When  is uivided by , the remainder
Hence

Eqn (1)-(2) we have
From

From eqn (3)
 
Eqn (5)
From eqn
Hence
When  is divided by  the remainder is
So  

Solve (1) and (2) simultaneously and check: whether you have  and .
Hence
Now apply your understanding of solved problem 8 , see whether you obtained  as a factor. Do long division and see whether your final solution is .

Simplifying … … …(1)

Simplifying … … …(2)
From eqn (1)

To factorize completely we carry out long division.

. Factorise
completely and hence state the set of values of  for which
Solution

If  is a factor of  then

To factorize completely, we carry out long division:

. Factorise
completely and hence state the set of values of  for which
Solution

If  is a factor of  then

To factorize completely, we carry out long division:

If  is a factor of  then

So
Expressing  in partial fractions is left as a partial exercise for you dear students.
Recali that the equivalent form of the partial fraction is

Complete and check your solution with

If  is a factor of  then
So


By long division we shall obtain the other factors and hence express  as required.

Here no factor is given. In such a case we find the factors of the constant term since the coefficient of the highest term is 1. We use the factors of the constant term to find a factor of  and hence factorise  completely.
Factors of
We see the  does not change sign hence a positive root to  does not exist. As a result? we use negative values.

Hence  is a factor of .
Now we cary out long division to determine the other factors.

if  is a factor of  then

if  is a factor them

Here we may not be able to do factorization by sight. So we do long division to find the other factor.
Since  and  are factors of

 is a factor of
Hence the other factor is .

If  is a factor of  then  and  are also factors of .

Equation
Equation