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)
3𝑥 3 −17𝑥 2 +15𝑥−25
𝑥−5
a) I need to change the sign of the (-5) to positive for my synthetic division
3
-17
15
-2
5
3
Answer:
𝟑𝒙𝟑 −𝟏𝟕𝒙𝟐 +𝟏𝟓𝒙−𝟐𝟓
=
𝒙−𝟓
15
-10
5
-25
25
0
3x2 – 2x + 5 remainder 0
1b) since the remainder is 0 I need to factor the numerator. Synthetic division tells me this is true
3𝑥 3 − 17𝑥 2 + 15𝑥 − 25 = (x-5)(3x2 – 2x + 5)
I just need to factor a bit more
Answer: 𝟑𝒙𝟑 − 𝟏𝟕𝒙𝟐 + 𝟏𝟓𝒙 − 𝟐𝟓 = (x-5)(3x-5)(x+1)
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
4
-2
4
Answer:
𝟒𝒙𝟑 +𝟖𝒙𝟐 −𝟗𝒙−𝟏𝟖
=
𝒙+𝟐
8
-8
0
-9
0
-9
-18
18
0
4x2 – 9 remainder 0
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)(4x2 – 9) I just need to factor more
Answer: 𝟒𝒙𝟑 + 𝟖𝒙𝟐 − 𝟗𝒙 − 𝟏𝟖= (x+2)(2x+3)(2x-3)
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 3x3 – 16x2 + 0x – 72.
3
-16
18
2
6
3
Answer:
𝟑𝒙𝟑 −𝟏𝟔𝒙𝟐 −𝟕𝟐
=
𝒙−𝟔
0
12
12
-72
72
0
3x2 + 2x + 12 remainder 0
5b) since the remainder is 0 I need to factor the numerator. Synthetic division tells me this is true
3x3 – 16x2 – 72 = (x-6)(3x2 + 2x + 12)
The 3x2 + 2x + 12 is prime, so I can’t factor more.
Answer: 3x3 – 16x2 – 72 = (x-6)(3x2 + 2x + 12)
7)
(5𝑥 3 + 6𝑥 + 8) ÷ (𝑥 + 2) this is the same as
5𝑥 3 +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 0x2 term in
the numerator 5x3 + 0x2 + 6x + 8
5
-2
5
0
-10
-10
6
20
26
Answer: (𝟓𝒙𝟑 + 𝟔𝒙 + 𝟖) ÷ (𝒙 + 𝟐) = 5x2 – 10x + 26 remainder 44
7b) skip this part since the remainder is not 0.
8
-52
44
9) (𝑥 3 − 27) ÷ (𝑥 − 3)
this is the same as
𝑥 3 −27
𝑥−3
a) I need to change the sign of the (-3) to positive for my synthetic division
I need to insert a 0x2 and a 0x.
(x3 + 0x2 + 0x -27)÷(x-3)
1
3
1
0
3
3
0
9
9
-27
27
0
Answer: (𝒙𝟑 − 𝟐𝟕) ÷ (𝒙 − 𝟑) = x2 + 3x + 9 remainder of 0
9b) since the remainder is 0 I need to factor the numerator. Synthetic division tells me this is true
x3 – 27 = (x-3)(x2 + 3x + 9)
The x2 + 3x + 9 is prime
Answer: x3 – 27 = (x-3)(x2 + 3x + 9)
#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) = x3 + 2x2 – 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.
1
2
-1
1
1
The result of my synthetic division gives me
-1
𝑥 3 +2𝑥 2 −5𝑥−6
𝑥+1
= 𝑥 2 + 𝑥 − 6 𝑟𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟 0
so now I can factor f(x)
f(x) = x3 + 2x2 – 5x – 6 = (x+1)(x2+x-6)
11b) Answer: f(x) = (x+1)(x-2)(x+3) )
-5
-1
-6
-6
6
0
13) f(x) = 2x3 – 13x2 + 24x – 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)
2
3
2
-13
6
-7
The result of the synthetic division tells me
24
-21
3
2𝑥 3 −13𝑥 2 +24𝑥−9
𝑥−2
Now I can factor f(x) = 2x3 – 13x2 + 24x – 9
Answer: f(x) = (x-3)(x-3)(2x-1) or (x-3)2(2x-1)
-9
9
0
= 2𝑥 2 − 7𝑥 + 3 𝑟𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟 0
= (x-3)(2x2 – 7x + 3)
15) f(x) = x4 + x3 – 3x2 – 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 results of my synthetic division should help me get more factors of f(x)
1
1
-2
-1
-2
1
1
1
1
-3
2
-1
-1
1
0
-1
2
1
-1
0
-1
2
-2
0
1
-1
0
The result of my double synthetic division tells me
f(x) = x4 + x3 – 3x2 – x + 2= (x+2)(x-1)(x2 – 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)
17) f(x) = 2x4 + 17x3 + 35x2 – 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)
2
17
-10
7
-5
2
2
1
2
35
-35
0
7
2
9
The result of the double synthetic division tells me
f(x) = 2x4 + 17x3 + 35x2 – 9x – 45 = (x+5)(x-1)(2x2 + 9x + 9)
Answer: f(x) = (x+5)(x-1)(2x+3)(x+3)
-9
0
-9
0
9
9
-45
45
0
-9
9
0
19) f(x) = x4 + 7x2 – 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)
1
0
1
1
1
1
1
-1
1
7
1
8
1
-1
0
0
8
8
8
0
8
The result of the double synthetic division tells me
Answer: f(x) = x4 + 7x2 – 8 = (x+1)(x-1)(x2 + 8) this is the answer as the x2 + 8 is prime
-8
8
0
8
-8
0
#21-30: Solve
21) x4 – x3 + 2x2 – 4x – 8 = 0
here is a graph of f(x) = x4 – x3 + 2x2 – 4x – 8
First I will factor f(x) = x4 – x3 + 2x2 – 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 results of my synthetic division should help me get more factors of f(x)
1
-1
2
1
2
1
1
2
2
4
1
-1
0
-1
1
-4
8
4
4
0
4
the result of the double synthetic division tells me
f(x) =x4 – x3 + 2x2 – 4x – 8 = (x+1)(x-2)(x2 + 4) I will use this to solve the problem.
x4 – x3 + 2x2 – 4x – 8 = 0 (factor)
(x+1)(x-2)(x2 + 4) = 0 set each factor equal to 0
x+1 = 0
x- 2 = 0
x2 + 4 = 0
x = -1
x=2
x2 = -4 ( 𝑥 = ±√−4)
Answer: x = -1, 2, ±𝟐𝒊
-8
8
0
4
-4
0
23) 4x3 +8x2 -9x-18 = 0
here is a graph of f(x) = 4x3 +8x2 -9x-18
First I will factor f(x) = 4x3 +8x2 -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 results of my synthetic division should help me get more factors of f(x)
4
-2
4
8
-8
0
-9
0
-9
The result of my synthetic division tells me
f(x) = 4x3 +8x2 -9x-18 = (x+2)(4x2 – 9)
I will use this to solve the problem
4x3 +8x2 -9x-18 = 0
(x+2)(4x2 -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, ±
𝟑
𝟐
-18
18
0
25) 2x4 +7x3 -4x2 -27x-18 = 0
Here is a graph of f(x) = 2x4 +7x3 -4x2 -27x-18
First I will factor f(x) = 2x4 +7x3 -4x2 -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)
the results of my synthetic division should help me get more factors of f(x)
2
7
-6
1
-3
2
2
-4
-3
-7
1
4
5
2
2
-27
21
-6
-7
10
3
My synthetic division tells me f(x) = 2x4 +7x3 -4x2 -27x-18 = (x+3)(x-2)(2x2 + 5x + 3)
I will use this to solve the problem
2x4 +7x3 -4x2 -27x-18 = 0
(x+3)(x-2)(2x2 + 5x + 3 ) = 0
(x+3)(x-2)(2x+3)(x+1) = 0
x+3 = 0
x-2 = 0
x = -3
x=2
Answer: x = -3, 2, -1, -3/2
2x + 3 = 0
x = -3/2
x+1=0
x = -1
-18
18
0
-6
6
0
27) x4 +4x3 + 2x2 – x + 6 = 0
Here is a graph of f(x) = x4 +4x3 + 2x2 – x + 6
First I will factor f(x) = x4 +4x3 + 2x2 – 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)
the results of my synthetic division should help me get more factors of f(x)
1
4
-3
1
-3
1
2
-3
-1
-1
3
2
1
1
-1
-2
2
1
-1
1
My synthetic division tells me: f(x) = x4 +4x3 + 2x2 – x + 6 = (x+3)(x+2)(x2-x+1)
I will use this to solve the problem
-2
x4 +4x3 + 2x2 – x + 6 = 0
(x2 – x + 1)(x+3)(x+2) = 0
x2- x + 1 = 0 (I need quad formula)
a = 1 b = -1 c = 1
𝑥=
1±√(−1)2 −4(1)(1)
2∗1
Answer: x = -3, -2,
=
1±√−3
2
𝟏±𝒊√𝟑
𝟐
x+3 = 0
x = -3
x+2=0
x = -2
6
-6
0
2
-2
0
29) x4 + 7x2 – 8 = 0
Here is a graph of f(x) = x4 + 7x2 – 8
First I will factor f(x) = x4 + 7x2 – 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)
the results of my synthetic division should help me get more factors of f(x)
1
0
1
1
1
1
1
-1
1
7
1
8
1
-1
0
0
8
8
8
0
8
my synthetic division tells me f(x) =x4 + 7x2 – 8 = (x+1)(x-1)(x2 + 8)
I will use this to solve the problem
x4 + 7x2 – 8 = 0
(x+1)(x-1)(x2 + 8) = 0
x+1 = 0
x–1=0
x2 + 8 = 0
x = -1
x=1
x2 = -8
𝑥 = ±√−8 = ±2𝑖√2
answer: x = -1, 1, ±𝟐𝒊√𝟐
-8
8
0
8
-8
0