Question

The cubic polynomial   is denoted by . It is given that  is a factor of .

    i.       Find the value of .

   ii.       When  has this value, solve the equation .

Solution

     i.
 

We are given that;

We are also given that is a factor of .

When a polynomial, , is divided by , and  is factor of , then the remainder is ZERO i.e. .

Therefore;

  ii.
 

We are required to solve the equation;

We are given that;

Therefore;

We have found in (i) that , therefore;

We are also given that is a factor of .

When a polynomial, , is divided by , and  is factor of , then the remainder is ZERO i.e. .

Therefore, division of with will yield a quadratic factor with ZERO remainder.

Therefore;

Hence;

Now we have two options.

One is a liner factor while other is quadratic factor and we need to solve both of them for . 

First we solve linear factor.

Now we solve quadratic factor.

Standard form of quadratic equation is;

Solution of a quadratic equation is expressed as;

Now we have two options.

Hence, there are following 03 solutions of ;