A:

Simplifying … … …(1)

Simplifying … … …(2)
From eqn (1)

To factorize completely we carry out long division.

A:

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  

A:

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 .

A:

Simplifying … … …(1)

Simplifying … … …(2)
From eqn (1)

To factorize completely we carry out long division.

A:

. 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:

A:

. 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:

A:

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

A:

If  is a factor of  then
So


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

A:

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.

A:

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 .

A:

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

Equation
Equation

A:

Let the remainder

Equation
From equation

Hence remainder

A:

If  is a factor then

Equation (2) - (1) we have
From equation (2)

A:

3. 
   Whout doing long division, we can rearrange this polynomials and get the required proof.

Here we see that  is a factor or the Syromial.
Factorising completely we have

If you cannot see the factors easily, do your long division of polynomials.

A:

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A:

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A:

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A:

Here we carry out long division first

Therefore

We now express  in partial fractions thus
At this level dear student; I am convinced you can solve or the constants  and .
Solve and see whether your result is

A:

Here we carry out long division first

Therefore

We now express  in partial fractions thus
At this level dear student; I am convinced you can solve or the constants  and .
Solve and see whether your result is

A:

We may also proceed as follows

Now complete the solution and check your result with