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Warm Up

Preview. Warm Up. California Standards. Lesson Presentation. Warm Up Simplify. 1. 4 2 3. (–2) 2 4. ( x ) 2 5. –(5 y 2 ). 49. 2 . 7 2. 16. x 2. 4. m 4. –5 y 2. 6. ( m 2 ) 2. 8. 7. 2(6 xy ). 12 xy. 2(8 x 2 ). 16 x 2. California Standards.

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Warm Up

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  1. Preview Warm Up California Standards Lesson Presentation

  2. Warm Up Simplify. 1.42 3. (–2)2 4. (x)2 5. –(5y2) 49 2.72 16 x2 4 m4 –5y2 6. (m2)2 8. 7. 2(6xy) 12xy 2(8x2) 16x2

  3. California Standards 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Student solve multistep problems, including word problems, by using these techniques.

  4. Vocabulary perfect-square trinomial difference of two squares

  5. 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.

  6. F L I This means that (a + b)2=a2+ 2ab + b2. You can use the FOIL method to verify this: (a + b)2 = (a + b)(a + b) = a2+ ab + ab+ b2 = 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.

  7. Additional 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.

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

  9. 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.

  10. Check It Out! Example 1 Multiply. Use the rule for (a + b)2. c. (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

  11. F L I 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 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).

  12. Additional Example 2: Finding Products in the Form (a – b)2 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 Simplify. = x2 – 12x + 36 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) + 102 Simplify. = 16m2 – 80m + 100

  13. Additional Example 2: Finding Products in the Form (a – b)2 Multiply. C. (2x – 5y )2 Use the rule for (a – b)2. (a – b)2 =a2 – 2ab + b2 Identify a and b: a = 2x and b = 5y. (2x – 5y)2 =(2x)2 – 2(2x)(5y) +(5y)2 Simplify. = 4x2 – 20xy +25y2 D. (7 – r3)2 Use the rule for (a – b)2. Identify a and b: a = 7 and b = r3. (a – b)2 =a2 – 2ab + b2 (7 – r3)2 =72 – 2(7)(r3) + (r3)2 = 49 – 14r3 + r6 Simplify.

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

  15. Check It Out! Example 2 Multiply. c. (a2 – 4)2 Use the rule for (a – b)2. (a – b)2 =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.

  16. You can use an area model to see that (a + b)(a – b) = a2 – b2. Then remove the smaller rectangle on the bottom. Turn it 90 and slide it up next to the top rectangle. Begin with a square with area a2. Remove a square with area b2. The area of the new figure is a2 – b2. The new arrange- ment is a rectangle with length a + b and width a – b. Its area is (a + b)(a – b).

  17. So (a + b)(a – b) = a2 – b2. A binomial of the form a2 – b2 is called a difference of two squares.

  18. Additional 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 Simplify. = x2 – 16 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 Simplify. = p4 – 64q2

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

  20. 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 Simplify. = 9 – 4y4

  21. 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.

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

  23. 1 Understand the Problem Additional Example 4 Continued 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.

  24. Make a Plan 2 Additional Example 4 Continued 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.

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

  26. 3 Area of yard total area area of pool Solve – = a – (x2 – 4) x2 + 10x + 25 = Additional Example 4 Continued Step 3 Find the area of the yard. Identify like terms. = x2+ 10x + 25 – x2 + 4 Group like terms together = (x2 – x2)+ 10x+ ( 25 + 4) = 10x + 29 The area of the yard is 10x + 29.

  27. 4 Additional Example 4 Continued 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. 

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

  29. 1 Understand the Problem Check It Out! Example 4 Continued 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.

  30. Make a Plan 2 Check It Out! Example 4 Continued 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.

  31. 3 Solve = –x2 + 25 = x x  = x2 Check It Out! Example 4 Continued 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 Step 2 Find the area lower square.

  32. 3 Solve Check It Out! Example 4 Continued 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.

  33. 4 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. 

  34. 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

  35. 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

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