1.   If is a factor of , Find the values of and
Solution
Let  .
Since is a factor of , the remainder when is divided by is 0.
By polynomial long division we can find the remainder.

Hence we have

and .
Similarly, we can say ,
When , and
When , .
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2.   If is divisible by , prove that .
Solution
Since is divisible by , The remainder when is divided by = is zero.
By polynomial long division,

and .
.
(a)    Let be a triangle with right angle and hypotenuse .
(See the figure.)
If the inscribed circle touches the hypotenuse at D,
show that .
(b)   If  , express the radius of the inscribed circle in terms  of and
(c)   If  is fixed and  varies, find the maximum value of .
Solution
Let   be the centre of the circle and and be points of
tangency of and respectively.
(given)
Draw and

Since and , is a square.
Therefore
Let .

(a)

(b)   Draw . Since is incentre, bisects .
In right ,

In right , By Pythagoras Theorem,

(c)   Since is fixed and  varies, is a function of and the rate of
change of with respect to is

has stationary value when  .

and .

When ,
.
will be maximum value when .