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.) 1) a) I need to change the sign of the (-5) to positive for my synthetic division Answer: = 3x 2 2x + 5 remainder 0 1b) since the remainder is 0 I need to factor the numerator. Synthetic division tells me this is true = (x-5)(3x 2 2x + 5) I just need to factor a bit more Answer: = (x-5)(3x-5)(x+1) 3) 3a) I need to change the sign of the 2 to negative for my synthetic division Answer: = 4x 2 9 remainder 0 3b) since the remainder is 0 I need to factor the numerator. Synthetic division tells me this is true Answer: = (x+2)(4x 2 9) I just need to factor more = (x+2)(2x+3)(2x-3)
2 5) 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: = 3x 2 + 2x + 12 remainder 0 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) ( ) ( ) this is the same as 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 2 10x + 26 remainder 44 7b) skip this part since the remainder is not 0.
3 9) ( ) ( ) this is the same as 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 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) 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 so now I can factor f(x) f(x) = x 3 + 2x 2 5x 6 = (x+1)(x 2 +x-6) 11b) Answer: f(x) = (x+1)(x-2)(x+3) )
5 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 Now I can factor f(x) = 2x 3 13x x 9 = (x-3)(2x 2 7x + 3) Answer: f(x) = (x-3)(x-3)(2x-1) or (x-3) 2 (2x-1)
6 15) f(x) = x 4 + x 3 3x 2 x + 2 here is a graph of f(x) 15a) I will use (-2) and 1 and perform double synthetic division 15b) since x = (-2) is a zero I know (x+2) is a factor of f(x) since x = 1 is a zero I know (x-1) is a factor of f(x) The result of my double synthetic division tells me f(x) = x 4 + x 3 3x 2 x + 2 = (x+2)(x-1)(x 2 1) (I took the second synthetic division results) Answer: f(x) = (x+2)(x-1)(x-1)(x+1) or (x-1) 2 (x+2)(x+1)
7 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 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)
8 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 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
9 #21-30: Solve 21) x 4 x 3 + 2x 2 4x 8 = 0 here is a graph of f(x) = x 4 x 3 + 2x 2 4x 8 First I will factor f(x) = x 4 x 3 + 2x 2 4x 8, the factoring will help me solve the problem. I will use (-1) and 2 and perform double synthetic division since x = (-1) is a zero I know (x+1) is a factor of f(x) since x = 2 is a zero I know (x-2) is a factor of f(x) the result of the double synthetic division tells me f(x) =x 4 x 3 + 2x 2 4x 8 = (x+1)(x-2)(x 2 + 4) I will use this to solve the problem. x 4 x 3 + 2x 2 4x 8 = 0 (factor) (x+1)(x-2)(x 2 + 4) = 0 set each factor equal to 0 x+1 = 0 x- 2 = 0 x = 0 x = -1 x = 2 x 2 = -4 ( ) Answer: x = -1, 2,
10 23) 4x 3 +8x 2-9x-18 = 0 here is a graph of f(x) = 4x 3 +8x 2-9x-18 First I will factor f(x) = 4x 3 +8x 2-9x-18, the factoring will help me solve the problem. I will use (-2) and perform synthetic division since x = (-2) is a zero I know (x+2) is a factor of f(x) The result of my synthetic division tells me f(x) = 4x 3 +8x 2-9x-18 = (x+2)(4x 2 9) I will use this to solve the problem 4x 3 +8x 2-9x-18 = 0 (x+2)(4x 2-9)=0 (x+2)(2x+3)(2x-3) = 0 x+2 = 0 2x + 3 = 0 2x 3 = 0 x = -2 2x = -3 2x = 3 Answer x = -2,
11 25) 2x 4 +7x 3-4x 2-27x-18 = 0 Here is a graph of f(x) = 2x 4 +7x 3-4x 2-27x-18 First I will factor f(x) = 2x 4 +7x 3-4x 2-27x-18, the factoring will help me solve the problem. I will use (-3) and (2) and perform double synthetic division since x = (-3) is a zero I know (x+3) is a factor of f(x) since x = 2 is a zero I know (x-2) is a factor of f(x) My synthetic division tells me f(x) = 2x 4 +7x 3-4x 2-27x-18 = (x+3)(x-2)(2x 2 + 5x + 3) I will use this to solve the problem 2x 4 +7x 3-4x 2-27x-18 = 0 (x+3)(x-2)(2x 2 + 5x + 3 ) = 0 (x+3)(x-2)(2x+3)(x+1) = 0 x+3 = 0 x-2 = 0 2x + 3 = 0 x + 1 = 0 x = -3 x = 2 x = -3/2 x = -1 Answer: x = -3, 2, -1, -3/2
12 27) x 4 +4x 3 + 2x 2 x + 6 = 0 Here is a graph of f(x) = x 4 +4x 3 + 2x 2 x + 6 First I will factor f(x) = x 4 +4x 3 + 2x 2 x + 6, the factoring will help me solve the problem. I will use (-3) and (-2) and perform double synthetic division since x = (-3) is a zero I know (x+3) is a factor of f(x) since x = -2 is a zero I know (x+2) is a factor of f(x) My synthetic division tells me: f(x) = x 4 +4x 3 + 2x 2 x + 6 = (x+3)(x+2)(x 2 -x+1) I will use this to solve the problem x 4 +4x 3 + 2x 2 x + 6 = 0 (x 2 x + 1)(x+3)(x+2) = 0 x 2 - x + 1 = 0 (I need quad formula) x+3 = 0 x + 2 = 0 a = 1 b = -1 c = 1 x = -3 x = -2 ( ) ( )( ) Answer: x = -3, -2,
13 29) x 4 + 7x 2 8 = 0 Here is a graph of f(x) = x 4 + 7x 2 8 First I will factor f(x) = x 4 + 7x 2 8, the factoring will help me solve the problem. 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) my synthetic division tells me f(x) =x 4 + 7x 2 8 = (x+1)(x-1)(x 2 + 8) I will use this to solve the problem x 4 + 7x 2 8 = 0 (x+1)(x-1)(x 2 + 8) = 0 x+1 = 0 x 1 = 0 x = 0 x = -1 x = 1 x 2 = -8 answer: x = -1, 1,
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