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: