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Remainder Theorem Worksheet

Remainder Theorem Worksheet
  • Page 1
 1.  
Check if (x + 5) is a factor of the given polynomial 5x3 + x2 - 7x - 5.
a.
(x + 5) is a factor
b.
(x + 5) is not a factor
c.
Can not be determined


Solution:

P(x) = 5x3 + x2 - 7x - 5

P(-5) = 5(-5)3 + (-5)2 - 7(-5) - 5
[Substitute the values.]

= -625 + 25 + 35 - 5

= -570
[Simplify.]

Since the remainder is not zero, (x + 5) is not a factor of 5x3 + x2 - 7x - 5.


Correct answer : (2)
 2.  
Evaluate f(x) at x = 1, by synthetic division.
f(x) = x4 - 3x3 + 5x2 + 4x - 5
a.
1
b.
5
c.
2
d.
3


Solution:

f (x) = x4 - 3x3 + 5x2 + 4x - 5

Use the synthetic division.

1|  1  - 3   5    4   - 5
          1  -2    3     7
----------------------------
     1  -2   3    7    2
----------------------------

f(1) = 2


Correct answer : (3)
 3.  
Check if (x - 4) is a factor of the given polynomial x3 + x2 - 16x - 16.
a.
(x - 4) is a factor
b.
Can not be determined
c.
(x - 4) is not a factor


Solution:

P(x) = x3 + x2 - 16x - 16

P(4) = (4)3 + (4)2 - 16(4) - 16
[Substitute the values.]

= 64 + 16 - 64 - 16

= 0
[Simplify.]

Since the remainder is zero, (x - 4) is a factor of x3 + x2 - 16x - 16.


Correct answer : (1)
 4.  
Evaluate f(x) at x = -3, by synthetic division.
f(x ) = x3 - 3x2 + 2x + 2
a.
-8
b.
-58
c.
-43
d.
-52


Solution:

f(x) = x³ - 3x² + 2x + 2

Use the synthetic division.

-3|  1  -3   2     2
          -3  18  -60
----------------------------
      1  -6   20  -58
----------------------------

f(-3) = -58


Correct answer : (2)
 5.  
x + 3 is a factor of the polynomial x3 - 8x2 - 9x + 72. Find the other factors.
a.
(x + 3), (x - 9)
b.
(x - 3), (x - 8)
c.
(x - 4), (x + 8)
d.
(x + 3), (x - 8)


Solution:

x3 - 8x2 - 9x + 72

x + 3 is a factor of the given polynomial.

Use the synthetic division.

- 3|     1    -8     -9      72
                -3       33     -72
     ------------------------------
          1    -11     24       0
     ------------------------------

x3 - 8x2 - 9x + 72 = (x + 3)(x2 - 11x + 24)

= (x + 3)(x - 3)(x - 8)
[Factor.]

So, the factors of x3 - 8x2 - 9x + 72 are (x + 3), (x - 3) and (x - 8).


Correct answer : (2)
 6.  
If y - 2 is a factor of the polynomial, then factor the polynomial, 4y3 - 8y2 - 9y + 18 completely.
a.
(y - 2)(2y + 3)(2y - 3)
b.
(y - 2)(2y + 3)
c.
(y - 2)(2y - 3)(2y - 3)
d.
(y - 2)(3y + 2)(3y - 2)


Solution:

4y3 - 8y2 - 9y + 18

y - 2 is a factor of the given polynomial.

Use the synthetic division.

2|     4    -8     -9     18
               8       0     -18
     --------------------------
        4       0     -9       0
     --------------------------

4y3 - 8y2 - 9y + 18 = (y - 2)(4y2 - 9)

= (y - 2)(2y + 3)(2y - 3)
[Factor.]

So, the completely factored form of 4y3 - 8y2 - 9y + 18 = (y - 2)(2y + 3)(2y - 3).


Correct answer : (1)
 7.  
a + 7 is a factor of the polynomial 36a3 + 312a2 + 445a + 175. Find the other factors.
a.
(6a - 5), (6a - 5)
b.
(6a + 5), (6a + 5)
c.
(6a + 5)
d.
(6a - 5), (6a + 5)


Solution:

36a3 + 312a2 + 445a + 175

a + 7 is a factor of the given polynomial.

Use the synthetic division.

-7|     36     312      445      175
               -252    -420     -175
     -----------------------------
         36     60        25        0
     -----------------------------

= (a + 7)(36a2 + 60a + 25)
36a3 + 312a2 + 445a + 175

= (a + 7)(6a + 5)2
[Factor.]

So, the factors of 36a3 + 312a2 + 445a + 175 are (a + 7), (6a + 5), and (6a + 5).


Correct answer : (2)
 8.  
If m + 20 is a factor of the polynomial, then factor m3 + 21m2 - 400 completely.
a.
(m + 20)(m + 4)(m + 5)
b.
(m + 20)(m + 5)(m - 4)
c.
(m + 20)(m - 5)(m + 4)
d.
(m + 20)(m - 4)(m - 5)


Solution:

m3 + 21m2 - 400

m + 20 is a factor of the given polynomial.

Use the synthetic division.

-20|     1     21      0    -400
               -20    -20      400
     --------------------------
         1     1     -20       0
     --------------------------

= (m + 20)(m2 + m - 20)
m3 + 21m2 - 400

= (m + 20)(m + 5)(m - 4)
[Factor.]

So, the completely factored form of m3 + 21m2 - 400 = (m + 20)(m + 5)(m - 4).


Correct answer : (2)
 9.  
t + 6 and t - 7 are the factors of the polynomial, t4 - 2t3 - 61t2 + 62t + 840. Factor the polynomial completely.
a.
(t + 4)(t + 5)(t + 6)(t + 7)
b.
(t + 4)(t - 5)(t + 6)(t - 7)
c.
(t - 4)(t - 5)(t + 6)(t - 7)
d.
(t + 4)(t - 5)(t - 6)(t - 7)


Solution:

t4 - 2t3 - 61t2 + 62t + 840

t + 6 is a factor of the given polynomial.

Use the synthetic division.

-6|     1    -2    -61    62     840
               -6     48    78    -840
     ------------------------------
         1   -8      -13      140       0
     ------------------------------

So, t4 - 2t3 - 61t2 + 62t + 840 = (t + 6)(t3 - 8t2 - 13t + 140).

t - 7 is another factor of the given polynomial.

Use the synthetic division once again for t3 - 8t2 - 13t + 140.

7|     1    -8      -13      140
                 7     -7    -140
     --------------------------
        1    -1    -20      0
     --------------------------

t4 - 2t3 - 61t2 + 62t + 840 = (t + 6)(t - 7)(t2 - t - 20)

= (t + 6)(t - 7)(t - 5)(t + 4)
[Factor.]

So, the completely factored form of t4 - 2t3 - 13t2 + 14t + 24 = (t + 4)(t - 5)(t + 6)(t - 7).


Correct answer : (2)
 10.  
Which of the following is a polynomial whose factors are 3x - 1, x + 2 and x - 2?
a.
3x3 - 4
b.
3x3 + 4
c.
3x3 + x2 - 12x - 4
d.
3x3 - x2 - 12x + 4


Solution:

f(x) = (3x - 1)(x + 2)(x - 2)

= (3x - 1)(x2 - 4)
[Use FOIL.]

= 3x3 - x2 - 12x + 4
[Use FOIL.]


Correct answer : (4)

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