warm UpDetermine whether the following are perfect squares. If so, find the square root.1. 64

yes;7p5 no

2. 36

yes; y4 3. 45 4. x2 5. y8 6. 4x6 7. 9y7 8. 49p10

yes; 2x3

yes; 8

no yes; x

yes; 6

11.0 Students apply basic factoring techniques to second-and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials.

California Standards

ObjectivesFactor Perfect Square Trinomials

Factor the difference of two squares

A trinomial is a perfect square if: • The first and last terms are perfect squares.

• The middle term is two times one factor from the first term and one factor from the last term.

9x2 + 12x + 4

3x 3x 2(3x 2) 2 2• ••

Example 1

Recognizing and Factoring Perfect-Square Trinomials

Determine whether each trinomial is a perfect square. If so, factor. If not, explain.

9x2 – 15x + 64

9x2 – 15x + 64

2(3x 8) ≠ –15x.

9x2 – 15x + 64 is not a perfect-square trinomial because –15x ≠ 2(3x 8).

8 83x 3x 2(3x 8)

Example 2

Recognizing and Factoring Perfect-Square Trinomials

Determine whether each trinomial is a perfect square. If so, factor. If not, explain.

81x2 + 90x + 25

81x2 + 90x + 25

The trinomial is a perfect square. Factor.

5 59x 9x 2(9x 5)● ●●

Example 3

Recognizing and Factoring Perfect-Square Trinomials

Determine whether each trinomial is a perfect square. If so, factor. If not, explain.

36x2 – 10x + 14

The trinomial is not a perfect-square because 14 is not a perfect square.

36x2 – 10x + 14

36x2 – 10x + 14 is not a perfect-square trinomial.

You can check your answer by using the FOIL method.For example 1, (9x + 5)2 = (9x + 5)(9x + 5) = 81x2 + 45x + 45x + 25 = 81x2 + 90x + 25

Remember!

Determine whether each trinomial is a perfect square. If so, factor. If not explain.

Factor.

(x + 2)(x + 2)

Example 4

x2 + 4x + 4

Factors of 4 Sum

(1 and 4) 5

(2 and 2) 4

Determine whether each trinomial is a perfect square. If so, factor. If not explain.

Factor.

Example 5

x2 – 14x + 49

(x – 7)(x – 7)

Factors of 49 Sum

(–1 and –49) –50

(–7 and –7) –14

A rectangular piece of cloth must be cut to make a tablecloth. The area needed is (16x2 – 24x + 9) in2. The dimensions of the cloth are of the form cx – d, where c and d are whole numbers. Find an expression for the perimeter of the cloth. Find the perimeter when x = 11 inches.

Problem-Solving Application

11 Understand the Problem

The answer will be an expression for the perimeter of the cloth and the value of the expression when x = 11.

List the important information:

• The tablecloth is a rectangle with area (16x2 – 24x + 9) in2.

• The side length of the tablecloth is in the form cx – d, where c and d are whole numbers.

Problem-Solving Application Continued

22 Make a Plan

The formula for the area of a rectangle is Area = length × width.

Problem-Solving Application Continued

Factor 16x2 – 24x + 9 to find the length and width of the tablecloth. Write a formula for the perimeter of the tablecloth, and evaluate the expression for x = 11.

Solve33

16x2 – 24x + 9

(4x)2 – 2(4x)(3) + 32

(4x – 3)2

16x2 – 24x + 9 = (4x – 3)(4x – 3)

a = 4x, b = 3

Write the trinomial as a2 – 2ab + b2.

Write the trinomial as (a – b)2.

Each side length of the tablecloth is (4x – 3) in. The tablecloth is a square.

Problem-Solving Application Continued

Write a formula for the perimeter of the tablecloth.

P = 4s

= 4(4x – 3)

= 16x – 12

An expression for the perimeter of the tablecloth in inches is 16x – 12.

Write the formula for the perimeter of a square.

Substitute the side length for s.

Distribute 4.

Problem-Solving Application Continued

Evaluate the expression when x = 11.

P = 16x – 12

= 16(11) – 12

= 164

When x = 11 in. the perimeter of the tablecloth is 164 in.

Substitute 11 for x.

Problem-Solving Application Continued

Look Back44

For a square with a perimeter of 164, the side length is and the area is 412 = 1681 in2.

Evaluate 16x2 – 24x + 9 for x = 11.

16(11)2 – 24(11) + 9

1936 – 264 + 9

1681

Additional Example 2 Continued

.

Review…

Recognizing and Factoring the Difference of Two Squares

Determine whether each binomial is a difference of two squares. If so, factor. If not, explain.

3p2 – 9q4

3p2 – 9q4

3q2 3q2 3p2 is not a perfect square.

3p2 – 9q4 is not the difference of two squares because 3p2 is not a perfect square.

Recognizing and Factoring the Difference of Two Squares

Determine whether each binomial is a difference of two squares. If so, factor. If not, explain.

100x2 – 4y2

Write the polynomial as (a + b)(a – b).

a = 10x, b = 2y

The polynomial is a difference of two squares.

100x2 – 4y2

2y 2y10x 10x

(10x)2 – (2y)2 (10x + 2y)(10x – 2y)

100x2 – 4y2 = (10x + 2y)(10x – 2y)

Lesson Quiz: Part I

Determine whether each trinomial is a perfect square. If so factor. If not, explain.

1. 64x2 – 40x + 25

2. 121x2 – 44x + 4

3. 49x2 + 140x + 100

4. A fence will be built around a garden with an area of (49x2 + 56x + 16) ft2. The dimensions of the garden are cx + d, where c and d are whole numbers. Find an expression for the perimeter of the garden. Find the perimeter when x = 5 feet.

P = 28x + 16; 156 ft

(7x + 10)2

(11x – 2)2

not a perfect-square trinomial because –40x ≠ 2(8x 5)

Lesson Quiz: Part II

Determine whether the binomial is a difference of two squares. If so, factor. If not, explain.

5. 9x2 – 144y4

6. 30x2 – 64y2

7. 121x2 – 4y8

(3x + 12y2)(3x – 12y2)

(11x + 2y4)(11x – 2y4)

not a difference of two squares; 30x2 is not a perfect square