Synthetic Division Worksheet

Synthetic Division Worksheet
  • Page 1
 1.  
Divide using synthetic division. Express your answer as:
(dividenddivisor) = quotient + (remainderdivisor).
(x3 - 9x2 - 5x + 4) ÷ (x - 3)

a.
(x2 - 6x - 23) - 73x - 3
b.
(x2 - 12x + 23) - 73x - 3
c.
(x2 - 6x - 23) + 65x - 3
d.
(x2 - 6x - 23) - 65x - 3


Solution:

x3 - 9x2 - 5x + 4

x - 3 is the divisor.

Use the synthetic division.

3|     1     - 9       - 5       4
                  3     - 18    - 69
     -----------------------------
       1       - 6     - 23    - 65
     -----------------------------

x3-9x2-5x+4x - 3 = (x2 - 6x - 23) - 65x - 3


Correct answer : (4)
 2.  
Divide using synthetic division. Express your answer as:
(dividenddivisor) = quotient + (remainderdivisor).
(4x3 - 15x + 17) ÷ (x + 8)

a.
(4x2 - 32x + 241) + 1911x + 8
b.
(4x2 - 32x + 241) - 1911x + 8
c.
(4x2 + 32x + 241) + 1945x + 8
d.
none of these


Solution:

4x3 - 15x + 17

x + 8 is the divisor.

Use the synthetic division.

- 8|     4     0       - 15      17
               - 32      256      - 1928
     -------------------------------
         4    - 32       241     - 1911
     -------------------------------
[The x2 term is missing, so insert 0 for 0x2.]

4x3 - 15x + 17x + 8 = (4x2 - 32x + 241) - 1911x + 8


Correct answer : (2)
 3.  
Divide using synthetic division. Express your answer as:
dividenddivisor = quotient + remainderdivisor.
(x4 - 14x2 + 15x - 16) ÷ (x + 3)

a.
(x3 + 3x2 - 5x + 0) - 60x - 3
b.
(x3 - 3x2 + 5x - 30) - 106x + 3
c.
(x3 - 3x2 - 5x + 30) - 106x + 3
d.
none of these


Solution:

x4 - 14x2 + 15x - 16

x + 3 is the divisor.

Use the synthetic division.

- 3|     1    0     - 14      15    - 16
              - 3      9       15    - 90
     -------------------------------
         1   - 3      - 5    30    - 106
     -------------------------------
[The x3 term is missing, so insert 0 for 0x3.]

x4 - 14x2 + 15x - 16x + 3 = (x3 - 3x2 - 5x + 30) - 106x + 3


Correct answer : (3)
 4.  
Divide using synthetic division. Express your answer as:
(dividenddivisor) = quotient + (remainderdivisor).
(x4 - 4x3 + 4x2 - 2x + 3) ÷ (x - 2)

a.
(x3 - 2x2 - 2) + 1x - 2
b.
(x3 - 2x2 - 2) - 1x - 2
c.
(x3 - 6x2 + 4x - 2) + 7x - 2
d.
(x3 + 2x2 + 2) - 1x - 2


Solution:

x4 - 4x3 + 4x2 - 2x + 3

x - 2 is the divisor.

Use the synthetic division.

2|     1   - 4     4     - 2     3
              2    - 4      0    - 4
     --------------------------
        1   - 2     0    - 2     - 1
     --------------------------

x4 - 4x3 + 4x2 - 2x + 3x - 2 = (x3 - 2x2 - 2) - 1x - 2


Correct answer : (2)
 5.  
Divide using synthetic division. Express your answer as:
(dividenddivisor) = quotient + (remainderdivisor).
(x4 - x3 - 5x2 + 4x - 6) ÷ (x + 8)
a.
(x3 - 9x2 + 67x + 532) + 4250x + 8
b.
(x3 + 5x2 + 77x + 540) + 4256x + 8
c.
(x3 - 9x2 + 67x - 532) + 4250x + 8
d.
none of these


Solution:

x4 - x3 - 5x2 + 4x - 6

x + 8 is the divisor.

Use the synthetic division.

- 8|     1    -1     - 5        4       - 6
               - 8     72   - 536    4256
       ----------------------------------
         1    - 9     67   - 532    4250
       ---------------------------------

x4-x3-5x2+4x - 6x + 8 = (x3 - 9x2 + 67x - 532) + 4250x + 8


Correct answer : (3)
 6.  
Divide using synthetic division. Express your answer as:
(dividenddivisor) = quotient + (remainderdivisor).
(y4 - 32) ÷ (y + 2)
a.
(y3 - 2y2 + 4y - 8) - 16y + 2
b.
(y3 + 2y2 + 4y + 8) - 16y + 2
c.
(y3 - 2y2 + 4y - 8) + 16y + 2
d.
none of these


Solution:

y4 - 32

y + 2 is the divisor.

Use the synthetic division.

- 2|     1    0     0       0    - 32
              - 2     4   - 8     16
      ----------------------------
         1   - 2     4   - 8    - 16
      ----------------------------
[The x3, x2 and x terms are missing, so insert 0 for 0x3, 0x2 and 0x.]

y4 - 32y + 2 = (y3 - 2y2 + 4y - 8) - 16y + 2.


Correct answer : (1)
 7.  
Find the value of c that will make the given divisor a factor of the dividend.
x3 - 4x2 - x + cx - 3
a.
9
b.
15
c.
12


Solution:

x3 - 4x2 - x + c

x - 3 is the divisor.

A divisor of a polynomial is called a factor, when it divides the polynomial without leaving a remainder

Use the synthetic division

3|     1     - 4     -1     c
                3     - 3     - 12
     -----------------------
        1     - 1    - 4      0
     -----------------------

c - 12 = 0
[Equate remainder to zero.]

c = 12
[Add 12 to both sides of the equation.]


Correct answer : (4)
 8.  
Find the value of d that will make the given divisor a factor of the dividend.
4x3+4x2 - 76x - dx - 4
a.
16
b.
- 80
c.
160


Solution:

4x3 + 4x2 - 76x - d

x - 4 is the divisor.

A divisor of a polynomial is called a factor, when it divides the polynomial without leaving a remainder.

Use the synthetic division.

4|     4     4   - 76   - d
               16   80     16
   -----------------------
       4    20     4   0
    -----------------------

- d + 16 = 0
[Equate remainder to zero.]

- d = - 16
[Subtracting 16 from the two sides of the equation.]

d = 16
[Multiply throughout by - 1.]


Correct answer : (2)
 9.  
Divide using synthetic division. Express your answer as:
(dividenddivisor) = quotient + (remainderdivisor).
(x3 + 125) ÷ (x - 5)

a.
(x2 + 5x + 25) - 250x - 5
b.
(x2 + 5x + 25) + 250x - 5
c.
(x2 - 5x + 25) - 250x - 5
d.
(x2 - 5x + 25)


Solution:

x3 + 125

x - 5 is the divisor.

Use the synthetic division.

5|     1     0      0      125
               5      25      125
       ------------------------
        1     5      25    250
       -----------------------
[The x2, x terms are missing, so insert 0 for 0x2, 0x.]

x3 + 125x - 5 = (x2 + 5x + 25) + 250x - 5


Correct answer : (2)
 10.  
Divide using synthetic division. Express your answer as:
dividenddivisor = quotient + remainderdivisor.
(9x3 + 9x2 + 11x - 4) ÷ (3x + 1)
a.
9x2 +6x + 9 + 73x +  1
b.
3x2 +2x + 3 + 73x +  1
c.
9x2 +6x + 9 - 73x +  1
d.
3x2 +2x + 3 - 73x +  1


Solution:

9x3 + 9x2 + 11x - 43x + 1 = 1 / 3(9x3 + 9x2 +11x - 4 x +13)

9x3 + 9x2 + 11x - 4

x + 1 / 3 is the divisor.

Use the synthetic division.

- 1/3|     9     9     11    -4
                 -3      -2     -3
       -------------------------
             9     6      9     -7
       -------------------------

Then 9x3 + 9x2 + 11x - 43x + 1 = 1 / 3(9x2 + 6x + 9 -  7 x +13)

= 3x2 + 2x + 3 - 7 3x + 1

So, 9x3 + 9x2 + 11x - 43x + 1 = 3x2 + 2x + 3 - 7 3x + 1


Correct answer : (4)

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