WORKSHEET ON REMAINDER THEOREM

Problem 1 :

Without performing division, find the remainder when

x3 + 2x- 7x + 5 is divided by x - 1

Solution

Problem 2 :

Without performing division, find the remainder when

x4 - 2x2 + 3x - 1 is divided by x + 2

Solution

Problem 3 :

Find a given that:

when x3 - 2x + a is divided by x - 2, the remainder is 7.

Solution

Problem 4 :

Find a given that:

when 2x3 + x2 + ax - 5 is divided by x + 1, the remainder is -8

Solution

Problem 5 :

Find a and b given that when x3 + 2x2 + ax + b is divided by x - 1 the remainder is 4, and when divided by x + 2 the remainder is 16.

Solution

Problem 6 :

2xn + ax2 - 6 leaves a remainder of -7 when divided by x - 1, and 129 when divided by x + 3. Find a and n given that n∈Z+.

Solution

Problem 7 :

When P(z) is divided by z2 - 3z + 2 the remainder is 4z - 7. Find the remainder when P(z) is divided by:

a. z - 1    b. z - 2

Solution

Problem 8 : 

When P(z) is divided by z + 1 the remainder is -8 and when divided by z - 3 the remainder is 4. Find the remainder when P(z) is divided by (z - 3) (z + 1).

Solution

Answer Key

1) The remainder is 1.

2) The remainder is 1.

3) a = 3

4) The value of a is 2.

5)  So, the values of a and b is -5 and 6.

6)  The values of a and n is -3 and 4.

7)  a) P(1) = -3      b) P(2) = 1

8)  remainder is 3z - 5.

Use the remainder theorem to find f(k).

Problem 1 :

k = 2; f(x) = x² - 2x + 5

A) -5     B) -3    C) -13     D) 5

Solution

Problem 2 :

k = -3; f(x) = x² + 2x + 2

A) 1     B) -13     C) 5     D) -17

Solution

Problem 3 :

k = -2; f(x) = 3x³ - 7x² - 3x + 3

A) 14     B) -55     C) -43     D) -5

Solution

Problem 4 :

k = 4; f(x) = x³ - 2x² + 5x - 2

A) 54     B) 50     C) -78     D) -76

Solution

Problem 5 :

k = 2; f(x) = 9x4 + 10x³ + 6x² - 6x + 16

A) 360     B) 500     C) 252     D) 36

Solution

Problem 6 :

k = 5; f(x) = x³ - 3x² - 4x - 5

A) 35     B) 25    C) -225     D) -220

Solution

Problem 7 :

Using the remainder theorem find the remainders obtained when x3 + (kx + 8)x + k is divided by x + 1 and x - 2. Hence, find k if the sum of the two remainders is 1.

Solution

Answer Key

1)  Remainder = 5

2)  Remainder = 5

3)  Remainder = -43

4)  Remainder = 50

5)  Remainder = 252

6)  Remainder = 25

7)  k = -2

Problem 1 :

Given that x − 2 is a factor of the polynomial

x3 − kx2 − 24x + 28

find k and the roots of this polynomial.

Solution

Problem 2 :

Find the quadratic whose roots are −1 and 1/3 nd whose value at x = 2 is 10.

Solution

Problem 3 :

Consider the polynomial p(x) = x3 − 4x2 + ax − 3.

(a) Find a if, when p(x) is divided by x + 1, the remainder is −12.

(b) Find all the factors of p(x).

Solution

Problem 4 :

Consider the polynomial

h(x) = 3x3 − kx2 − 6x + 8

(a) Given that x − 4 is a factor of h(x), find k and find the other factors of h(x).

Solution

Problem 5 :

Find the quadratic which has a remainder of −6 when divided by x − 1, a remainder of −4 when divided by x − 3 and no remainder when divided by x + 1

Solution

Problem 6 :

Find the value of a if x − 3 is a factor of f(x)= x3 - 11x + a

Solution

Problem 7 :

Find the value of k if f(x) = 3(x2 + 3x - 4) - 8(x - k) is divisible by x.

Solution

Problem 8 :

If x − 2 is a factor of polynomial 

p(x) = a(x3 - 2x) + b(x2 - 5)

which of the following must be true ?

a)  a + b = 0    b) 2a - b = 0    c) 2a + b = 0  d)  4a - b = 0

Solution

Answer Key

1)  k = -2

2) p(x) = 2(x + 1) (x - 1/3)

3)  (x + 1) and (x2 - 5x - 3) are factors.

4)  k = 11

5)  p(x) = x2 - 3x - 4

6)  a = 6

7)  k = 3/2

8)  4a - b = 0

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