Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2005 | May-Jun | (P2-9709/02) | Q#5

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  Question

The polynomial  is denoted by . It is given that  is a factor of and that when  is divided by  the remainder is 12.

    i.       Find the values of a and b.

   ii.       When  and  have these values, factorise .

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. .

We can write the factor in standard form as;

Therefore;

We are also given that when is divided by the remainder is 12.

When a polynomial, , is divided by a , the remainder is the constant

Therefore;

From we can substitute in above equation;

Substitution of in any of these two equations yields value of . We choose;

   ii.
 

We are required to factorise .

We are given that;

We have found in (i) that and  therefore;

We are also given that is a factor of . Therefore, division of with will yield a  quadratic factor with ZERO remainder.

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Therefore;

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