﻿ Remainder Theorem Worksheet | Problems & Solutions

# Remainder Theorem Worksheet

Remainder Theorem Worksheet
• Page 1
1.
Check if ($x$ + 5) is a factor of the given polynomial 5$x$3 + $x$2 - 7$x$ - 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.

2.
Evaluate $f$($x$) at $x$ = 1, by synthetic division.
$f$($x$) = $x$4 - 3$x$3 + 5$x$2 + 4$x$ - 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

3.
Check if ($x$ - 4) is a factor of the given polynomial $x$3 + $x$2 - 16$x$ - 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.

4.
Evaluate $\mathrm{f\left(x}$) at $x$ = -3, by synthetic division.
$\mathrm{f\left(x}$ ) = $x$3 - 3$x$2 + 2$x$ + 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

5.
$x$ + 3 is a factor of the polynomial $x$3 - 8$x$2 - 9$x$ + 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).

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

7.
$a$ + 7 is a factor of the polynomial 36$a$3 + 312$a$2 + 445$a$ + 175. Find the other factors.
 a. (6$a$ - 5), (6$a$ - 5) b. (6$a$ + 5), (6$a$ + 5) c. (6$a$ + 5) d. (6$a$ - 5), (6$a$ + 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).

8.
If $m$ + 20 is a factor of the polynomial, then factor $m$3 + 21$m$2 - 400 completely.
 a. ($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).

9.
$t$ + 6 and $t$ - 7 are the factors of the polynomial, $t$4 - 2$t$3 - 61$t$2 + 62$t$ + 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).

10.
Which of the following is a polynomial whose factors are 3$x$ - 1, $x$ + 2 and $x$ - 2?
 a. 3$x$3 - 4 b. 3$x$3 + 4 c. 3$x$3 + $x$2 - 12$x$ - 4 d. 3$x$3 - $x$2 - 12$x$ + 4

#### Solution:

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

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

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