Question 1: Show is a factor of
. Factorize the polynomial.
Answer:
For ,
Remainder:
Hence is a factor of
Hence
Question 2: Using remainder theorem, factorize completely. [2014]
Answer:
For ,
Remainder:
Hence is a factor of
Hence
Question 3: Find the value of when
is divided by
, the remainder is
.
Answer:
When
Remainder
Question 4: What should be subtracted from , so that the resulting expression has
as a factor.
Answer:
Let be subtracted
When
Remainder
Question 5: If and
are factors of
, find the values of
. Factorize the polynomial also.
Answer:
When
Remainder:
… … … … … … i)
When
Remainder:
… … … … … … ii)
Solving i) and ii) we get
Substituting the values in the polynomial we get
Given is a factor of
Hence
Question 6: If is a factor of
and
, find the values of
.
Answer:
When
Remainder:
Given
Therefore
Solving for
Hence
Question 7: Using remainder theorem, factorize completely.
Answer:
For ,
Remainder:
Hence is a factor of
Hence
Question 8: Find the value of , if
and
leave the same remainder when each is divided by
.
Answer:
When
Remainder 1
Remainder 2
Given Remainder 1 = Remainder 2
Question 9: The polynomial is completely divisible by
. Find the value of
. Also for these values of
, factorize the given polynomial completely.
Answer:
is a factor of
is a factor of
are factors of
When
… … … … … i)
When
… … … … … ii)
Solving i) and ii) we get
Question 10: Find the number which should be added to so that the resulting polynomials completely divisible by
.
Answer:
Let be added to the polynomial.
When
Remainder:
Question 11: When the polynomial is divided by
, the remainder is
. When the polynomial
is divided by
, the remainder is
. Find the value of
is
.
Answer:
When
Remainder:
… … … … … i)
When
Remainder:
… … … … … ii)
Given
Question 12: is the factor of the polynomial
. Find the value of
and factorize the give polynomial.
Answer:
When
Remainder:
Hence is a factor of
Hence
Question 13: When divided by the polynomials
and
leave the same remainder. Find the value of
. [2010]
Answer:
When
Remainder1
Remainder2
Given Remainder1 = Remainder 2
Question 14: Use the remainder theorem to factorize the following expression: . [2010]
Answer:
Let
Remainder
Hence is a factor of
Hence