Warm Up 01/18/17 Simplify. 1. 42 2. (–2)2 3. –(5y2) 4. 2(6xy)

Essential Question How do you find special products of binomials?

Vocabulary perfect-square trinomial difference of two squares

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. F L (a + b)2 = (a + b)(a + b) = a2 + ab + ab + b2 I O = a2 + 2ab + b2

Example 1: Finding Products in the Form (a + b)2 A. (x +3)2 (a + b)2 = a2 + 2ab + b2 (x + 3)2 = x2 + 2(x)(3) + 32 = 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

F L (a – b)2 = (a – b)(a – b) = a2 – ab – ab + b2 I O = 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 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 (a – b)2 = a2 – 2ab + b2 (4m – 10)2 = (4m)2 – 2(4m)(10) + (10)2 = 16m2 – 80m + 100

You can use an area model to see that (a + b)(a–b)= a2 – b2. 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) (a + b)(a – b) = a2 – b2 (x + 4)(x – 4) = x2 – 42 = x2 – 16 B. (p2 + 8q)(p2 – 8q) (a + b)(a – b) = a2 – b2 (p2 + 8q)(p2 – 8q) = (p2)2 – (8q)2 = p4 – 64q2

Check It Out! Multiply. c. (9 + r)(9 – r) (a + b)(a – b) = a2 – b2 (9 + r)(9 – r) = 92 – r2 = 81 – r2

Example 4: Problem-Solving Application Write a polynomial that represents the area of the yard around the pool shown below.

Example 4 Continued Solve 3 Step 1 Find the total area. (x +5)2 = x2 + 2(x)(5) + 52 Use the rule for (a + b)2: a = x and b = 5. x2 + 10x + 25 = Step 2 Find the area of the pool. (x + 2)(x – 2) = x2 – 2x + 2x – 4 Use the rule for (a + b)(a – b): a = x and b = 2. x2 – 4 =

Example 4 Continued Solve 3 Step 3 Find the area of the yard around the pool. Area of yard = total area – area of pool a = x2 + 10x + 25 (x2 – 4) – Identify like terms. = x2 + 10x + 25 – x2 + 4 = (x2 – x2) + 10x + ( 25 + 4) Group like terms together = 10x + 29 The area of the yard around the pool is 10x + 29.

Lesson Quiz: Part I Multiply. 1. (x + 7)2 2. (x – 2)2 3. (5x + 2y)2 4. (2x – 9y)2

Lesson Quiz: Part II 5. Write a polynomial that represents the shaded area of the figure below. x + 6 x – 7 x – 6 x – 7