Question 1: Find the remainder when is divided by

Answer:

Question 2: Find the value of if the division of by leaves a remainder of 5.

Answer:

The value of at is

Question 3: When the polynomial is divided by, the remainder is 14. Find the value of

Answer:

Given. remainder is 14, therefore:

Question 4: The polynomials and leave the same remainder when divided by Find the value of

Answer:

Since, the given polynomials leave the same remainder when divided by

Value of polynomial is the same as the value of polynomial

Question 5: When is divided by the remainder is zero and when divided by the remainder is Find the values of and

Answer:

Since,

Given that on dividing the remainder

Given that on dividing the remainder

Solving i) and ii) we get and

Question 6: What number should be added to so that when the resulting polynomial is divided by the remainder is ?

Answer:

Let the number added be so the resulting polynomial is

Given, when this polynomial is divided by the remainder

Therefore the required number to be added

Question 7: Determine whether is a factor of or not?

Answer:

Therefore when given polynomial is divided by the remainder

which is not equal to zero.

Therefore is not a factor of the given polynomial.

Question 8: If is a factor of find the value of a.

Answer:

Since, is a factor of polynomial

Therefore remainder

Question 9: Find the value of if is a factor of Hence, determine whether is also a factor. [ICSE Board 2011]

Answer:

is a factor and

Therefore The value of given expression is zero at

On substituting the given expression becomes

Now to check whether is also a factor or not, find the value of the given expression for

Therefore

Since, the remainder is

is a factor.

Question 10: Given that and are factors of calculate the values of and

Answer:

Given, is a factor of

Again, given that : is a factor of

Solving i) and ii) we get and

Question 11: Polynomial leaves remainder when divided by and is a factor of it. Find the values of and

Answer:

On dividing by the polynomial leaves remainder

is a factor of polynomial

Solving i) and ii) we get and

Question 12: Using the Factor Theorem, show that is a factor of Hence, factorize the given expression.

Answer:

Since

Therefore Remainder = The value of at

is a factor of

Now dividing by we get quotient

Therefore

Question 13: Show that is a factor of Hence factorize the given expression completely, using the factor theorem. [ICSE Board 2006].

Answer:

Question 14: Using the Remainder Theorem, factorize the expression completely.

Answer:

First Step : For the value of given expression

is a factor of

Second Step:

Question 15: Find the values of and so that the polynomial has and as its factors. For the values of and , as obtained above, factorize the given polynomial completely.

Answer:

is a factor of given polynomial

is a factor of given polynomial

On solving equations i) and ii), we get : and

Therefore the given polynomial

Now divide this polynomial by we get

Question 16: If is a factor of

(i) find the value of

(ii) with the value of factorize the above expression completely

Answer:

(i)

Since, is a factor of given expression

(ii)

On dividing by we get: