Question

The polynomial  is given by .

a.   Use the Remainder Theorem to find the remainder when  is divided by .

b.   Use the Factor Theorem to show that  is a factor of .

c.    

                    i.       Express  in the form , where b and c are integers.

                  ii.       Hence show that the equation  has exactly one real root.

Solution

a.
 

Remainder theorem states that if  is divided by  then;

For the given case   is divided by .

Here  and . Hence;

b.
 

Factor theorem states that if  is a factor of   then;

For the given case  is factor of .

We can write the divisor in standard form as;

Here  and . Hence;

Hence,  is factor of .

 

c.
 

                            i.
 

If  is a polynomial of degree  then  will have exactly  factors, some of which may repeat.

We are given a polynomial of degree ,

Therefore, it will have 03 factors some of which may repeat.

From (b) we already have 01 factor  of .

We may divide the given polynomial with any of the already known linear  factor(s) to get a quadratic factor and then factorize the obtained quadratic factor to find the 2^nd and 3^rd linear factors.

For the given case;

Already known factor from (b) is .

We may divide the given polynomial by this factor .

Therefore, we get the quadratic factor  for given polynomial.

Hence,   can be written as product of linear and quadratic factors as follows;

 

                          ii.
 

We are required to show that the equation   has only one real root.

We have found in (c:i) that  can be expressed as;

Therefore, one root of the given equation can be found from this factor;

Now we need to verify that given equation has only one real root.

Any other root, if exists will come from;

We recognize that it is a quadratic equation.

For a quadratic equation , the expression for solution is;

Where  is called discriminant.

If , the equation will have two roots.

If , the equation will have two identical/repeated roots.

If , the equation will have no roots.

If there is no other real root of the given equation then for this quadratic factor of given equation  should not have any real root. Therefore;

Hence,  has no root and, therefore,    has only one real root which is -1.