# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2006 | May-Jun | (P2-9709/02) | Q#4

Question

The cubic polynomial , where  and  are constants, is denoted by . It is given that   and  are factors of .

i.       Find the values of  and .

ii.       When  and  have these values, find the other linear factor of .

Solution

i.

We are given that;

We are also given that and are a factors of .

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

We can write the factors in standard form as;

Therefore;

We can substitute from into .

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

ii.

We are required to find other linear factor of .

We are given that;

We have found in (i) that and  therefore;

We are also given that and are factors of . Therefore, division of with any  of these factors will yield a quadratic factor with ZERO remainder.

We divide by  by .

Therefore;

Hence, is another linear factor of .