WORKSHEET ON REMAINDER THEOREM

Problem 1 :

Without performing division, find the remainder when

x³ + 2x² - 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

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

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