Remainder Theorem:
If is a polynomial in
, and is divided by
; then the remainder is
.
So for example, If is divided by
; then the remainder is
. All you have to do is to substitute
and calculate the value of the function.
Factor Theorem:
When a polynomial is divided by
; then the remainder is
. And if the remainder
, then
is a factor of the polynomial
Let’s do a couple of examples to make it more clear.
Example 1:
Find the value of if the division of
by
leaves a remainder of
.
Answer:
It is given that the remainder is .
Therefore, if you substitute , then the remainder should be
.
Hence
Example 2:
If is a factor of
, then find the value of
.
Answer:
Since is a factor, therefore when we substitute
into the polynomial, the remainder would be
.
Therefore
Example 3:
Find such that the polynomial
has
as its factors. Once you find the value of the
, factorize the polynomial.
Answer:
is a factor of the given polynomial,
Similarly,
is a factor of the given polynomial,
On solving (i) and (ii) simultaneously, we get .
Then the polynomial becomes the following:
To factorize:
Step 1: Divide the polynomial by as shown
Therefore