The graphs coincide. Therefore, the trinomial has been factored correctly. The factors are (x + 2)(x + 12).

Transcription

1 8-6 Solving x^ + bx + c = 0 Factor each polynomial. Confirm your answers using a graphing calculator. 1. x + 14x + 4 In this trinomial, b = 14 and c = 4, so m + p is positive and mp is positive. Therefore, m and p must both be positive. List the factors of 4 and identify the factors with a sum of 14. Factors of 4 1, 4 5, , , 6 10 The correct factors are and 1. Confirm by graphing Y1 = x +14x + 4 and Y = (x + )(x + 1) on the same screen. The graphs coincide. Therefore, the trinomial has been factored correctly. The factors are (x + )(x + 1).. y 7y 30 In this trinomial, b = 7 and c = 30, so m + p is negative and mp is negative. Therefore, m and p must have different signs. List the factors of 30 and identify the factors with a sum of 7. Factors of , 30 13, , , 6 1 6, , , 9 30, 1 The correct factors are 3 and 10. Confirm by graphing Y1 = y 7y 30 + and Y = (x 10)(x + 3) on the same screen. Page 1

2 The graphs coincide. Therefore, the trinomial has been factored correctly. 8-6 Solving x^are + bx + )(x c = 0+ 1). The factors (x +. y 7y 30 In this trinomial, b = 7 and c = 30, so m + p is negative and mp is negative. Therefore, m and p must have different signs. List the factors of 30 and identify the factors with a sum of 7. Factors of , 30 13, , , 6 1 6, , , 9 30, 1 The correct factors are 3 and 10. Confirm by graphing Y1 = y 7y 30 + and Y = (x 10)(x + 3) on the same screen. The graphs coincide. Therefore, the trinomial has been factored correctly. The factors are (x 10)(x + 3). 3. n + 4n 1 In this trinomial, b = 4 and c = 1, so m + p is positive and mp is negative. Therefore, m and p must have different signs. List the factors of 1 and identify the factors with a sum of 4. Factors of 1 0 1, 1 4 3, 7 4 7, 3 1, 1 0 The correct factors are 7 and 3. Confirm by graphing Y1 = n + 4n 1 + and Y = (n 3)(n + 7) on the same screen. Page

3 The graphs the trinomial has been factored correctly. 8-6 Solving x^coincide. + bx + ctherefore, =0 The factors are (x 10)(x + 3). 3. n + 4n 1 In this trinomial, b = 4 and c = 1, so m + p is positive and mp is negative. Therefore, m and p must have different signs. List the factors of 1 and identify the factors with a sum of 4. Factors of 1 0 1, 1 4 3, 7 4 7, 3 1, 1 0 The correct factors are 7 and 3. Confirm by graphing Y1 = n + 4n 1 + and Y = (n 3)(n + 7) on the same screen. The graphs coincide. Therefore, the trinomial has been factored correctly. The factors are (n 3)(n + 7). 4. m 15m + 50 In this trinomial, b = 15 and c = 50, so m + p is negative and mp is positive. Therefore, m and p must both be negative. List the factors of 50 and identify the factors with a sum of 15. Factors of 50 1, 50 51, 5 7 5, The correct factors are 5 and 10. Confirm by graphing Y1 = m 15m and Y = (m 5)(m 10) on the same screen. The graphs coincide. Therefore, the trinomial has been factored correctly. Page 3

4 The graphs the trinomial has been factored correctly. 8-6 Solving x^coincide. + bx + ctherefore, =0 The factors are (n 3)(n + 7). 4. m 15m + 50 In this trinomial, b = 15 and c = 50, so m + p is negative and mp is positive. Therefore, m and p must both be negative. List the factors of 50 and identify the factors with a sum of 15. Factors of 50 1, 50 51, 5 7 5, The correct factors are 5 and 10. Confirm by graphing Y1 = m 15m and Y = (m 5)(m 10) on the same screen. The graphs coincide. Therefore, the trinomial has been factored correctly. The factors are (m 5)(m 10). Solve each equation. Check your solutions. 5. x 4x 1 = 0 List the factors of 1 and identify the factors with a sum of 4. Factors of 1 1, 1 0 1, , 7 4 3, 7 4 The roots are 3 and 7. Check by substituting 3 and 7 in for x in the original equation. and Page 4

5 The graphs the trinomial has been factored correctly. 8-6 Solving x^coincide. + bx + ctherefore, =0 The factors are (m 5)(m 10). Solve each equation. Check your solutions. 5. x 4x 1 = 0 List the factors of 1 and identify the factors with a sum of 4. Factors of 1 1, 1 0 1, , 7 4 3, 7 4 The roots are 3 and 7. Check by substituting 3 and 7 in for x in the original equation. and The solutions are 3 and n 3n + = 0 List the factors of and identify the factors with a sum of 3. Factors of 1, 3 1, 3 The roots are 1 and. Check by substituting 1 and in for n in the original equation. and Page 5

6 8-6 Solving x^ +are bx 3 + and c = 7. 0 The solutions 6. n 3n + = 0 List the factors of and identify the factors with a sum of 3. Factors of 1, 3 1, 3 The roots are 1 and. Check by substituting 1 and in for n in the original equation. and The solutions are 1 and. 7. x 15x + 54 = 0 List the factors of 54 and identify the factors with a sum of 15. Factors of 54 1, , 54 55, 7 9, 7 9 3, , , , 9 15 The roots are 9 and 6. Check by substituting 9 and 6 in for x in the original equation. Page 6

7 8-6 Solving x^ + bx + c = 0 The solutions are 1 and. 7. x 15x + 54 = 0 List the factors of 54 and identify the factors with a sum of 15. Factors of 54 1, , 54 55, 7 9, 7 9 3, , , , 9 15 The roots are 9 and 6. Check by substituting 9 and 6 in for x in the original equation. and The solutions are 6 and x + 1x = 3 Rewrite the equation with 0 on the right side. List the factors of 3 and identify the factors with a sum of 1. Factors of 3 1, 3 33, , 8 1 Page 7

8 8-6 Solving x^ + bx + c = 0 The solutions are 6 and x + 1x = 3 Rewrite the equation with 0 on the right side. List the factors of 3 and identify the factors with a sum of 1. Factors of 3 1, 3 33, , 8 1 The roots are 8 and 4. Check by substituting 8 and 4 in for x in the original equation. and The solutions are 8 and x x 7 = 0 List the factors of 7 and identify the factors with a sum of 1. Factors of 7 1, , 7 71, 36 34, , 4 1 3, 4 1 4, , , 1 6, 1 6 8, 9 1 Page 8

9 8-6 Solving x^ + bx + c = 0 The solutions are 8 and x x 7 = 0 List the factors of 7 and identify the factors with a sum of 1. Factors of 7 1, , 7 71, 36 34, , 4 1 3, 4 1 4, , , 1 6 6, 1 6 8, 9 1 8, 9 1 The roots are 8 and 9. Check by substituting 8 and 9 in for x in the original equation. and The solutions are 8 and x 10x = 4 Rewrite the equation with 0 on the right side.. List the factors of 4 and identify the factors with a sum of 10. Factors of 4 1, 4 5, , 8 11 Page 9

10 8-6 Solving x^ + bx + c = 0 The solutions are 8 and x 10x = 4 Rewrite the equation with 0 on the right side.. List the factors of 4 and identify the factors with a sum of 10. Factors of 4 1, 4 5, , , 6 10 The roots are 6 and 4. Check by substituting 6 and 4 in for x in the original equation. and The solutions are 6 and FRAMING Tina bought a frame for a photo, but the photo is too big for the frame. Tina needs to reduce the width and length of the photo by the same amount. The area of the photo should be reduced to half the original area. If the original photo is 1 inches by 16 inches, what will be the dimensions of the smaller photo? Let x be the amount that Tina should reduce the photo. So the dimensions are now (1 x)(16 x). The original area was 1(16) = 19 square inches. Since the area is to be reduced by half, the new area will be 96 square inches. Now solve the equation. Page 10

11 8-6 Solving x^ + bx + c = 0 The solutions are 6 and FRAMING Tina bought a frame for a photo, but the photo is too big for the frame. Tina needs to reduce the width and length of the photo by the same amount. The area of the photo should be reduced to half the original area. If the original photo is 1 inches by 16 inches, what will be the dimensions of the smaller photo? Let x be the amount that Tina should reduce the photo. So the dimensions are now (1 x)(16 x). The original area was 1(16) = 19 square inches. Since the area is to be reduced by half, the new area will be 96 square inches. Now solve the equation. The answer x = 4 does not make sense because it would result in a new length of 1 4, or 1 inches. Therefore, the length and the width of the photo must both be reduced by 4 inches. So, the new dimensions are 8 inches by 1 inches. Factor each polynomial. Confirm your answers using a graphing calculator. 1. x + 17x + 4 In this trinomial, b = 17 and c = 4, so m + p is positive and mp is positive. Therefore, m and p must both be positive. List the factors of 4 and identify the factors with a sum of 17. Factors of 4 1, 4 43, 1 3 3, , 7 13 The correct factors are 3 and 14. Confirm by graphing Y1 = x + 17x + 4 and Y = (x + 3)(x + 14) on the same screen. The graphs coincide. Therefore, the trinomial has been factored correctly. The factors are (x + 3)(x + 14). 13. y 17y + 7 Page 11

12 The answer x = 4 does not make sense because it would result in a new length of 1 4, or 1 inches. 8-6 Solving x^ bx + cand = 0the width of the photo must both be reduced by 4 inches. So, the new dimensions are 8 Therefore, the+length inches by 1 inches. Factor each polynomial. Confirm your answers using a graphing calculator. 1. x + 17x + 4 In this trinomial, b = 17 and c = 4, so m + p is positive and mp is positive. Therefore, m and p must both be positive. List the factors of 4 and identify the factors with a sum of 17. Factors of 4 1, 4 43, 1 3 3, , 7 13 The correct factors are 3 and 14. Confirm by graphing Y1 = x + 17x + 4 and Y = (x + 3)(x + 14) on the same screen. The graphs coincide. Therefore, the trinomial has been factored correctly. The factors are (x + 3)(x + 14). 13. y 17y + 7 In this trinomial, b = 17 and c = 7, so m + p is negative and mp is positive. Therefore, m and p must both be negative. List the factors of 7 and identify the factors with a sum of 17. Factors of , 7 38, , 4 4, , , 9 The correct factors are 8 and 9. Confirm by graphing Y1 = y 17y + 7 and Y = (y 8)(y 9) on the same screen. Page 1

13 The graphs coincide. Therefore, the trinomial has been factored correctly. 8-6 Solving x^are + bx + 3)(x c = 0+ 14). The factors (x y 17y + 7 In this trinomial, b = 17 and c = 7, so m + p is negative and mp is positive. Therefore, m and p must both be negative. List the factors of 7 and identify the factors with a sum of 17. Factors of , 7 38, , 4 4, , , 9 The correct factors are 8 and 9. Confirm by graphing Y1 = y 17y + 7 and Y = (y 8)(y 9) on the same screen. The graphs coincide. Therefore, the trinomial has been factored correctly. The factors are (y 8)(y 9). 14. a + 8a 48 In this trinomial, b = 8 and c = 48, so m + p is positive and mp is negative. Therefore, m and p must have opposite signs. List the factors of 48 and identify the factors with a sum of 8. Factors of 48 1, 48 47, 4 3, , 1 8 6, 8 8, 6 1, , , 48, 1 47 The correct factors are 4 and 1. Confirm by graphing Y1 = a + 8a 48 and Y = (a 4)(a + 1) on the same screen. Page 13

14 The graphs the trinomial has been factored correctly. 8-6 Solving x^coincide. + bx + ctherefore, =0 The factors are (y 8)(y 9). 14. a + 8a 48 In this trinomial, b = 8 and c = 48, so m + p is positive and mp is negative. Therefore, m and p must have opposite signs. List the factors of 48 and identify the factors with a sum of 8. Factors of 48 1, 48 47, 4 3, , 1 8 6, 8 8, 6 1, , , 48, 1 47 The correct factors are 4 and 1. Confirm by graphing Y1 = a + 8a 48 and Y = (a 4)(a + 1) on the same screen. The graphs coincide. Therefore, the trinomial has been factored correctly. The factors are (a 4)(a + 1). 15. n n 35 In this trinomial, b = and c = 35, so m + p is negative and mp is negative. Therefore, m and p must have opposite signs. List the factors of 35 and identify the factors with a sum of. Factors of 35 1, , 7 7, 5 35, 1 34 The correct factors are 7 and 5. Confirm by graphing Y1 = n n 35 and Y = (n 7)(n + 5) on the same screen. Page 14

15 The graphs the trinomial has been factored correctly. 8-6 Solving x^coincide. + bx + ctherefore, =0 The factors are (a 4)(a + 1). 15. n n 35 In this trinomial, b = and c = 35, so m + p is negative and mp is negative. Therefore, m and p must have opposite signs. List the factors of 35 and identify the factors with a sum of. Factors of 35 1, , 7 7, 5 35, 1 34 The correct factors are 7 and 5. Confirm by graphing Y1 = n n 35 and Y = (n 7)(n + 5) on the same screen. The graphs coincide. Therefore, the trinomial has been factored correctly. The factors are (n 7)(n + 5) h + h First rearrange the polynomial in decreasing order, h + 15h In this trinomial, b = 15 and c = 44, so m + p is positive and mp is positive. Therefore, m and p must both be positive. List the factors of 44 and identify the factors with a sum of 15. Factors of 44 1, 44 45, 4 4, The correct factors are 4 and 11. Confirm by graphing Y1 = h + h and Y = (h + 4)(h + 11) on the same screen. The graphs coincide. Therefore, the trinomial has been factored correctly. Page 15

16 The graphs the trinomial has been factored correctly. 8-6 Solving x^coincide. + bx + ctherefore, =0 The factors are (n 7)(n + 5) h + h First rearrange the polynomial in decreasing order, h + 15h In this trinomial, b = 15 and c = 44, so m + p is positive and mp is positive. Therefore, m and p must both be positive. List the factors of 44 and identify the factors with a sum of 15. Factors of 44 1, 44 45, 4 4, The correct factors are 4 and 11. Confirm by graphing Y1 = h + h and Y = (h + 4)(h + 11) on the same screen. The graphs coincide. Therefore, the trinomial has been factored correctly. The factors are (h + 4)(h + 11) x + x First rearrange the polynomial in decreasing order, x x In this trinomial, b = and c = 40, so m + p is negative and mp is positive. Therefore, m and p must both be negative. List the factors of 40 and identify the factors with a sum of. Factors of , 40, , , 8 The correct factors are and 0. Confirm by graphing Y1 = 40 x + x and Y = (x )(x 0) on the same screen. Page 16

17 The graphs the trinomial has been factored correctly. 8-6 Solving x^coincide. + bx + ctherefore, =0 The factors are (h + 4)(h + 11) x + x First rearrange the polynomial in decreasing order, x x In this trinomial, b = and c = 40, so m + p is negative and mp is positive. Therefore, m and p must both be negative. List the factors of 40 and identify the factors with a sum of. Factors of , 40, , , 8 The correct factors are and 0. Confirm by graphing Y1 = 40 x + x and Y = (x )(x 0) on the same screen. The graphs coincide. Therefore, the trinomial has been factored correctly. The factors are (x )(x 0) x + x First rearrange the polynomial in decreasing order x 10x 4. In this trinomial, b = 10 and c = 4, so m + p is negative and mp is negative. Therefore, m and p must have opposite signs. List the factors of 4 and identify the factors with a sum of 10. Factors of 4 1, 4 3, , 8 5 4, 6 6, 4 8, 3 5 1, 10 4, 1 3 The correct factors are 1 and. Confirm by graphing Y1 = 4 10x + x and Y = (x + )(x 1) on the same screen. Page 17

18 The graphs the trinomial has been factored correctly. 8-6 Solving x^coincide. + bx + ctherefore, =0 The factors are (x )(x 0) x + x First rearrange the polynomial in decreasing order x 10x 4. In this trinomial, b = 10 and c = 4, so m + p is negative and mp is negative. Therefore, m and p must have opposite signs. List the factors of 4 and identify the factors with a sum of 10. Factors of 4 1, 4 3, , 8 5 4, 6 6, 4 8, 3 5 1, 10 4, 1 3 The correct factors are 1 and. Confirm by graphing Y1 = 4 10x + x and Y = (x + )(x 1) on the same screen. The graphs coincide. Therefore, the trinomial has been factored correctly. The factors are (x + )(x 1) m + m First rearrange the polynomial in decreasing order m m 4. In this trinomial, b = 1 and c = 4, so m + p is negative and mp is negative. Therefore, m and p must have opposite signs. List the factors of 4 and identify the factors with a sum of 1. Factors of 4 1, 4 41, , , 7 1 7, , , 19 4, 1 41 The correct factors are 7 and 6. Page 18 Confirm by graphing Y1 = 4 m + m and Y = (m + 6)(m 7) on the same screen.

19 The graphs the trinomial has been factored correctly. 8-6 Solving x^coincide. + bx + ctherefore, =0 The factors are (x + )(x 1) m + m First rearrange the polynomial in decreasing order m m 4. In this trinomial, b = 1 and c = 4, so m + p is negative and mp is negative. Therefore, m and p must have opposite signs. List the factors of 4 and identify the factors with a sum of 1. Factors of 4 1, 4 41, , , 7 1 7, , , 19 4, 1 41 The correct factors are 7 and 6. Confirm by graphing Y1 = 4 m + m and Y = (m + 6)(m 7) on the same screen. The graphs coincide. Therefore, the trinomial has been factored correctly. The factors are (m + 6)(m 7). Solve each equation. Check your solutions. 0. x 7x + 1 = 0 List the factors of 1 and identify the factors with a sum of 7. Factors of 1 1, 1 13, 6 8 3, 4 7 The roots are 3 and 4. Check by substituting 3 and 4 in for x in the original equation. Page 19

20 The graphs the trinomial has been factored correctly. 8-6 Solving x^coincide. + bx + ctherefore, =0 The factors are (m + 6)(m 7). Solve each equation. Check your solutions. 0. x 7x + 1 = 0 List the factors of 1 and identify the factors with a sum of 7. Factors of 1 1, 1 13, 6 8 3, 4 7 The roots are 3 and 4. Check by substituting 3 and 4 in for x in the original equation. and The solutions are 3 and y + y = 0 Rewrite the equation with 0 on the right side. List the factors of 0 and identify the factors with a sum of 1. Factors of 0 1, , , 10 8, , 5 1 4, 5 Page 0

21 8-6 Solving x^ +are bx3+and c =4.0 The solutions 1. y + y = 0 Rewrite the equation with 0 on the right side. List the factors of 0 and identify the factors with a sum of 1. Factors of 0 1, , , 10 8, , 5 4, 5 1 The roots are 5 and 4. Check by substituting 5 and 4 in for y in the original equation. and The solutions are 5 and 4.. x 6x = 7 Rewrite the equation with 0 on the right side. List the factors of 7 and identify the factors with a sum of 6. Factors of 7 1, 7 6 1, 7 6 3, 9 6 Page 1

22 8-6 Solving x^ + bx + c = 0 The solutions are 5 and 4.. x 6x = 7 Rewrite the equation with 0 on the right side. List the factors of 7 and identify the factors with a sum of 6. Factors of 7 1, 7 6 1, 7 6 3, 9 6 3, 9 6 The roots are 3 and 9. Check by substituting 3 and 9 in for x in the original equation. and The solutions are 3 and a + 11a = 18 Rewrite the equation with 0 on the right side. List the factors of 18 and identify the factors with a sum of 11. Factors of 18 1, 18 19, , 6 19 Page

23 8-6 Solving x^ + bx + c = 0 The solutions are 3 and a + 11a = 18 Rewrite the equation with 0 on the right side. List the factors of 18 and identify the factors with a sum of 11. Factors of 18 1, 18 19, , 6 19 The roots are and 9. Check by substituting and 9 in for a in the original equation. and The solutions are and c + 10c + 9 = 0 List the factors of 9 and identify the factors with a sum of 10. Factors of 9 1, , 3 6 The roots are 1 and 9. Check by substituting 1 and 9 in for c in the original equation. Page 3

24 8-6 Solving x^ + bx + c = 0 The solutions are and c + 10c + 9 = 0 List the factors of 9 and identify the factors with a sum of 10. Factors of 9 1, , 3 6 The roots are 1 and 9. Check by substituting 1 and 9 in for c in the original equation. and The solutions are 1 and x 18x = 3 Rewrite the equation with 0 on the right side. List the factors of 3 and identify the factors with a sum of 18. Factors of 3 1, 3 33, , 8 1 The roots are 16 and. Check by substituting 16 and in for x in the original equation. Page 4

25 8-6 Solving x^ + bx + c = 0 The solutions are 1 and x 18x = 3 Rewrite the equation with 0 on the right side. List the factors of 3 and identify the factors with a sum of 18. Factors of 3 1, 3 33, , 8 1 The roots are 16 and. Check by substituting 16 and in for x in the original equation. and The solutions are and n 10 = 7n Rewrite the equation with 0 on the right side. List the factors of 10 and identify the factors with a sum of 7. Factors of 10 1, , , 60 58, , esolutions Manual Powered by Cognero 37 3, , , 30 Page 5

26 List the factors of 10 and identify the factors with a sum of 7. Factors of 10 1, , Solving x^ + bx + c = 0, 60 58, , , , , , , , , 0 7 8, , 15 1, 10 1, 10 The roots are 15 and 8. Check by substituting 15 and 8 in for n in the original equation. and The solutions are 8 and d + 56 = 18d Rewrite the equation with 0 on the right side. List the factors of 56 and identify the factors with a sum of 18. Factors of 56 1, 56 57, , , 8- Powered by Cognero15 esolutions Manual Page 6

27 8-6 Solving x^ + bx + c = 0 The solutions are 8 and d + 56 = 18d Rewrite the equation with 0 on the right side. List the factors of 56 and identify the factors with a sum of 18. Factors of 56 1, 56 57, , , 8 15 The roots are 4 and 14. Check by substituting 4 and 14 in for d in the original equation. and The solutions are 4 and y 90 = 13y Rewrite the equation with 0 on the right side. List the factors of 90 and identify the factors with a sum of 13. Factors of 90 1, , 90 89, 45 43, , 30 3, 30 7 Page 7

28 8-6 Solving x^ + bx + c = 0 The solutions are 4 and y 90 = 13y Rewrite the equation with 0 on the right side. List the factors of 90 and identify the factors with a sum of 13. Factors of 90 1, , 90 89, 45 43, , , , , , , , , 10 The roots are 5 and 18. Check by substituting 5 and 18 in for y in the original equation. and The solutions are 5 and h + 48 = 16h Rewrite the equation with 0 on the right side. Page 8

29 8-6 Solving x^ + bx + c = 0 The solutions are 5 and h + 48 = 16h Rewrite the equation with 0 on the right side. List the factors of 48 and identify the factors with a sum of 16. Factors of 48 1, 48 49, 4 6 3, , , 8 The roots are 4 and 1. Check by substituting 4 and 1 in for h in the original equation. and The solutions are 4 and GEOMETRY A triangle has an area of 36 square feet. If the height of the triangle is 6 feet more than its base, what are its height and base? Let b represent the base. Then b + 6 is the height. List the factors of 7 and identify the factors with a sum of 6. Page 9

30 8-6 Solving x^ + bx + c = 0 The solutions are 4 and GEOMETRY A triangle has an area of 36 square feet. If the height of the triangle is 6 feet more than its base, what are its height and base? Let b represent the base. Then b + 6 is the height. List the factors of 7 and identify the factors with a sum of 6. Factors of 7 1, , 7 71, 36 34, , 4 1 3, , , , 1 6 6, 1 1 8, 9 1 8, 9 However, the base cannot be negative, so the base is 6 feet. The height is b + 6 = = 1 feet. 31. GEOMETRY A rectangle has an area represented by x 4x 1 square feet. If the length is x + feet, what is the width of the rectangle? The area of the rectangle is x 4x 1. List the factors of 1 and identify the factors with a sum of 4. Factors of 1 1, , 1 11, 6 4, -6Powered by Cognero 4 esolutions Manual 1 3, 4 1 3, 4 Page 30

31 8-6 Solving x^ + bx + c = 0 However, the base cannot be negative, so the base is 6 feet. The height is b + 6 = = 1 feet. 31. GEOMETRY A rectangle has an area represented by x 4x 1 square feet. If the length is x + feet, what is the width of the rectangle? The area of the rectangle is x 4x 1. List the factors of 1 and identify the factors with a sum of 4. Factors of 1 1, , 1 11, 6 4, , 4 1 3, 4 Then area of the rectangle is (x + )(x 6). Area is found by multiplying the length by the width. Because the length is x +, the width must be x SOCCER The width of a high school soccer field is 45 yards shorter than its length. a. Define a variable, and write an expression for the area of the field. b. The area of the field is 9000 square yards. Find the dimensions. a. Let = length. The area of the field is the length times the width, or b. ( 45). Then length cannot be negative, so it is 10 yd and the width is 10 45, or 75 yards. CCSS STRUCTURE Factor each polynomial. 33. q + 11qr + 18r In this trinomial, b = 11 and c = 18, so m + p is positive and mp is positive. Therefore, m and p must both be positive. List the factors of 18 and identify the factors with a sum of 11. Factors of 18 1, 18 esolutions Manual - Powered by Cognero, 9 3, Page 31

32 8-6 Solving x^ + bx + c = 0 Then length cannot be negative, so it is 10 yd and the width is 10 45, or 75 yards. CCSS STRUCTURE Factor each polynomial. 33. q + 11qr + 18r In this trinomial, b = 11 and c = 18, so m + p is positive and mp is positive. Therefore, m and p must both be positive. List the factors of 18 and identify the factors with a sum of 11. Factors of 18 1, 18, 9 3, The correct factors are and 9. The trinomial has two variables q and r. The first term in each binomial will have q's, the second term will have the r along with the factors. 34. x 14xy 51y In this trinomial, b = 14 and c = 51, so m + p is negative and mp is negative. Therefore, m and p must have different signs. List the factors of 51 and identify the factors with a sum of 14. Factors of 51 1, , , , 1 50 The correct factors are 3 and 17. The trinomial has two variables x and y. The first term in each binomial will have x's, the second term will have the y along with the factors. 35. x 6xy + 5y In this trinomial, b = 6 and c = 5, so m + p is negative and mp is positive. Therefore, m and p must both be negative. List the negative factors of 5 and identify the factors with a sum of 6. Factors of 5 6 1, 5 The correct factors are 1 and 6. The trinomial has two variables x and y. The first term in each binomial will have x's, the second term will have the y along with the factors. 36. a + 10ab 39b Page 3

33 The trinomial has two variables x and y. The first term in each binomial will have x's, the second term will have the y along with the factors. 8-6 Solving x^ + bx + c = x 6xy + 5y In this trinomial, b = 6 and c = 5, so m + p is negative and mp is positive. Therefore, m and p must both be negative. List the negative factors of 5 and identify the factors with a sum of 6. Factors of 5 6 1, 5 The correct factors are 1 and 6. The trinomial has two variables x and y. The first term in each binomial will have x's, the second term will have the y along with the factors. 36. a + 10ab 39b In this trinomial, b = 10 and c = 39, so m + p is positive and mp is negative. Therefore, m and p must have different signs. List the factors of 39 and identify the factors with a sum of 10. Factors of , , 13 3, , 1 38 The correct factors are 3 and 13. The trinomial has two variables a and b. The first term in each binomial will have as, the second term will have the b along with the factors. 37. SWIMMING The length of a rectangular swimming pool is 0 feet greater than its width. The area of the pool is 55 square feet. a. Define a variable and write an equation for the area of the pool. b. Solve the equation. c. Interpret the solutions. Do both solutions make sense? Explain. a. Sample answer: Let w = width. Then the length is 0 feet greater than its width, so = w + 0. The area, which is 55 square feet, is found by multiplying the length times the width. Therefore the area is (w + 0)w = 55. b. List the factors of 55 and identify the factors with a sum of 0. Factors of 55 1, , esolutions Manual - Powered by Cognero 5, , , 75 Page 33

34 The trinomial has two variables a and b. The first term in each binomial will have as, the second term will have the b along with the factors. 8-6 Solving x^ + bx + c = SWIMMING The length of a rectangular swimming pool is 0 feet greater than its width. The area of the pool is 55 square feet. a. Define a variable and write an equation for the area of the pool. b. Solve the equation. c. Interpret the solutions. Do both solutions make sense? Explain. a. Sample answer: Let w = width. Then the length is 0 feet greater than its width, so = w + 0. The area, which is 55 square feet, is found by multiplying the length times the width. Therefore the area is (w + 0)w = 55. b. List the factors of 55 and identify the factors with a sum of 0. Factors of 55 1, , , , , , , , , 5 4 1, 5 The width cannot be negative, therefore it is 15 feet. The length is , 35 feet. c. The solution of 15 means that the width is 15 ft. The solution 35 does not make sense because the width cannot be negative. GEOMETRY Find an expression for the perimeter of a rectangle with the given area. 38. A = x + 4x 81 The area of the rectangle is x + 4x 81. In this trinomial, b = 4 and c = 4, so m + p is positive and mp is positive. Therefore, m and p must both be positive. List the factors of 81 and identify the factors with a sum of 4. Factors of 81 1, , , 7 4 Page 34 3, 7 4 9, 9 0

35 The width cannot be negative, therefore it is 15 feet. The length is , 35 feet. c. The solution 8-6 Solving x^ + bx + cmeans = 0 that the width is 15 ft. The solution 35 does not make sense because the width cannot of 15 be negative. GEOMETRY Find an expression for the perimeter of a rectangle with the given area. 38. A = x + 4x 81 The area of the rectangle is x + 4x 81. In this trinomial, b = 4 and c = 4, so m + p is positive and mp is positive. Therefore, m and p must both be positive. List the factors of 81 and identify the factors with a sum of 4. Factors of 81 1, , , 7 4 3, 7 4 9, 9 0 Then area of the rectangle of x + 4x 81 can be factored to (x + 7)(x 3). Area is found by multiplying the length by the width, so the length is x + 7 and the width is x 3. So, an expression for the perimeter of the rectangle is 4x A = x + 13x 90 The area of the rectangle is x + 13x 90. In this trinomial, b = 13 and c = 90, so m + p is positive and mp is positive. Therefore, m and p must both be positive. List the factors of 90 and identify the factors with a sum of 13. Factors of 90 1, , 90 89, 45 43, , , , 18 5, , , , 10 9, 10 1 The area of the rectangle x + 13x 90 factors to (x + 18)(x 5). Area is found by multiplying the length by the width, so the length is x + 18 and the width is x 5. Page 35

36 8-6 Solving x^ + bxfor + cthe = 0perimeter of the rectangle is 4x So, an expression 39. A = x + 13x 90 The area of the rectangle is x + 13x 90. In this trinomial, b = 13 and c = 90, so m + p is positive and mp is positive. Therefore, m and p must both be positive. List the factors of 90 and identify the factors with a sum of 13. Factors of 90 1, , 90 89, 45 43, , , , , , , , , 10 The area of the rectangle x + 13x 90 factors to (x + 18)(x 5). Area is found by multiplying the length by the width, so the length is x + 18 and the width is x 5. So, an expression for the perimeter of the rectangle is 4x MULTIPLE REPRESENTATIONS In this problem, you will explore factoring when the leading coefficient is not 1. a. TABULAR Copy and complete the table below. b. ANALYTICAL How are m and p related to a and c? c. ANALYTICAL How are m and p related to b? d. VERBAL Describe a process you can use for factoring a polynomial of the form ax + bx + c. a. Use FOIL or the Distributive Property to find the product of the binomials. Page 36

37 8-6 Solving x^ + bx + c = 0 So, an expression for the perimeter of the rectangle is 4x MULTIPLE REPRESENTATIONS In this problem, you will explore factoring when the leading coefficient is not 1. a. TABULAR Copy and complete the table below. b. ANALYTICAL How are m and p related to a and c? c. ANALYTICAL How are m and p related to b? d. VERBAL Describe a process you can use for factoring a polynomial of the form ax + bx + c. a. Use FOIL or the Distributive Property to find the product of the binomials. b. ac is the product of the coefficients of the first and last terms. mp is the product of the coefficients of the middle two terms when foiling. Looking at the last two columns and you will see that mp = ac. c. Look at the second and third columns and you will see that m + p = b. The coefficients of the middle terms equal b. d. When factoring trinomials, we look for two integers, m and p, for which mp = ac and m + p = b. 41. ERROR ANALYSIS Jerome and Charles have factored x + 6x 16. Is either of them correct? Explain your reasoning. Charles is correct. In this trinomial, b = 6 and c = 16, so m + p is positive and mp is positive. Therefore, m and p must both be positive. List the factors of 16 and identify the factors with a sum of 6. Factors of 16 1, , 16 15, 8 6, 8 6 4, 4 0 The correct factors are, 8. Jerome s answer once multiplied is x 6x 16. The middle term should be positive. CCSS ARGUMENTS Find all values of k so that each polynomial can be factored using integers. 4. x + k x 19 Page 37

38 , 8 6, 8 6 4, 4 + bx + c = Solving x^ The correct factors are, 8. Jerome s answer once multiplied is x 6x 16. The middle term should be positive. CCSS ARGUMENTS Find all values of k so that each polynomial can be factored using integers. 4. x + k x 19 In this trinomial, b = k, c = 19, and mp is negative. Therefore, m + p must equal the sum of the factors of 19. List the factors of 19. The sum of m + p will equal k. of m + p Factors of , 19 19, 1 18 Therefore, k could have values of 18 or x + k x + 14 In this trinomial, b = k, c = 14 and mp is positive. Therefore, m + p must equal the sum of the factors of 14. List the factors of 14. The sum of m + p will equal k. Factors of 14 of m + p 1, 14 15, , 14 9, 7 Therefore, k could have values of 9, 15, 9 or x 8x + k, k > 0 In this trinomial, b = 8, c = k and m + p is negative. Therefore, mp must equal the product of the two numbers that add to 8. Because k > 0, this means that both m and p are negative. List the numbers that add to 8. The product of mp will equal k. Two numbers that add Product of mp to 8 7 1, 7 1, , , 4 Therefore, k could have values of 7, 1, 15, or x 5x + k, k > 0 In this trinomial, b = 5, c = k and m + p is negative. Therefore, mp must equal the product of the two numbers that add to 5. Because k > 0, this means that both m and p are negative. List the numbers that add to 5. The product of mp will equal k. Two numbers that add Product of mp to 5 4 1, 4 6, 3 Therefore, k could have values of 4 or 6. Page REASONING For any factorable trinomial, x + bx + c, will the absolute value of b sometimes, always, or never be less than the absolute value of c? Explain.

39 Two numbers that add Product of mp to 5 4 1, Solving x^ bx + c = 0 6,+ 3 Therefore, k could have values of 4 or REASONING For any factorable trinomial, x + bx + c, will the absolute value of b sometimes, always, or never be less than the absolute value of c? Explain. The absolute value of b will sometimes be less than the absolute value of c. Sample answer: The trinomial x + 10x + 9 = (x + 1)(x + 9) and 10 > 9. The trinomial x + 7x + 10 = (x + )(x + 5) and 7 < OPEN ENDED Give an example of a trinomial that can be factored using the factoring techniques presented in this lesson. Then factor the trinomial. Students answers will vary. Sample answer: x + 19x 0 In this trinomial, b = 19 and c = 0, so m + p is positive and mp is negative. Therefore, m and p must have different signs. List the factors of 0, and look for the pair of factors with a sum of 19. Factors of , 0 8, , 5 5, , 8 0, 1 19 The correct factors are 1 and CHALLENGE Factor (4y 5) + 3(4y 5) 70. The trinomial is written in x + bx + c = 0 form.we can substitute x for 4y 5 to get x + 3x 70 = 0. In this trinomial, b = 3 and c = 70, so m + p is positive and mp is negative. Therefore, m and p must have different signs. List the factors of 70, and look for the pair of factors with a sum of 3. Factors of , 70 33, , , 10 10, , , 33 70, 1 69 The correct factors are 7 and 10. Page WRITING IN MATH Explain how to factor trinomials of the form x + bx + c and how to determine the signs of

40 0, 1 The correct factors are 1 and Solving x^ + bx + c = CHALLENGE Factor (4y 5) + 3(4y 5) 70. The trinomial is written in x + bx + c = 0 form.we can substitute x for 4y 5 to get x + 3x 70 = 0. In this trinomial, b = 3 and c = 70, so m + p is positive and mp is negative. Therefore, m and p must have different signs. List the factors of 70, and look for the pair of factors with a sum of 3. Factors of , 70 33, , , 10 10, , , 33 70, 1 69 The correct factors are 7 and WRITING IN MATH Explain how to factor trinomials of the form x + bx + c and how to determine the signs of the factors of c. Find factors m and p such that m + p = b and mp = c. If b and c are positive, then m and p are positive. For example: If b is negative and c is positive, then m and p are negative. For example: When c is negative, m and p have different signs and the factor with the greatest absolute value has the same sign as b. For example: and 50. Which inequality is shown in the graph? A Manual - Powered by Cognero esolutions B Page 40

41 When c is negative, m and p have different signs and the factor with the greatest absolute value has the same sign as b. 8-6 Solving x^ + bx + c = 0 For example: and 50. Which inequality is shown in the graph? A B C D The line is dashed which means that choices A and D can be eliminated. So check (0, 0) in the inequalities given in choices B and C to determine which is the correct choice. True False So, the correct choice is C. 51. SHORT RESPONSE Olivia must earn more than $54 from selling candy bars in order to go on a trip with the National Honor Society. If each candy bar is sold for $1.5, what is the fewest candy bars she must sell? Let x represent the number of candy bars Olivia must sell. So, Olivia must sell at least 04 candy bars to go on the trip. 5. GEOMETRY Which expression represents the length of the rectangle? Page 41

42 8-6 Solving x^ + bx + c = 0 So, Olivia must sell at least 04 candy bars to go on the trip. 5. GEOMETRY Which expression represents the length of the rectangle? F x +5 G x +6 H x 6 J x 5 The area of the rectangle is x 3x 18. In this trinomial, b = 3and c = 18, so m + p is positive and mp is positive. Therefore, m and p must both be positive. List the factors of 18 and identify the factors with a sum of 3. Factors of 18 1, , 18 17, 9 6 6, 9 3 3, 6 3 3, 6 The area of the rectangle x 3x 18 factors to (x + 3)(x 6). Because the width is x + 3, the length must be x 6. So, the correct choice is H. 53. The difference of 1 and a number n is 6. Which equation shows the relationship? A 1 n = 6 B 1 + n = 6 C 1n = 6 D 6n = 1 The phrase is 6 means = 6, so choice D can be eliminated. The phrase difference of means subtraction, so A is the correct choice. Factor each polynomial a + 40a The greatest common factor is 10a + 40a = 10a(a + 4) or 10a x + 44x y Page 4

43 The greatest common factor is 8-6 Solving x^= +10a(a bx ++c 4) =0 10a + 40a or 10a x + 44x y The greatest common factor is 11x + 44x y = 11x(1 + 4xy) 3 or 11x. 56. m p 16mp + 8mp The greatest common factor is or mp. 3 m p 16mp + 8mp = mp(m p 8p + 4) 57. ax + 6xc + ba + 3bc Factor by grouping ac ad + 4bc bd Factor by grouping. 59. x xy xy + y Factor by grouping. esolutions Manual - Powered by Cognero a polynomial that represents 60. Write the area of the shaded region in the figure. Page 43

44 8-6 Solving x^ + bx + c = Write a polynomial that represents the area of the shaded region in the figure. Use elimination to solve each system of equations. 61. Because x and x have opposite coefficients, add the equations. Now, substitute 13 for y in either equation to find the value of x. Check the solution in each equation. Therefore, the solution is (4, 13). Page 44

45 8-6 Solving x^ + bx + c = 0 Use elimination to solve each system of equations. 61. Because x and x have opposite coefficients, add the equations. Now, substitute 13 for y in either equation to find the value of x. Check the solution in each equation. Therefore, the solution is (4, 13). 6. Because 5a and 5a have opposite coefficients, add the equations. Page 45

46 Therefore, the solution is (4, 13). 8-6 Solving x^ + bx + c = 0 6. Because 5a and 5a have opposite coefficients, add the equations. Now, substitute 3 for b in either equation to find the value of a. Check the solution in each equation. Therefore, the solution is (, 3). 63. Because d and d have opposite coefficients, add the equations. Now, substitute 3 for c in either equation to find the value of d. Page 46

47 Therefore, the solution is (, 3). 8-6 Solving x^ + bx + c = Because d and d have opposite coefficients, add the equations. Now, substitute 3 for c in either equation to find the value of d. Check the solution in each equation. Therefore, the solution is (3, 6). 64. Because y and y have opposite coefficients, add the equations. Now, substitute for x in either equation to find the value of y. Page 47

48 8-6 Solving x^ + bx + c = 0 Therefore, the solution is (3, 6). 64. Because y and y have opposite coefficients, add the equations. Now, substitute for x in either equation to find the value of y. Check the solution in each equation. Therefore, the solution is (, 1). 65. LANDSCAPING Kendrick is planning a circular flower garden with a low fence around the border. He has 38 feet of fence. What is the radius of the largest garden he can make? (Hint: C = πr) If he has 38 feet of fence, use the formula, C = πr, with C = 38. So, the largest radius he can make is about 6 feet. Factor each polynomial. Page 48

49 8-6 Solving x^ + bx + c = 0 Therefore, the solution is (, 1). 65. LANDSCAPING Kendrick is planning a circular flower garden with a low fence around the border. He has 38 feet of fence. What is the radius of the largest garden he can make? (Hint: C = πr) If he has 38 feet of fence, use the formula, C = πr, with C = 38. So, the largest radius he can make is about 6 feet. Factor each polynomial mx 4m + 3rx r 67. 3ax 6bx + 8b 4a 68. d g + fg + 4d h + 4fh Page 49