Question 1: In each of the following cases, find the remainder when

i) is divided by

ii) is divided by

iii) is divided by

Answer:

i) Required Remainder = Value of given polynomial for

Therefore Remainder

ii) Required Remainder = Value of given polynomial for

Therefore Remainder

iii) Required Remainder = Value of given polynomial for

Therefore Remainder

Question 2: Show that

i) is a factor of

ii) is a factor of

Answer:

i) If is a factor of , then the remainder should be for

Remainder

Hence is a factor of

ii) If is a factor of , then the remainder should be for

Remainder

Hence is a factor of

Question 3: Find which of the following is a factor of

i) ii) iii)

Answer:

i) If is a factor of , then the remainder should be for

Remainder =

Hence is a factor of

ii) If is a factor of , then the remainder should be for

Remainder =

Hence is NOT a factor of

iii) If is a factor of , then the remainder should be for

Remainder =

Hence is a factor of

Question 4: Find the value of if

i) is a factor of

ii) is a factor of

iii) is a factor of

iv) is a factor of

Answer:

i) is a factor

Therefore Remainder for

ii) is a factor

Therefore Remainder for

iii) If is a factor

Therefore Remainder for

iv) If is a factor

Therefore Remainder for

Question 5: Find the value of , when

i) and are both factors of

ii) and are both factors of

Answer:

i) If is a factor

… … … … … i)

Similarly, if is a factor

… … … … … ii)

Solving i) and ii) we get

ii) If is a factor

… … … … … i)

If is a factor

… … … … … ii)

Solving i) and ii)

Question 6: When is divided by , the remainder is . Find .

Answer:

When , Remainder

Question 7: Find the value of , if the division of by leaves a remainder of .

Answer:

When , Remainder

Question 8: If has as a factor and leaves a remainder of when divided by , find the value of . [2005]

Answer:

When , Remainder

… … … … … i)

When , Remainder

… … … … … ii)

Solving i) and ii)

Question 9: Find the value of , when leaves a remainder when divided by and respectively.

Answer:

When , Remainder

… … … … … i)

When , Remainder

… … … … … ii)

Solving i) and ii)

Question 10: What number should be added to , so that when it is divided by , the remainder is .

Answer:

Let be added to , so that when it is divided by , the remainder is

When , Remainder is

Question 11: What number should be subtracted to , so that when it is divided by , the remainder is .

Answer:

Let be subtracted to , so that when it is divided by , the remainder is

When , Remainder is

Question 12: The polynomials and leave the same remainder when divided by . Find the value of .

Answer:

For polynomial :

When , Remainder

For polynomial :

When , Remainder

Therefore

Question 13: If is a factor of the expression and when the expression is divided by , it leaves a remainder . Find the value of . [2013]

Answer:

When , Remainder

… … … … … i)

When , Remainder

… … … … … ii)

Solving i) and ii), we get