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

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1 Page 1 of 10 Attendance Problems. Determine whether the following are perfect squares. If so, find the square root x 2 5. y x y 7 49 p 10 I can factor perfect square trinomials. I can factor the difference of two squares. Common Core: CC.9-12.A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x 4 y 4 as (x 2 ) 2 - (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 ) (x 2 + y 2 ). 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. 9x x + 4 3x 3x 2(3x 2) 2 2

2 Page 2 of 10 Video Example 1: Determine whether each trinomial is a perfect square. If so, factor. If not explain. A) x 2 + 8x + 16 B) 9y 2-6y + 1 C) x x + 25

3 Page 3 of 10 1 Recognizing and Factoring Perfect-Square Trinomials Determine whether each trinomial is a perfect square. If so, factor. If not, explain. A x x + 36 x x + 36 x x 2 (x 6) 6 6 The trinomial is a perfect square. Factor. Method 1 Factor. Method 2 Use the rule. x x + 36 x x + 36 a = x, b = 6 Factors of 36 Sum 1 and and and and and 6 12 (x + 6)(x + 6) x (x) (6) (x + 6) 2 Write the trinomial as a 2 + 2ab + b 2. Write the trinomial as (a + b) 2. Determine whether each trinomial is a perfect square. If so, factor. If not, explain. B 4 x 2-12x x 2-12x x 2 x 2 (2x 3) 3 3 The trinomial is a perfect square. Factor. 4 x 2-12x + 9 a = 2x, b = 3 (2x) 2-2 (2x)(3) (2x - 3) 2 a 2-2ab + b 2 (a - b) 2 C x 2 + 9x + 16 x 2 + 9x + 16 x x 2 (x 4) (x 4) 9x x 2 + 9x + 16 is not a perfect-square trinomial because 9x 2 (x 4).

4 Page 4 of 10 Example 1. Determine whether each trinomial is a perfect square. If so, factor. If not explain. A. 9x 2 15x + 64 B. 81x x + 25 C. 36x 2 10x + 14 Guided Practice: Determine whether each trinomial is a perfect square. If so, factor. If not explain. 9. x 2 + 4x x 2 14x x 2 6x + 4

5 Page 5 of 10 Video Example 2. The garden is Annie s yard is in the shape of a rectangle. The area of the garden is 36x x + 25 ft 2. The dimensions of the garden are approximately cx + d, where c and d are whole numbers. Find an expression for the perimeter of the garden. Find the perimeter when x = 10 ft.

6 Algebra 7-5 Study Guide: Factoring Special Products (pp ) Page 6 of 10 2 Problem-Solving Application The park in the center of the Place des Vosges in Paris, France, is in the shape of a square. The area of the park is (25 x x + 49) ft 2. The side length of the park is in the form cx + d, where c and d are whole numbers. Find an expression in terms of x for the perimeter of the park. Find the perimeter when x = 8 ft. 1 Understand the Problem The answer will be an expression for the perimeter of the park and the value of the expression when x = 8. List the important information: The park is a square with area (25 x x + 49) ft 2. The side length of the park is in the form cx + d, where c and d are whole numbers. 2 Make a Plan The formula for the area of a square is area = (side) 2. Factor 25 x x + 49 to find the side length of the park. Write a formula for the perimeter of the park, and evaluate the expression for x = 8. 3 Solve 25 x x + 49 a = 5x, b = 7 (5x) (5x)(7) Write the trinomial as a 2 + 2ab + b 2. (5x + 7) 2 Write the trinomial as (a + b) x x + 49 = (5x + 7) (5x + 7) The side length of the park is (5x + 7) ft. Write a formula for the perimeter of the park. P = 4s Write the formula for the perimeter of a square. = 4 (5x + 7) Substitute the side length for s. = 20x + 28 Distribute 4. An expression for the perimeter of the park in feet is 20x Evaluate the expression when x = 8. P = 20x + 28 = 20 (8) + 28 Substitute 8 for x. = 188 When x = 8 ft, the perimeter of the park is 188 ft. 4 Look Back For a square with a perimeter of 188 ft, the side length is 188 = 47 ft and the 4 area is 47 2 = 2209 ft 2. Evaluate 25 x x + 49 for x = 8: 25 (8) (8)

7 Page 7 of 10 Example 2. A square piece of cloth must be cut to make a tablecloth. The area needed is (16x 2 24x + 9) in 2. 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. 12. Guided Practice: What if? A company produces square sheets of aluminum, each of which has an area of (9x 2 + 6x + 1) m 2. The side length of each sheet is in the form cx + d, where c and d are whole numbers. Find an expression in terms of x for the perimeter of a sheet. Find the perimeter when x = 3 m.

8 Page 8 of 10 In Chapter 7 you learned that the difference of two squares has the form a 2 b 2. The difference of two squares can be written as the product (a + b)(a b). You can use this pattern to factor some polynomials. A polynomial is a difference of two squares if: There are two terms, one subtracted from the other. Both terms are perfect squares. 4x 2 9 2x 2x 3 3 Video Example 3: Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. A) x B) 4w 2 9z 2 C) w 6 8z 4

9 Page 9 of 10 3 Recognizing and Factoring the Difference of Two Squares Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. A x 2-81 x 2-81 x x 9 9 The polynomial is a difference of two squares. x (x + 9)(x - 9) x 2-81 = (x + 9) (x - 9) a = x, b = 9 Write the polynomial as (a + b) (a - b). Example 3. Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. A. 3p 2 9q 4 B. 100x 2 4y 2 C. x 4 25y 6 B 9 p 4-16 q 2 9 p 4-16 q 2 3 p 2 3 p 2 4q 4q The polynomial is a difference of two squares. (3 p 2 ) 2 - (4q) 2 a = 3 p 2, b = 4q (3 p 2 + 4q) (3 p 2-4q) Write the polynomial as (a + b) (a - b). 9 p 4-16 q 2 = (3 p 2 + 4q) (3 p 2-4q) C x 6-7 y 2 x 3 x 3 x 6-7 y 2 7 y 2 is not a perfect square. x 6-7 y 2 is not the difference of two squares because 7 y 2 is not a perfect square.

10 Page 10 of 10 Guided Practice: Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. 13) 1 4x 2 14) p 8 49q 6 15) 16x 2 4y Factoring Special Products (p 494) 17, 19, 20, 23, 25, 29, 36-41, 45, 46. Question: What did Poly Nomial call her spoiled triplets. Answer: The perfect Tri Nomials.