Lesson Objective: I will be able to … Find special products of binomials Language Objective: I will be able to … Read, write, and listen about vocabulary, key concepts, and examples

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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.

This means that (a + b)2 = a2+ 2ab + b2. You can use the FOIL method to verify this: F L (a + b)2 = (a + b)(a + b) = a2 + ab + ab + b2 I = a2 + 2ab + b2 O A trinomial of the form a2 + 2ab + b2 is called a perfect-square trinomial. Page 37 A perfect-square trinomial is a trinomial that is the result of squaring a binomial.

Example 1: Finding Products in the Form (a + b)2 Page 38 Multiply. A. (x +3)2 Use the rule for (a + b)2. (a + b)2 = a2 + 2ab + b2 Identify a and b: a = x and b = 3. (x + 3)2 = x2 + 2(x)(3) + 32 = x2 + 6x + 9 Simplify. B. (4s + 3t)2 Use the rule for (a + b)2. (a + b)2 = a2 + 2ab + b2 Identify a and b: a = 4s and b = 3t. (4s + 3t)2 = (4s)2 + 2(4s)(3t) + (3t)2 = 16s2 + 24st + 9t2 Simplify.

Identify a and b: a = 5a and b = b. (5a + b)2 = (5a)2 + 2(5a)(b) + b2 Your Turn 1 Page 38 Multiply. (5a + b)2 (a + b)2 = a2 + 2ab + b2 Use the rule for (a + b)2. Identify a and b: a = 5a and b = b. (5a + b)2 = (5a)2 + 2(5a)(b) + b2 = 25a2 + 10ab + b2 Simplify.

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 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 Page 38 Multiply. A. (x – 6)2 Use the rule for (a – b)2. (a – b)2 = a2 – 2ab + b2 Identify a and b: a = x and b = 6. (x – 6)2 = x2 – 2x(6) + (6)2 = x2 – 12x + 36 Simplify. B. (4m – 10)2 Use the rule for (a – b)2. Identify a and b: a = 4m and b = 10. (a – b)2 = a2 – 2ab + b2 (4m – 10)2 = (4m)2 – 2(4m)(10) + (10)2 = 16m2 – 80m + 100 Simplify.

Identify a and b: a = 3b and b = 2c. Your Turn 2 Page 39 Multiply. (3b – 2c)2 (a – b) = a2 – 2ab + b2 Use the rule for (a – b)2. Identify a and b: a = 3b and b = 2c. (3b – 2c)2 = (3b)2 – 2(3b)(2c) + (2c)2 = 9b2 – 12bc + 4c2 Simplify.

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). Page 37 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) Page 39 Multiply. A. (x + 4)(x – 4) Use the rule for (a + b)(a – b). (a + b)(a – b) = a2 – b2 Identify a and b: a = x and b = 4. (x + 4)(x – 4) = x2 – 42 = x2 – 16 Simplify. B. (p2 + 8q)(p2 – 8q) Use the rule for (a + b)(a – b). (a + b)(a – b) = a2 – b2 Identify a and b: a = p2 and b = 8q. (p2 + 8q)(p2 – 8q) = (p2)2 – (8q)2 = p4 – 64q2 Simplify.

Use the rule for (a + b)(a – b). Your Turn 3 Page 39 Multiply. (3 + 2y2)(3 – 2y2) (a + b)(a – b) = a2 – b2 Use the rule for (a + b)(a – b). Identify a and b: a = 3 and b = 2y2. (3 + 2y2)(3 – 2y2) = 32 – (2y2)2 = 9 – 4y4 Simplify.

Classwork Assignment #7 Holt 7-8 #2 - 13