Question 1: Using the factor show that:
i) is a factor of
. Hence factorise the polynomial
.
ii) is a factor of
. Hence factorise the polynomial
.
iii) is a factor of
. Hence factorise the polynomial
.
iv) is a factor of
. Hence factorise the polynomial
.
Answer:
i) Since
Remainder
Therefore
is a factor of
.
Dividing by
Therefore
ii) Since
Remainder
Therefore
is a factor of
.
Dividing by
Hence
iii) Since
Remainder
Therefore is a factor of
Dividing by
Hence
iv) Since
Remainder
Therefore is a factor of
Dividing by
Hence
Question 2: Factorise using factor theorem:
i) [2012]
ii)
iii)
iv)
v) [2004]
vi)
vii)
Answer:
i)
For ,
Expression:
Hence is a factor of
Hence
ii)
For ,
Expression:
Hence is a factor of
Hence
iii)
For ,
Expression:
Therefore is a factor of
Hence
iv)
For ,
Expression:
Therefore is a factor of
Hence
v)
For
Expression:
Therefore is a factor of
Hence
vi)
For
Expression:
Therefore is a factor of
Hence
vii)
For
Expression:
Therefore is a factor of
Hence
Question 3: If and
are factors of
, calculate the values of
. Then factorise.
Answer:
When
… … … … … i)
When
… … … … … ii)
Solving i) and ii) we get
Question 4: When is divided by
and
the remainder left is
respectively. Calculate the values of
and then factorise the given polynomial.
Answer:
When , Remainder
… … … … … i)
… … … … … i)
Solving i) and ii) we get
Question 5: If is a common factor of expressions
and
; show that
Answer:
When , Remainder
… … … i)
… … … ii)
From i) and ii)
Question 6: When and
are divided by
, the remainder is the same. Find the value of
.
Answer:
When
Question 7: Find the value of , if
is a factor of
. [2003]
Answer:
When , Remainder
Therefore
Question 8: What number should be subtracted from , so that
is a factor of the given polynomial.
Answer:
Let be subtracted from the polynomial.
When , Remainder
Therefore