b) since the remainder is 0 I need to factor the numerator. Synthetic division tells me this is true

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1 Section 5.2 solutions #1-10: a) Perform the division using synthetic division. b) if the remainder is 0 use the result to completely factor the dividend (this is the numerator or the polynomial to the left of the division sign.) 3x 3 17x 2 +15x 25 1) x 5 a) I need to change the sign of the (-5) to positive for my synthetic division Answer: 3x3 17x 2 +15x 25 = 3x 2 2x + 5 remainder 0 x 5 1b) since the remainder is 0 I need to factor the numerator. Synthetic division tells me this is true 3x 3 17x x 25 = (x-5)(3x 2 2x + 5) I the 3x 2 2x + 5 is prime, so this is completely factored. Answer: 3x 3 17x x 25 = (x-5)(3x 2 2x + 5) 3) 4x 3 +8x 2 9x 18 x+2 3a) I need to change the sign of the 2 to negative for my synthetic division Answer: 4x3 +8x 2 9x 18 = 4x 2 9 remainder 0 x+2 3b) since the remainder is 0 I need to factor the numerator. Synthetic division tells me this is true 4x 3 + 8x 2 9x 18= (x+2)(4x 2 9) I just need to factor more Answer: 4x 3 + 8x 2 9x 18= (x+2)(2x+3)(2x-3)

2 5) 3x 3 16x 2 72 x 6 5a) I need to change the sign of the (-6) to positive for my synthetic division. I need to think of the numerator having the form 3x 3 16x 2 + 0x Answer: 3x3 16x 2 72 = 3x 2 + 2x + 12 remainder 0 x 6 5b) since the remainder is 0 I need to factor the numerator. Synthetic division tells me this is true 3x 3 16x 2 72 = (x-6)(3x 2 + 2x + 12) The 3x 2 + 2x + 12 is prime, so I can t factor more. Answer: 3x 3 16x 2 72 = (x-6)(3x 2 + 2x + 12) 7) (5x 3 + 6x + 8) (x + 2) this is the same as 5x3 +6x+8 x+2 a) I need to change the sign of the 2 to negative for my synthetic division. I need to insert a 0x 2 term in the numerator 5x 3 + 0x 2 + 6x Answer: (5x 3 + 6x + 8) (x + 2) = 5x 2 10x + 26 remainder -44 7b) skip this part since the remainder is not 0.

3 9) (x 3 27) (x 3) this is the same as x3 27 x 3 a) I need to change the sign of the (-3) to positive for my synthetic division I need to insert a 0x 2 and a 0x. (x 3 + 0x 2 + 0x -27) (x-3) Answer: (x 3 27) (x 3) = x 2 + 3x + 9 remainder of 0 9b) since the remainder is 0 I need to factor the numerator. Synthetic division tells me this is true x 3 27 = (x-3)(x 2 + 3x + 9) The x 2 + 3x + 9 is prime Answer: x 3 27 = (x-3)(x 2 + 3x + 9)

4 #11 20: a) use your graphing calculator, or the rational root theorem to find a zero of the polynomial i) you need to find one zero for a third degree polynomial ii) you need to find two zeros for a fourth degree polynomial b) use synthetic division to completely factor the polynomial (use double synthetic division for fourth degree polynomials) c) Use your answer to part b to solve f(x) = 0 11) f(x) = x 3 + 2x 2 5x 6 here is a graph of f(x) 11a) Answer: I will use the numbers (-1) for my synthetic division, I could have also used 2. 11b) since x = -1 is a zero, I know (x+1) is a factor of f(x) the synthetic division will get me the remaining factors The result of my synthetic division gives me x 3 +2x 2 5x 6 x+1 = x 2 + x 6 remainder 0 so now I can factor f(x) f(x) = x 3 + 2x 2 5x 6 = (x+1)(x 2 +x-6) Answer #11b: f(x) = (x+1)(x-2)(x+3)

5 11c) Solve f(x) = 0 Just take answer to part b and set it equal to 0 and solve for x. (x+1)(x-2)(x+3) = 0 x + 1 = 0 x 2 = 0 x + 3 = 0 x = -1 x = 2 x = -3 Answer #11c: x = -1, 2, -3

6 13) f(x) = 2x 3 13x x 9 here is a graph of f(x) 13a) I will use 3 is the value for my synthetic division 13b) since x = 3 is a zero, I know (x-3) is a factor of f(x) the results of my synthetic division should help me get additional factors of f(x) The result of the synthetic division tells me 2x3 13x 2 +24x 9 x 2 Now I can factor f(x) = 2x 3 13x x 9 = (x-3)(2x 2 7x + 3) Answer 13b: f(x) = (x-3)(x-3)(2x-1) or (x-3) 2 (2x-1) = 2x 2 7x + 3 remainder 0

7 13c) Solve f(x) = 0 Just take answer to part b and set it equal to 0 and solve for x. (x-3)(x-3)(2x-1) = 0 x - 3 = 0 x 3 = 0 2x 1 = 0 x = 3 x = 3 2x = 1 x = ½ Answer 13c: x = 3, ½

8 15) f(x) =x 3 5x 2 4x - 20 here is a graph of f(x) 15a) I will use (-2), but I could have used any of the three x-intercepts. 15b) since x = (-2) is a zero I know (x+2) is a factor of f(x) the results of my synthetic division should help me get more factors of f(x) The result of my synthetic division tells me f(x) = x 3 5x 2 4x - 20 = (x+2)(x 2-7x+10) (I just need to factor the second parenthesis to get my answer) Answer #15b: f(x) = (x+2)(x-2)(x-5) 15c) Solve f(x) = 0 Just take answer to part b and set it equal to 0 and solve for x. (x+2)(x-2)(x-5) = 0 x + 2 = 0 x 2 = 0 x 5 = 0 x = -2 x = 2 x = 5 Answer #15c: x = -2,2,5

9 17) f(x) = 2x x x 2 9x 45 here is a graph of f(x) 17a) I will use (-5) and (1) and perform double synthetic division 17b) since x = (-5) is a zero I know (x+5) is a factor of f(x) since x = 1 is a zero I know (x-1) is a factor of f(x) the results of my synthetic division should help me get more factors of f(x) The result of the double synthetic division tells me f(x) = 2x x x 2 9x 45 = (x+5)(x-1)(2x 2 + 9x + 9) Answer: f(x) = (x+5)(x-1)(2x+3)(x+3)

10 17c) Solve f(x) = 0 Just take answer to part b and set it equal to 0 and solve for x. (x+5)(x-1)(2x+3)(x+3) = 0 x + 5 = 0 x 1 = 0 2x + 3 = 0 x + 3 = 0 x = -5 x = 1 2x = -3 x = -3 x = -3/2 Answer #17c: x = -5,1, 3 2,-3

11 19) f(x) = x 4 + 7x 2 8 here is a graph of f(x) 19a) I will use (-1) and (1) and perform double synthetic division since x = (-1) is a zero I know (x+1) is a factor of f(x) since x = 1 is a zero I know (x-1) is a factor of f(x) the results of my synthetic division should help me get more factors of f(x) The result of the double synthetic division tells me Answer: f(x) = x 4 + 7x 2 8 = (x+1)(x-1)(x 2 + 8) this is the answer as the x is prime

12 19c) Solve f(x) = 0 Just take answer to part b and set it equal to 0 and solve for x. (x+1)(x-1)(x 2 + 8) = 0 x+1 = 0 x 1 = 0 x = 0 x = -1 x = 1 x 2 = -8 x = ± 8 = ±2i 2 Answer #19c: x = 1, 1, ±2 2i

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