# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2004 | May-Jun | (P2-9709/02) | Q#3

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 ;