# 8-8 Differences of Squares. Factor each polynomial. 1. x 9 SOLUTION: 2. 4a 25 SOLUTION: 3. 9m 144 SOLUTION: 4. 2p 162p SOLUTION: 5.

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1 Factor each polynomial. 1.x 9 SOLUTION:.a 5 SOLUTION:.9m 1 SOLUTION:.p 16p SOLUTION: 5.u 81 SOLUTION: Page 1

2 5.u 81 SOLUTION: 6.d f SOLUTION: 7.0r 5n SOLUTION: 8.56n c SOLUTION: Page

3 8.56n c SOLUTION: 9.c + c c SOLUTION: 10.f f 9f + 6 SOLUTION: 11.t + t 8t SOLUTION: Page

4 11.t + t 8t SOLUTION: 1.w w 9w + 7 SOLUTION: EXTENDEDRESPONSEAfteranaccident,skidmarksmayresultfromsuddenbreaking.Theformula s = d approximates a vehicle s speed s in miles per hour given the length d in feet of the skid marks on dry concrete. 1.If skid marks on dry concrete are 5 feet long, how fast was the car traveling when the brakes were applied? SOLUTION: Page

5 EXTENDEDRESPONSEAfteranaccident,skidmarksmayresultfromsuddenbreaking.Theformula s = d approximates a vehicle s speed s in miles per hour given the length d in feet of the skid marks on dry concrete. 1.If skid marks on dry concrete are 5 feet long, how fast was the car traveling when the brakes were applied? SOLUTION: Since the speed cannot be negative, the car was traveling 6 mph when the brakes were applied. 1.If the skid marks on dry concrete are 150 feet long, how fast was the car traveling when the brakes were applied? SOLUTION: Since the speed cannot be negative, the car was traveling 60 mph when the brakes were applied. Factor each polynomial. 15.q 11 SOLUTION: Page 5

6 8-8 Differences of Squares Since the speed cannot be negative, the car was traveling 60 mph when the brakes were applied. Factor each polynomial. 15.q 11 SOLUTION: 16.r k SOLUTION: 17.6n 6 SOLUTION: 18.w 65 SOLUTION: 19.r 9t SOLUTION: Page 6

7 19.r 9t SOLUTION: 0.c d SOLUTION: 1.h 100h SOLUTION:.h 56 SOLUTION:.x x 16x + 81 SOLUTION:.x y Page 7

8 .x x 16x + 81 SOLUTION:.x y SOLUTION: 5.7h 7p SOLUTION: 6.c + c 17c 98 SOLUTION: 7.6k h 5k Page 8

9 6.c + c 17c 98 SOLUTION: 7.6k h 5k SOLUTION: 8.5a 0a SOLUTION: 9.f + f 6f 18 SOLUTION: 0.r 19r Page 9

10 9.f + f 6f 18 SOLUTION: 0.r 19r SOLUTION: 1.10q 110q SOLUTION:.xn 7x SOLUTION: 5.p r p r SOLUTION: Page 10

11 5.p r p r SOLUTION:.8c 8c SOLUTION: 5.r 5r 100r SOLUTION: 6.t 7t t + 7 SOLUTION: Page 11

12 6.t 7t t + 7 SOLUTION: 7.a 9 SOLUTION: 8.m + 9m 6m 81 SOLUTION: 9.m + SOLUTION: 0.x + x 75x 5 SOLUTION: Page 1

13 9.m + SOLUTION: 0.x + x 75x 5 SOLUTION: 1.1a + a 19a SOLUTION:.x + 6x 6x 16x SOLUTION:.15m + 1m 75m 00 SOLUTION: esolutions Manual - Powered by Cognero Page 1

14 .15m + 1m 75m 00 SOLUTION:.GEOMETRY The drawing shown is a square with a square cut out of it. a. Write an expression that represents the area of the shaded region. b. Find the dimensions of a rectangle with the same area as the shaded region in the drawing. Assume that the dimensions of the rectangle must be represented by binomials with integral coefficients. SOLUTION: a. Use the formula for the area of square A = s s to write an expression for the area of the shaded region. b. Write the area of the shaded region in factor form to find two binomial dimensions for a rectangle with the same area. The dimensions of the rectangle would be (n + 6) by (n ). 5.DECORATIONS arch decorated esolutions Manual - Powered byan Cognero with balloons was used to decorate the gym for the spring dance. The shape Page 1 of the arch can be modeled by the equation y = 0.5x +.5x, where x and y are measured in feet and the x-axis represents the floor.

15 8-8 Differences of Squares The dimensions of the rectangle would be (n + 6) by (n ). 5.DECORATIONS An arch decorated with balloons was used to decorate the gym for the spring dance. The shape of the arch can be modeled by the equation y = 0.5x +.5x, where x and y are measured in feet and the x-axis represents the floor. a. Write the expression that represents the height of the arch in factored form. b. How far apart are the two points where the arch touches the floor? c. Graph this equation on your calculator. What is the highest point of the arch? SOLUTION: a. b. Find the two places along the x-axis (the floor) where the height of the arch is 0. or Subtract the two values to find out how far apart they are: 9 0 = 9. The two points where the arch touches the floor are 9 ft. apart. c. Useagraphingcalculatortofindtheheightofthearchatitshighestpoint. [, 10] scl: by [, 10] scl: The arch is ft. high. 6.CCSS SENSE-MAKING Zelda is building a deck in her backyard. The plans for the deck show that it is to be feet by feet. Zelda wants to reduce one dimension by a number of feet and increase the other dimension by the same number of feet. If the area of the reduced deck is 51 square feet, what are the dimensions of the deck? SOLUTION: Let x be the number of feet added and subtracted to each dimension of the deck. Replace A with 51, l with + x, and w with - x. esolutions Manual - Powered by Cognero Page 15

16 [, 10] scl: by [, 10] scl: 8-8 Differences of Squares The arch is ft. high. 6.CCSS SENSE-MAKING Zelda is building a deck in her backyard. The plans for the deck show that it is to be feet by feet. Zelda wants to reduce one dimension by a number of feet and increase the other dimension by the same number of feet. If the area of the reduced deck is 51 square feet, what are the dimensions of the deck? SOLUTION: Let x be the number of feet added and subtracted to each dimension of the deck. Replace A with 51, l with + x, and w with - x. Since the amount added and subtracted to the dimensions cannot be negative, 8 feet is the amount that is added and subtracted to the dimensions. 8 = = Therefore, the dimensions of the deck are 16 feet by feet. 7.SALES The sales of a particular CD can be modeled by the equation S = 5m + 15m, where S is the number of CDs sold in thousands, and m is the number of months that it is on the market. a. In what month should the music store expect the CD to stop selling? b. In what month will CD sales peak? c. How many copies will the CD sell at its peak? SOLUTION: a. Find the values of m for which S equals 0. m =0orm = 5 esolutions Manual - Powered by Cognero The music store should expect the CD to stop selling in month 5. Page 16

17 m =0orm = 5 The music store should expect the CD to stop selling in month 5. b. Use a graphing calculator to find the maximum. [0, 10] scl: 1 by [0, 50] scl: 5 The maximum occurs at the point x =.5. So the CD sales should peak in month.5. c. The maximum on the graph is S = 156.5, where S is measured in the thousands. The peak number of copies sold is =156,50copies. Solve each equation by factoring. Confirm your answers using a graphing calculator. 8.6w = 11 SOLUTION: or The roots are and or about 1.8 and 1.8. Page 17 Confirm the roots using a graphing calculator. Let Y1 = 6w and Y = 11. Use the intersect option from the

18 c. The maximum on the graph is S = 156.5, where S is measured in the thousands. The peak number of copies sold is =156,50copies. 8-8 Differences of Squares Solve each equation by factoring. Confirm your answers using a graphing calculator. 8.6w = 11 SOLUTION: or The roots are and or about 1.8 and 1.8. Confirm the roots using a graphing calculator. Let Y1 = 6w and Y = 11. Use the intersect option from the CALC menu to find the points of intersection. Thus, the solutions are = 5x and. SOLUTION: Page 18

19 Thus, the solutions are and 8-8 Differences of Squares = 5x. SOLUTION: or The roots are and. Confirm the roots using a graphing calculator. Let Y1 = 100 and Y = 5x. Use the intersect option from the CALC menu to find the points of intersection. Thus, the solutions are and. 50.6x 1 = 0 SOLUTION: The roots are and or 0.15 and Page 19 Confirm the roots using a graphing calculator. Let Y1 = 6x 1 and Y = 0. Use the intersect option from the

20 8-8 Differences of Squares Thus, the solutions are and. 50.6x 1 = 0 SOLUTION: The roots are and or 0.15 and Confirm the roots using a graphing calculator. Let Y1 = 6x 1 and Y = 0. Use the intersect option from the CALC menu to find the points of intersection. Thus, the solutions are and. 51.y =0 SOLUTION: Page 0

21 Thus, the solutions are and. 51.y =0 SOLUTION: The roots are and or -0.75and0.75. Confirm the roots using a graphing calculator. Let Y1 = y and Y = 0. Use the intersect option from the CALC menu to find the points of intersection. Thus, the solutions are and. 5. b = 16 SOLUTION: Page 1

22 8-8 Differences of Squares Thus, the solutions are and. 5. b = 16 SOLUTION: The roots are 8and8. Confirm the roots using a graphing calculator. Let Y1 = andy = 16. Use the intersect option from the CALC menu to find the points of intersection. Thus, the solutions are 8 and x =0 SOLUTION: or Page

23 8-8 Differences of Squares Thus, the solutions are 8 and x =0 SOLUTION: or The roots are 5and5. Confirm the roots using a graphing calculator. Let Y1 = CALC menu to find the points of intersection. andy =0. Use the intersect option from the Thus, the solutions are 5 and d 81 = 0 SOLUTION: The roots are and. Page Confirm the roots using a graphing calculator. Let Y1 = 9d 81 and Y =0. Use the intersect option from the

24 8-8 Differences of Squares Thus, the solutions are 5 and d 81 = 0 SOLUTION: The roots are and. Confirm the roots using a graphing calculator. Let Y1 = 9d 81 and Y =0. Use the intersect option from the CALC menu to find the points of intersection. Thus, the solutions are and. 55.a = SOLUTION: The roots are and or and Page

25 8-8 Differences of Squares Thus, the solutions are and. 55.a = SOLUTION: The roots are and or and Confirm the roots using a graphing calculator. Let Y1 = a and Y = CALC menu to find the points of intersection.. Use the intersect option from the Thus, the solutions are and. 56.MULTIPLEREPRESENTATIONS In this problem, you will investigate perfect square trinomials. a.tabular Copy and complete the table below by factoring each polynomial. Then write the first and last terms of the given polynomials as perfect squares. Page 5

26 56.MULTIPLEREPRESENTATIONS In this problem, you will investigate perfect square trinomials. a.tabular Copy and complete the table below by factoring each polynomial. Then write the first and last 8-8 Differences of Squares terms of the given polynomials as perfect squares. b.analytical Write the middle term of each polynomial using the square roots of the perfect squares of the first and last terms. c.algebraic Write the pattern for a perfect square trinomial. d.verbal What conditions must be met for a trinomial to be classified as a perfect square trinomial? SOLUTION: a. In this trinomial, a = 9, b = and c = 16, so m + p is negative and mp is negative. Therefore, m and p must have different signs. List the factors of 9(16) or 1 with a sum of. Factors of 1 Sum 1, 1 15, 7 7, 8 51, 6 0 6, 0 8, , , 1 The correct factors are 1 and 1. In this trinomial, a =, b = 0 and c = 5, so m + p is negative and mp is negative. Therefore, m and p must have different signs. List the factors of (5) or 100 with a sum of 0. Factors of 100 1, 100, 50, 5 10, 10 Sum The correct factors are 10 and 10. Page 6

27 different signs. List the factors of (5) or 100 with a sum of 0. Factors of 100 1, Differences of Squares, 50, 5 10, 10 Sum The correct factors are 10 and 10. In this trinomial, a = 16, b = 0 and c = 9, so m + p is negative and mp is negative. Therefore, m and p must have different signs. List the factors of 16(9) or 1 with a sum of. Factors of 1 1, 1, 7, 8, 6 6, 8, 18 9, 16 1, 1 Sum The correct factors are 1 and 1. In this trinomial, a = 5, b = 0 and c =, so m + p is negative and mp is negative. Therefore, m and p must have different signs. List the factors of 5() or 100 with a sum of 0. Factors of 100 1, 100, 50, 5 10, 10 Sum esolutions Manual - Powered by Cognero The correct factors are 10 and 10. Page 7

28 Factors of 100 Sum 1, , 50 5, , 10 0 The correct factors are 10 and 10. b. See table. c. (a + b)(a + b) = a + ab + b and (a b)(a b) = a ab + b d. The first and last terms must be perfect squares and the middle term must be times the square roots of the first and last terms. 57.ERRORANALYSIS Elizabeth and Lorenzo are factoring an expression. Is either of them correct? Explain your reasoning. SOLUTION: Lorenzo is correct. Elizabeth s answer multiplies to 16x 5y. The exponent on x should be. 58.CHALLENGE Factor and simplify 9 (k + ), a difference of squares. SOLUTION: Page 8

29 8-8 Differences of Squares Elizabeth s answer multiplies to 16x 5y. The exponent on x should be. 58.CHALLENGE Factor and simplify 9 (k + ), a difference of squares. SOLUTION: Thus 9 (k + ) factor to [ + (k + )][ (k + )] and simplifies to k 6k. 59.CCSSPERSEVERANCE Factor x SOLUTION: 60.REASONING Write and factor a binomial that is the difference of two perfect squares and that has a greatest common factor of 5mk. SOLUTION: Begin with the GCF of 5mk and the difference of squares. 5mk(a b ) Distribute the GCF to obtain the binomial of 5mka 5mkb. 5mka 5mkb = 5mk(a b ) = 5mk(a b)(a + b) 61.REASONING Determine whether the following statement is true or false. Give an example or counterexample to justify your answer. All binomials that have a perfect square in each of the two terms can be factored. SOLUTION: false; The two squares cannot be added together in the binomial. For example, a + b cannot be factored. 6.OPENENDED Write a binomial in which the difference of squares pattern must be repeated to factor it completely. Then factor the binomial. SOLUTION: Ifyouplantorepeatthethefactoringtwice,youneedtofindthedifferenceoftwotermstothethpower. esolutions Manual - Powered by Cognero Page 9

30 SOLUTION: false; The two squares cannot be added together in the binomial. 8-8 Differences of Squares For example, a + b cannot be factored. 6.OPENENDED Write a binomial in which the difference of squares pattern must be repeated to factor it completely. Then factor the binomial. SOLUTION: Ifyouplantorepeatthethefactoringtwice,youneedtofindthedifferenceoftwotermstothethpower. 6.WRITINGINMATH Describe why the difference of squares pattern has no middle term with a variable. SOLUTION: When the difference of squares pattern is multiplied together using the FOIL method, the outer and inner terms are opposites of each other. When these terms are added together, the sum is zero. Consider the example. 6.One of the roots of x + 1x = is 8. What is the other root? A B C D SOLUTION: Page 0 The correct choice is B. 65.Which of the following is the sum of both solutions of the equation x + x = 5?

31 6.One of the roots of x + 1x = is 8. What is the other root? A B C D SOLUTION: The correct choice is B. 65.Which of the following is the sum of both solutions of the equation x + x = 5? F 1 G H J 1 SOLUTION: or =. The correct choice is G. esolutions Manual - Powered by Cognero are the x-intercepts of the 66.What graph of y = x + 7x + 0? Page 1

32 9 + 6 =. 8-8 Differences of Squares The correct choice is G. 66.What are the x-intercepts of the graph of y = x + 7x + 0? A, B, C, D, SOLUTION: To find the x-intercepts, find the zeros or roots of the related equation. or The correct choice is C. 67.EXTENDEDRESPONSE Two cars leave Cleveland at the same time from different parts of the city and both drive to Cincinnati. The distance in miles of the cars from the center of Cleveland can be represented by the two equations below, where t represents the time in hours. Car A: 65t + 15 Car B: 60t + 5 a. Which car is faster? Explain. b. Find an expression that models the distance between the two cars. c. How far apart are the cars after hours? SOLUTION: a. The slope represents the speed of the car. Car A has a speed of 65 mph and Car B has a speed of 60 mph. Thus, Car A is traveling at a greater speed. b. c. Substitute in for t. Page

33 or 8-8 Differences of Squares The correct choice is C. 67.EXTENDEDRESPONSE Two cars leave Cleveland at the same time from different parts of the city and both drive to Cincinnati. The distance in miles of the cars from the center of Cleveland can be represented by the two equations below, where t represents the time in hours. Car A: 65t + 15 Car B: 60t + 5 a. Which car is faster? Explain. b. Find an expression that models the distance between the two cars. c. How far apart are the cars after hours? SOLUTION: a. The slope represents the speed of the car. Car A has a speed of 65 mph and Car B has a speed of 60 mph. Thus, Car A is traveling at a greater speed. b. c. Substitute in for t. After hours,thecarsare.5milesapart. Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime. 68.5x 17x + 1 SOLUTION: In this trinomial, a = 5, b = 17 and c = 1, so m + p is negative and mp is positive. Therefore, m and p must both be negative. List the negative factors of 5(1) or 70, and look for the pair of factors with a sum of 17. Factors of 70, 5 5, 1 7, 10 Sum The correct factors are 7 and 10. Page

34 8-8 Differences of Squares After hours,thecarsare.5milesapart. Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime. 68.5x 17x + 1 SOLUTION: In this trinomial, a = 5, b = 17 and c = 1, so m + p is negative and mp is positive. Therefore, m and p must both be negative. List the negative factors of 5(1) or 70, and look for the pair of factors with a sum of 17. Factors of 70, 5 5, 1 7, 10 Sum The correct factors are 7 and a a + 15 SOLUTION: In this trinomial, a = 5, b = and c = 15, so m + p is negative and mp is positive. Therefore, m and p must both be negative. List the negative factors of 5(15) or 75, and look for the pair of factors with a sum of. Factors of 75, 5 5, 15 Sum 8 0 There are no factors of 75 with a sum of. So, the trinomial is prime x 0xy + 10y SOLUTION: In this trinomial, a = 10, b = 0 and c = 10, so m + p is negative and mp is positive. Therefore, m and p must both be negative. List the negative factors of 10(10) or 100, and look for the pair of factors with a sum of 0. Factors of 100, 50 5, 0 10, 10 Sum The correct factors are 10 and 10. Page

35 5, 15 0 There are no factors of 75 with a sum of. 8-8 Differences of Squares So, the trinomial is prime x 0xy + 10y SOLUTION: In this trinomial, a = 10, b = 0 and c = 10, so m + p is negative and mp is positive. Therefore, m and p must both be negative. List the negative factors of 10(10) or 100, and look for the pair of factors with a sum of 0. Factors of 100, 50 5, 0 10, 10 Sum The correct factors are 10 and 10. Solve each equation. Check your solutions. 71.n 9n = 18 SOLUTION: The roots are and 6. Check by substituting and 6 in for n in the original equation. and The solutions are and 6. Page 5

36 Solve each equation. Check your solutions. 71.n 9n = 18 SOLUTION: The roots are and 6. Check by substituting and 6 in for n in the original equation. and The solutions are and a = 7a SOLUTION: The roots are and 5. Check by substituting and 5 in for a in the original equation. and The Manual solutions are byand 5. esolutions - Powered Cognero 7.x x = 96 Page 6

37 8-8 Differences Squares The solutionsofare and a = 7a SOLUTION: The roots are and 5. Check by substituting and 5 in for a in the original equation. and The solutions are and 5. 7.x x = 96 SOLUTION: The roots are 6 and 16. Check by substituting 6 and 16 in for x in the original equation. and The solutions are 6 and 16. esolutions Manual - Powered Cognero Victoriabyand Trey 7.SAVINGS Page 7 each want to buy a scooter. In how many weeks will Victoria and Trey have saved the same amount of money, and how much will each of them have saved?

38 8-8 Differences of Squares The solutions are 6 and SAVINGS Victoria and Trey each want to buy a scooter. In how many weeks will Victoria and Trey have saved the same amount of money, and how much will each of them have saved? SOLUTION: Victoria and Trey will have saved the same amount of money at the end of week () = 0 The amount of money they will have saved is \$0. Solve each inequality. Graph the solution set on a number line. 75.t SOLUTION: The solution is {t t }. 76.d +5 7 SOLUTION: The solution is {d d } k > 1 SOLUTION: Page 8

39 The solution is {d d } k > 1 SOLUTION: The solution is {k k > } < + g SOLUTION: The solution is {g g > } m SOLUTION: The solution is {m m }. 80.y > 8 + y SOLUTION: The solution is {y y > 8}. esolutions Manual - Powered Cognero Silvia isbybeginning 81.FITNESS Page 9 an exercise program that calls for 0 minutes of walking each day for the first week. Each week thereafter, she has to increase her daily walking for the week by 7 minutes. In which week will she first walk over an hour a day?

40 The solution is {y y > 8}. 81.FITNESS Silvia is beginning an exercise program that calls for 0 minutes of walking each day for the first week. Each week thereafter, she has to increase her daily walking for the week by 7 minutes. In which week will she first walk over an hour a day? SOLUTION: Silvia's time exercising can be modeled by an arithmetic sequence. 0, 7,, 1, 8, 55,... a 1 is the initial value or 0. d is the difference or 7. Substitute the values into the formula for the nth term of an arithmetic sequence: The first week in which she will walk over an hour a day is the seventh week. Find each product. 8.(x 6) SOLUTION: 8.(x )(x ) SOLUTION: 8.(x + )(x + ) SOLUTION: 85.(x 5) SOLUTION: Page 0

41 8.(x + )(x + ) SOLUTION: 85.(x 5) SOLUTION: 86.(6x 1) SOLUTION: 87.(x + 5)(x + 5) SOLUTION: Page 1

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