Special Products of Binomials 7-8 Special Products of Binomials Warm Up Lesson Presentation Lesson Quiz Holt Algebra 1

Warm Up Simplify. 1. 42 3. (–2)2 4. (x)2 5. –(5y2) 16 2. 72 49 4 x2 –25y2 6. (m2)2 m4 7. 2(6xy) 12xy 8. 2(8x2) 16x2

Objective Find special products of binomials.

Vocabulary perfect-square trinomial difference of two squares

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. 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 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.

Example 1C: Finding Products in the Form (a + b)2 Multiply. C. (5 + m2)2 Use the rule for (a + b)2. (a + b)2 = a2 + 2ab + b2 Identify a and b: a = 5 and b = m2. (5 + m2)2 = 52 + 2(5)(m2) + (m2)2 = 25 + 10m2 + m4 Simplify.

Check It Out! Example 1 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 + 2(x)(6) + 62 = x2 + 12x + 36 Simplify. B. (5a + b)2 Use the rule for (a + b)2. (a + b)2 = a2 + 2ab + b2 Identify a and b: a = 5a and b = b. (5a + b)2 = (5a)2 + 2(5a)(b) + b2 = 25a2 + 10ab + b2 Simplify.

Check It Out! Example 1C Multiply. Use the rule for (a + b)2. (1 + c3)2 (a + b)2 = a2 + 2ab + b2 Identify a and b: a = 1 and b = c3. (1 + c3)2 = 12 + 2(1)(c3) + (c3)2 Simplify. = 1 + 2c3 + c6

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 Multiply. A. (x – 6)2 Use the rule for (a – b)2. (a – b) = a2 – 2ab + b2 Identify a and b: a = x and b = 6. (x – 6) = x2 – 2x(6) + (6)2 = x – 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) = a2 – 2ab + b2 (4m – 10) = (4m)2 – 2(4m)(10) + 102 = 16m2 – 80m + 100 Simplify.

Example 2: Finding Products in the Form (a – b)2 Multiply. C. (2x – 5y )2 Use the rule for (a – b)2. (a – b) = a2 – 2ab + b2 Identify a and b: a = 2x and b = 5y. (2x – 5y)2 = (2x)2 – 2(2x)(5y) + (5y)2 = 2x2 – 20xy +25y2 Simplify. D. (7 – r3)2 Use the rule for (a – b)2. (a – b) = a2 – 2ab + b2 Identify a and b: a = 7 and b = r3. (7 – r3) = 72 – 2(7)(r3) + (r3)2 = 49 – 14r3 + r6 Simplify.

Check It Out! Example 2 Multiply. a. (x – 7)2 Use the rule for (a – b)2. (a – b) = a2 – 2ab + b2 Identify a and b: a = x and b = 7. (x – 7)2 = x2 – 2(x)(7) + (7)2 = x2 – 14x + 49 Simplify. b. (3b – 2c)2 Use the rule for (a – b)2. (a – b) = a2 – 2ab + b2 Identify a and b: a = 3b and b = 2c. (3b – 2c)2 = (3b)2 – 2(3b)(2c) + (2c)2 = 9b2 – 12bc + 4c2 Simplify.

Check It Out! Example 2c Multiply. (a2 – 4)2 Use the rule for (a – b)2. (a – b) = a2 – 2ab + b2 Identify a and b: a = a2 and b = 4. (a2 – 4)2 = a4 – 2(a2)(4) + (4)2 = a4 – 8a2 + 16 Simplify.

You can use an area model to see that (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). 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) 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) = p4 – (8q)2 = p4 – 64q2 Simplify.

Example 3: Finding Products in the Form (a + b)(a – b) Multiply. C. (10 + b)(10 – b) Use the rule for (a + b)(a – b). (a + b)(a – b) = a2 – b2 Identify a and b: a = 10 and b = b. (10 + b)(10 – b) = 102 – b2 = 100 – b2 Simplify.

Check It Out! Example 3 Multiply. a. (x + 8)(x – 8) Use the rule for (a + b)(a – b). (a + b)(a – b) = a2 – b2 Identify a and b: a = x and b = 8. (x + 8)(x – 8) = x2 – 82 = x2 – 64 Simplify. b. (3 + 2y2)(3 – 2y2) Use the rule for (a + b)(a – b). (a + b)(a – b) = a2 – b2 Identify a and b: a = 3 and b = 2y2. (3 + 2y2)(3 – 2y2) = 32 – (2y2)2 = 9 – 4y4 Simplify.

Check It Out! Example 3 Multiply. c. (9 + r)(9 – r) Use the rule for (a + b)(a – b). (a + b)(a – b) = a2 – b2 Identify a and b: a = 9 and b = r. (9 + r)(9 – r) = 92 – r2 = 81 – r2 Simplify.

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

Understand the Problem Example 4 Continued 1 Understand the Problem The answer will be an expression that shows the area of the yard less the area of the pool. List important information: The yard is a square with a side length of x + 5. The pool has side lengths of x + 2 and x – 2.

Example 4 Continued 2 Make a Plan The area of the yard is (x + 5)2 less the area of the pool. The area of the pool is (x + 2)(x – 2). You can subtract the area of the pool from the yard to find the area of the yard surrounding the pool.

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. 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 is 10x + 29.

 Example 4 Continued 4 Look Back Suppose that x = 20. Then the total area in the back yard would be 252 or 625. The area of the pool would be 22  18 or 396. The area of the yard around the pool would be 625 – 396 = 229. According to the solution, the area of the pool is 10x + 29. If x = 20, then 10x +29 = 10(20) + 29 = 229. 

To subtract a polynomial, add the opposite of each term. Remember!

Check It Out! Example 4 Write an expression that represents the area of the swimming pool.

Understand the Problem Check It Out! Example 4 Continued 1 Understand the Problem The answer will be an expression that shows the area of the two rectangles combined. List important information: The upper rectangle has side lengths of 5 + x and 5 – x . The lower rectangle is a square with side length of x.

Check It Out! Example 4 Continued 2 Make a Plan The area of the upper rectangle is (5 + x)(5 – x). The area of the lower square is x2. Added together they give the total area of the pool.

Check It Out! Example 4 Continued Solve 3 Step 1 Find the area of the upper rectangle. Use the rule for (a + b) (a – b): a = 5 and b = x. (5 + x)(5 – x) = 25 – 5x + 5x – x2 –x2 + 25 = Step 2 Find the area lower square. = x x  x2 =

Check It Out! Example 4 Continued Solve 3 Step 3 Find the area of the pool. Area of pool = rectangle area + square area a = –x2 + 25 + x2 = –x2 + 25 + x2 Identify like terms. = (x2 – x2) + 25 Group like terms together = 25 The area of the pool is 25.

Check It Out! Example 4 Continued Look Back Suppose that x = 2. Then the area of the upper rectangle would be 21. The area of the lower square would be 4. The area of the pool would be 21 + 4 = 25. According to the solution, the area of the pool is 25. 

Lesson Quiz: Part I Multiply. 1. (x + 7)2 2. (x – 2)2 3. (5x + 2y)2 4. (2x – 9y)2 5. (4x + 5y)(4x – 5y) 6. (m2 + 2n)(m2 – 2n) x2 + 14x + 49 x2 – 4x + 4 25x2 + 20xy + 4y2 4x2 – 36xy + 81y2 16x2 – 25y2 m4 – 4n2

Lesson Quiz: Part II 7. Write a polynomial that represents the shaded area of the figure below. x + 6 x – 7 x – 6 x – 7 14x – 85